[User] Please transcribe this text

[Assistant] And the ordering for this, usually in the US and usually in the Valley, is that startups will first raise money on a unpriced round, they'll raise a little bit of money, and then a couple of years later, uhm, they'll, they'll hopefully raise money on a priced round.


[User] Please transcribe this text

[Assistant] The rest of the world hasn't quite caught up to this idea of, uhm, unpriced rounds. So generally we see when, when companies are coming to us from around the world that they actually raise money immediately on priced rounds.


[User] Please transcribe this text

[Assistant] So let's go through an example of a priced round, uhm, and you can see how this works.


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[Assistant] So as I say, the valuation of the round sets the price, uhm, that the investor will pay to buy a share.


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[Assistant] So let's, let's work through this, uhm, with an example. So let's assume we have the same company as before.


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[Assistant] Our founders have 9 million shares and they've done really well and they're going to raise 2 million dollars at an 8 million dollar valuation.


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[Assistant] Now that's eight million dollar valuation is pre-money is the terminology. So that basically means before the, before the investor puts any money in, the valuation of the company is eight million.


[User] Please transcribe this text

[Assistant] And so the way that you calculate the price for the shares that the investor will buy is you just divide the 8 million valuation by the pre-money shares, so the 9 million shares, which gives a price of 89 cents.


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[Assistant] And so then they will be able to buy 2.25 million shares for that 2 million dollars.


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[Assistant] The post-money valuation at this point is 10 million dollars, because it was the 8 million dollar valuation before the investor put money in, and now there's 2 million dollars sitting in the bank account as well, so it's increased the value of the company up by 2 million. So the post-money is 10 million. And just, you'll hear those terminologies just to get them straight in your mind.


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[Assistant] So now this is what the cap table looks like. So we still have our three founders. They still own three million shares each.


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[Assistant] But we've now created a new class of shares, which have been sold to the investor. And now the investor owns 20% of the company.


[User] Please transcribe this text

[Assistant] And each of the founders have been reduced from 33% of the company to 26% of the company.


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[Assistant] Makes it all sound really simple. And actually it's not. Uhm, there's various other complexities in here that I'm not going to talk about. And there's a lot of documents that are involved in here with a lot of things that need to be negotiated. Uhm, so if, if


[User] Please transcribe this text

[Assistant] If a company is doing a priced round, then you would need to use a lawyer. You'd need to hire a lawyer to help you with all of that. And they'd be able to talk you through the whole process.


[User] Please transcribe this text

[Assistant] So, moving on to the un-priced rounds, and for most of, uhm, companies who raise money now here, out here in the Bay Area, is they, they will use an instrument called a SAFE, which stands for Simple Agreement for Future Equity.


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[Assistant] And the simple is really important because it actually is simple. Uhm, it's only a few pages long so it's easy to understand. Uh, we also have these documents on the Y Combinator website and a really great primer that explains a lot more about them. Uhm, so you can look at that for some more details. And basically what it's saying is


[User] Please transcribe this text

[Assistant] The investor is giving the company money now in return for the right to receive shares in a future round.


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[Assistant] But the investor's giving the money now, and so they don't want to receive shares in a future round at the price of the future round. Because when they invest now, they're probably investing at a much riskier stage, it's a much earlier startup, and so they want some kind of bonus or deal to reflect that.


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[Assistant] And so the way that that's built in is the safe includes what's called a evaluation cap.


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[Assistant] And what this cap does is it sets a valuation upper bound for the calculation of working out the price for the shares.


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[Assistant] So again, as an example, uhm, we'll use our, our same company, 9 million shares for the founders, but this time in April of this year they raised $400,000 on safes with a $4 million cap, and then


[User] Please transcribe this text

[Assistant] Fast forward some time, we're going to the future. December 2017, they raised a 2 million dollar round, that's an 8 million. Free money valuation.


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[Assistant] So this time, there's actually two calculations that needs to happen because there's the safe conversion price and there's the priced round price. And you can see here that the safe conversion price uses the four million dollar cap.


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[Assistant] As the valuation, whereas the round price uses the 8 million dollar.


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[Assistant] valuation even though all these shares are being actually given at the same time


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[Assistant] And so as a result, the safe holder here has a-


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[Assistant] A price per share of half the Series A investors, uh, the price round investors.


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[Assistant] So that means that actually for their money, because they came in early, they're getting twice as many shares as somebody who was putting in the same amount of money in the price round. So that's how they get their, their sort of bonus in all of this.


[User] Please transcribe this text

[Assistant] So again, the cap table, by the time we get to the December priced round, the cap table looks like this. So now we also have the safe investor in there on the cap table and they own the same shares, the same class of shares. Yes, question.


[User] Please transcribe this text

[Assistant] Most safes are on a K, on a cap. There is a separate, uhm, kind of safe that has a discount. So are you, are you asking does the cap kick in if the, the valuation of the round is lower than the cap?


[User] Please transcribe this text

[Assistant] Generally not, uhm, so generally you wouldn't have a cap and a discount because that's kind of cherry picking for the investor. So you would try very hard to avoid that situation. Uhm, in the situation where you had a four million dollar cap but your round was at three million dollars for instance, everybody would just convert to the three million dollar round.


[User] Please transcribe this text

[Assistant] Uh, yeah, so again, we have, uhm, the founders are now earning, owning less of the company because they've sold some of the company to their investors.


[User] Please transcribe this text

[Assistant] Yeah, so the benefits of a safe over a priced round is that it's a lot simpler. So you can just sign one safe immediately and get the money. Uhm, it doesn't require lawyers. Uhm, and I'll come into a few more details, but basically it's simple. And that's the important thing.


[User] Please transcribe this text

[Assistant] Okay, so, let's talk about dilution and then we'll talk a little bit more about that.


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[Assistant] So the idea is, is that at the early stage of your company, if you think about your company as a pie, where the, the size of the pie is the value of the company. In the early days, you have a massive slice of a very small pie.


[User] Please transcribe this text

[Assistant] And then as the company grows, the size of the pie grows, but your slice becomes smaller because you're giving away or you're selling.


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[Assistant] One thing that we see founders make a mistake early on is that they sell too much of a company early on at a very low valuation. And really in the early days of a company,


[User] Please transcribe this text

[Assistant] You know, you're still trying to figure out what products you're building, you're still trying to figure out what you're going to do, so...


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[Assistant] What do you need to spend the money on? Well mostly money when you spend money when you've raised is on hiring. But if you're at that stage you probably don't want to hire. So why raise more money than you need if you can wait six months or a year?


[User] Please transcribe this text

[Assistant] And raise it to higher valuation. So, just something to bear in mind as a way to not sell too much of the company in the early days.


[User] Please transcribe this text

[Assistant] Okay, so in terms of the logistics, as I was saying, you know, a price round is complex, a safe is simple.


[User] Please transcribe this text

[Assistant] Again with safes, uh, you can use Klerky. They have a fundraising product as well. So they have the standard YC templates built into their system. So you can just


[User] Please transcribe this text

[Assistant] Plug in the investor details, plug in the amount of money, send it off to your investors to be signed. They get the, uh, bank wire details as well so they can send the money to the company.


[User] Please transcribe this text

[Assistant] Which is also an important point. You do actually want the money back from the investors. Uhm, it's very easy to say, oh yeaah, we've got investors in, it's great. And our first question is,


[User] Please transcribe this text

[Assistant] Have you got the money in the bank? And the founders say, oh no, because, some excuse. And we're like, you've got to, you've got to keep, keep working on your investors until the money is in the bank.


[User] Please transcribe this text

[Assistant] The other thing that's a slightly uhm, just a minor point here is that because you're selling equity in the company, the company will need to complete a board consent for part of this, even with a safe. And that's a legal document with a specific form. It's not enough just for the founders to email each other and say, we're doing this right.


[User] Please transcribe this text

[Assistant] And again, Clerky will help you with that, or if you're using lawyers to do any of this, they'll help you with it as well. So it's just something to bear in mind.


[User] Please transcribe this text

[Assistant] Alright, so now you've raised some money. What are you going to do with that money? Well, probably you're going to spend most of that money on hiring.


[User] Please transcribe this text

[Assistant] Again, hiring is a complex area. It's, uhm, it's governed by a lot of laws and regulations and I'm not a specialist. Don't expect you as founders to be specialists. The important thing is that you understand the basics and that you understand that there are people out there who can help you with all of this.


[User] Please transcribe this text

[Assistant] Okay, so the first part of this is deciding if you're going to hire somebody, what role they're going to be. So, in the US again, the uhm,


[User] Please transcribe this text

[Assistant] You can hire somebody either as a contractor or as an employee. And they have slightly different characteristics and slightly different results in terms of how you pay them.


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[Assistant] So in both cases, a contractor and an employee will assign the IP that they create to the company. So a contractor will sign a consulting agreement, an employee will sign an IP assignment agreement.


[User] Please transcribe this text

[Assistant] And the, for the contractor, they are, uhm, there's, there's some requirements for them. So generally they set their own work hours, they, uhm, work on a specific project with an end goal that's, that's very defined, uh, they use their own equipment.


[User] Please transcribe this text

[Assistant] A month long project and they deliver you a website and you say great this is hap- this is just what we need and off they go onto their next project.


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[Assistant] On the other hand, employees will be tend to use company equipment, so the company generally will provide them with a laptop.


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[Assistant] Uhm, they will usually be working company hours and in the company location. Of course with startups that gets a little bit great. Uhm, there's more supervision, they have less autonomy over their decisions about how they make, how they actually build their, or you know, how they do their work.


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[Assistant] Uhm, and importantly for the US, for an employee to work in the US, they need to have work authorization. Whereas a contractor, you could hire to work remotely and they wouldn't need work authorization.


[User] Please transcribe this text

[Assistant] So for contractors, if you decide that you, uhm, are hiring a contractor, then you...


[User] Please transcribe this text

[Assistant] Would agree in the contract how you would actually pay them. And usually that's either time or milestone based. So milestone based would be, in our example of a web designer, maybe you provide 25% of the fees upfront and you provide the other 75% of the fees when the designer provides the final design for the website.


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[Assistant] It's really straightforward for the company to pay them because they don't have to worry about any taxes being withheld or anything else like that, so you just pay the contractor.


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[Assistant] And then at the end of the year, the company will create a form that's called a Form 1099, which you send to the contractor so that they can include the income in their personal tax returns and also the company sends one to the IRS.


[User] Please transcribe this text

[Assistant] Obviously that's a US regulation, so if you have a company that's not in the US, then you wouldn't need to file a 1099. Uhm, and generally if you're working with a contractor who's not in the US, you wouldn't need to file a 1099 either. But things get a little bit more complex there, so take advice.


[User] Please transcribe this text

[Assistant] On the flip side, paying employees is more complex for the company. So, the company pays their employees, but they have to withhold, or the company withholds tax on the employee's behalf, and pays this over to various agencies. So, IRS, to state agencies, and sometimes to city agencies as well.


[User] Please transcribe this text

[Assistant] So there's a lot of calculations in there. It gets very complex, very fast if you have employees all over in different states. And it's more of a headache for the company.


[User] Please transcribe this text

[Assistant] But the IRS prefer it because they get their tax every time the employee gets paid.


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[Assistant] And then so what happens at the end of the year is the company provides a form W2 to each of the employees and then the employee uses that to show how much tax has already been withheld on their behalf and how much income they have.


[User] Please transcribe this text

[Assistant] So this is a situation where you should absolutely not be the person calculating all of this. You should use a payroll service provider.


[User] Please transcribe this text

[Assistant] It's totally worth the money, it's not very expensive, and they basically take all of this hassle away from you. And the company that a lot of YC startups use, who is a YC company themselves, is Gusto.


[User] Please transcribe this text

[Assistant] The other thing to bear in mind is that under California laws, in particular, but other laws as well,


[User] Please transcribe this text

[Assistant] Employees are required to be paid minimum wage.


[User] Please transcribe this text

[Assistant] So you can't have somebody working for you for free and expecting them to be producing, uhm, producing work for the company. So do just bear in mind that when you are hiring people, you do need to pay them.


[User] Please transcribe this text

[Assistant] The other part of a startup employee's compensation package is made up of equity. And usually this is, this is again the part that uhm,


[User] Please transcribe this text

[Assistant] Is the way that they possibly create more wealth for themselves. Uhm, it's really important that startup founders do give employees equity because it helps to incentivize everybody to work towards the same thing, which is creating a successful company.


[User] Please transcribe this text

[Assistant] And often in the early days, for employees they're being paid below market rate. Still above minimum wage, but below market rate.


[User] Please transcribe this text

[Assistant] Uhm, and so giving them stock in the company is a way to compensate them for that, uhm, reduction in cash that they're potentially seeing. So it increases their upside.


[User] Please transcribe this text

[Assistant] They are going to be with you for a long time, they're going to be working really hard in the same way that the founders are, they're going to be setting the, uhm, you know, how the company goes forward, they're going to be doing a lot of stuff and so it's important that they are motivated in the same way.


[User] Please transcribe this text

[Assistant] And as a rule of thumb, you can think about giving maybe 10% of the company to the first 10 employees, but obviously on a sliding scale. So employee one gets more than employee two, who gets more than employee three, etc.


[User] Please transcribe this text

[Assistant] Uhm, and the reason why it's not that much of a gamble to be generous is that all of these will have vesting. All of these employees will have vesting on their shares. So if something doesn't actually work out and you have to fire the employee, which...


[User] Please transcribe this text

[Assistant] When you're hiring people, generally startups make a lot of mistakes and have to fire a lot of people.


[User] Please transcribe this text

[Assistant] You're not going to be too burnt by that because the vesting means that the company has the right to repurchase their shares.


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[Assistant] Structure this is they create a stock plan, which is again another bunch of legal documents, uhm, which need to be signed and kept.


[User] Please transcribe this text

[Assistant] Uhm, and they, again, can be done on Clerky or you can use a lawyer to, to do those for you.


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[Assistant] And so what this means that's good for the employees is they don't have to pay the cash immediately. They can actually exercise their options in the future when the company is looking like it's successful or even in conjunction with uhm, an acquisition or something. So they don't even end up anyway in a out of pocket.


[User] Please transcribe this text

[Assistant] There are some tax consequences to that, which is too much detail to go into now, but there's a lot of resources online about options, about how to issue options, about the pros and cons of them, so if you're interested, there's lots out there.


[User] Please transcribe this text

[Assistant] Finally, the other thing on this is it's really important to communicate to your employees what they're getting in terms of equity. So first of all, you need to be clear to yourself what they're getting, which means that you need to understand the number of shares they're getting and what that represents in terms of percentage of the company.


[User] Please transcribe this text

[Assistant] Because if you only give them one and not the other, it gets very confusing because you can say, you have x percent of the company, but what does that mean, x percent of what? Whereas if you say, you have a hundred thousand shares, well, what does that mean? You know, is that, is that a lot of the company or is that a small amount of the company?


[User] Please transcribe this text

[Assistant] So, clear in your head and then be clear with the employee about what's going on.


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[Assistant] So it's important that you keep all of these documents that we're talking about in a safe place. You're going to need them at stressful times. You're going to need them during acquisitions. You're going to need them during fundraising due diligence. You're going to need them


[User] Please transcribe this text

[Assistant] So many times that if you don't have them all stored in signed format, it's just going to make your life really hard.


[User] Please transcribe this text

[Assistant] So keep all your documents in a safe place and just assign one of the co-founders to be responsible for this.


[User] Please transcribe this text

[Assistant] You should make sure you know your key metrics in the company, which again is going to be covered in a lot of more detail throughout the course. But it's the kind of thing that again you just want to have off the top of your head immediately. And also you should make sure that the money that you're spending that the investors have given to you


[User] Please transcribe this text

[Assistant] Is being spent sensibly. You know, spend money on things that will increase the chances of the company being a success.


[User] Please transcribe this text

[Assistant] These investors have trusted you with their money to try to multiply their money. They haven't given you their money to go off to Vegas, which happened to one of our companies, uh, a few years ago where one of the founders took quite a lot of company money, went to Vegas, had a great time, according to Facebook.


[User] Please transcribe this text

[Assistant] Okay, so in conclusion, that was a really fast run through of some of the basics.


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[Assistant] Uhm, don't expect you to be experts in any of this. Don't expect you to, to have all of this figured out. But it's just important that you know that there are some things that you need to do. And these items up here are sort of the really key takeaways uhm, for this to just, to just bear in mind as you go forward with your companies.


[User] Please transcribe this text

[Assistant] Well, so, you know, it depends, because it depends on the kind of hires that you're giving. So if, if your first hire is going to be a software engineer, that's going to be very different to if your first hire is a community manager, say. Uhm, I mean, for the, for the first ten is, is usually somewhere around ten percent. Maybe for the first five you want to be somewhere on the sort of seven, seven to eight mark.


[User] Please transcribe this text

[Assistant] You know, 30% of the company, and then employee number 2, 20% of the company, because you're gonna run out of, you're gonna run out of company to give them. So, it's part of, you know, planning ahead for your hiring needs.


[User] Please transcribe this text

[Assistant] Yeah, I mean in the early days, where it's just the founders, there should be 100% transparency. You know, everybody should know everything. Uhm, generally as the company grows, it becomes slightly less. You know, you wouldn't show a line by line table to your employees, for instance. But obviously this transparency of saying, you have this number of shares, and the total number of shares is this.


[User] Please transcribe this text

[Assistant] Uhm, I mean if it's, if you're looking at a startup that's, you know, the sort of general startup idea of hockey stick growth and most of the value is created by equity, it's a good idea to, to generally


[User] Please transcribe this text

[Assistant] If you're looking at more of something that's kind of more of a lifestyle business which can be a great business for, for, you know, two or three founders who don't take outside money and create a business that's very profitable from the start.


[User] Please transcribe this text

[Assistant] Then yes, maybe, maybe doing some kind of profit share or some kind of very generous bonus scheme or something like that is, is a better answer. But usually that's not the case for startups. Usually they're not profitable for a long time.


[User] Please transcribe this text

[Assistant] So non-profits are very different, uhm, because there isn't this idea of owning shares and things, so that's a whole different idea. Uhm, and there's also rules around how much you can pay employees in non-profits. On the social good companies, it kind of depends on what sort of a company it is, uhm,


[User] Please transcribe this text

[Assistant] You know, again, you need to, you need to decide in a way of like, what's your mentality and is your mentality still to be like a startup? Even though you're going to do some...


[User] Please transcribe this text

[Assistant] Some something that, you know, gives back. You can still have a startup mentality in that. And so you'd probably still want to go down that path of, of...


[User] Please transcribe this text

[Assistant] Having equity and having that compensation in that way. And you can still fundraise as well as a social good kind of company. It's a little bit harder, but you can still do it.


[User] Please transcribe this text

[Assistant] Well, this gets a little bit complex, but the simple answer is that subsequent rounds of fundraising will dilute all existing shareholders in relation, in proportion to their existing holding. It's a little bit more complex than that, but that's the general.


[User] Please transcribe this text

[Assistant] But by the time they get to an exit, it might represent 0.01% of the company. But in that respect, they'd probably have been given later, they'd probably be given more later on. And this idea of the pie being much bigger, so even, you know, 1% of a company that's worth 0 is still less.


[User] Please transcribe this text

[Assistant] Uhm, yeaah, so investors get preferred shares and as the cycle suggests they have some preferences. Uhm, and I'm not a lawyer, I'm a finance person so I can do this in a very top level.


[User] Please transcribe this text

[Assistant] To get the amount of the money that they invested back before anybody else does. So if, for example, if investors, uhm, invested $2 million into a company and then it got sold, but the cash was, and, and, sorry, they invested $2 million into the company and they own 20%.


[User] Please transcribe this text

[Assistant] But the cash that came out of an exit further down the road was only two million dollars. They would see all their money.


[User] Please transcribe this text

[Assistant] Before any of the common holders, i.e. the founders, would see this. So that's one. Uhm, they have, there's, there's various other different, uhm, protections that they have in there. That's probably the key one that, that's the financial basis. But then there's, there's voting rights, there's rights of first refusal.


[User] Please transcribe this text

[Assistant] Uhm, there's lots of different things and that's why it's a much more complex negotiation because you have to negotiate all of these rights within there.


[User] Please transcribe this text

[Assistant] I mean, I think, I think the companies that do this well are the ones that are transparent, uhm, and the ones that, that sit down with their employees and really talk through, like, what, what these things mean and what options are and, and how vesting works and, you know, here's a scenario where, you know, a lot of, some companies will have


[User] Please transcribe this text

[Assistant] Whereas if the company exists for this much money, how will you see? And so, you know, they've done all of these calculations and they really make sure that employees understand. And I think that's a really good practice to have. And what that means, or what that needs first, is the founders need to understand it, which is actually surprisingly not common.


[User] Please transcribe this text

[Assistant] I mean, the, you know, the first money that companies raise will be the most dilutive, so if an employee comes in and there's already been three rounds of money, then they're probably not going to be diluted as much as if they were the first employee in and they had to go through all of that dilution. But it's, it's hard to give a number.


[User] Please transcribe this text

[Assistant] They con- they convert usually into preference shares. Uhm, because again the investors want all of the rights and privileges that come with those preferred shares.


[User] Please transcribe this text

[Assistant] There are some circumstances where they might convert into a combination of preferred and common, but that's less usual.


[User] Please transcribe this text

[Assistant] I sometimes see companies push it out to five years, uhm, and also sometimes companies don't have the cliff at all and they just do monthly vesting over the period. Uhm, but that's, it's a harder sell for the five year, the longer the timeline, it's a harder sell for the employees. Uhm, and so you've got to be mindful of that.


[User] Please transcribe this text

[Assistant] But on the flip side, what usually happens is, you know, if you've had an employee who's been with you for four years, then they've been fairly diluted because there's going to have been funding rounds, there's going to have been subsequent employees, so you would actually give them a new stock grant.


[User] Please transcribe this text

[Assistant] Priced around documents, this allows preferred shares to be converted into common in various different circumstances.


[User] Please transcribe this text

[Assistant] Uhm, but that's usually around exits and calculation of, of proceeds and things like that. Or is it, or are you trying to get at, uhm, like founders preferred shares?


[User] Please transcribe this text

[Assistant] Ugh. Oh, my head. Oh!


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[Assistant] The following content is provided under a Creative Commons license.


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[Assistant] Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.


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[Assistant] To make a donation or view additional materials from hundreds of MIT courses.


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[Assistant] Visit MIT OpenCourseWare at ocw.mit.edu.


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[Assistant] Now generally speaking, a proof is going to have seven characteristics that you want to keep in mind.


[User] Please transcribe this text

[Assistant] You gotta have all the details there, all the key steps have to be there. They should be clear, so we can understand what's going on.


[User] Please transcribe this text

[Assistant] You know, you don't want to crush somebody with all the details. You want to get to the key points and be crisp.


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[Assistant] That means clever. It's the mathematician's notion of beauty. Like you go to the art museum and the artist will say, wow, that's a beautiful painting.


[User] Please transcribe this text

[Assistant] In mathematics you say, wow, that's an elegant proof.


[User] Please transcribe this text

[Assistant] It's crisp, clever, short, to the point, and it's really the highest compliment you can get.


[User] Please transcribe this text

[Assistant] Now, from a mathematician anyway, uh, and there's a lot of judgment that goes into that.


[User] Please transcribe this text

[Assistant] Just like in art, there's judgment over what is great art. The proof should be well organized.


[User] Please transcribe this text

[Assistant] You know, for example, use lemmas the same way you would use subroutines in writing code. Helps to make it clear. And the proof should be in order.


[User] Please transcribe this text

[Assistant] You know, sometimes you'll see proofs where things are done in a haphazard fashion. All the pieces are there but they're in the wrong order.


[User] Please transcribe this text

[Assistant] Sometimes, and they teach this in some high schools, they teach you to do proofs backwards.


[User] Please transcribe this text

[Assistant] You know, and the classic thing there is, say you're trying to prove A equals B.


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[Assistant] Well the proof sometimes I'll start with what you're trying to prove and then they'll do a bunch of steps and then you'll end up with 1 equals 1 and you'll write a check. Because you went from A equals B to 1 equals 1. And that's of course true. Now,


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[Assistant] That's not a good thing to do. It can be correct if the implications go this way. Because really you're starting with a fact, one equals one.


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[Assistant] And deriving A equals B. So if in fact the implications work this way, then your proof is right.


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[Assistant] But most people, especially as you get farther along, think about it going from top to bottom. Okay, so don't use this technique, because it'll confuse people, you're liable to make a mistake.


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[Assistant] Just start with one equals one and work your way from there, top down, for being in order.


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[Assistant] Now good proofs are very much like good code.


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[Assistant] You know, in fact, one of the reasons we care so much about teaching you how to write a good proof in computer science is so that later on


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[Assistant] You'll be able to prove that your programs are doing what you expect, what they're supposed to do.


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[Assistant] Now there are some many famous examples where programs did not do what they were supposed to do with disastrous consequences. You know, the Airbus A300


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[Assistant] Was one of the first commercial jets totally operated by software. Could take off, fly and land totally by software.


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[Assistant] There's a major advance, you know, in the airline industry. The only problem was that on one of the first flights of the A300, the software accidentally opened the rear door just before landing, and the plane crashed as a result.


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[Assistant] It's the first plane crash in commercial history because of a software bug.


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[Assistant] There's a famous radiation device for cancer patients called the Therac-25.


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[Assistant] It's famous because it got into a race condition occasionally, which caused the device to just slam the patient with radiation so much so, killed the patient.


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[Assistant] They had multiple examples of this. Of course, a lot of lawsuits afterwards. How many of you all remember the 2000 election? Are you?


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[Assistant] Few of you do, okay. This will probably be the last class that remembers that. Uh, but that's where, you know, Al Gore was going up against George Bush. Very close election. All came down to recounting the votes in Florida. Uhm, but in fact, Al Gore got negative 16,000 votes in one county.


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[Assistant] Because of a software bug in the electronic voting, uh, booths, which I think they got rid of a lot of them in the election after that because the software was buggy.


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[Assistant] So poor Al had enough problems with Chad's and funny business in Florida, but getting negative 16,000 votes certainly didn't help his chances.


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[Assistant] Several years ago, a single faulty command in a computer system used by United and American Airlines grounded the entire fleets of both airlines for close to a day.


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[Assistant] Because they couldn't do anything. They're all run by a computer and the whole thing was screwed up.


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[Assistant] So there's lots more examples. We run into this issue all the time at Akamai.


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[Assistant] Uh, Akamai is a company started by MIT folks, by myself and Danny Lewin and a dozen undergrads in the late 1990s.


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[Assistant] And, uh, we deliver a lot of the content you get on the web. Facebook, you know, all the search engines.


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[Assistant] Uh, a lot of the stuff you go to comes from our servers. So we gotta be very careful that we don't have software bugs. Now in fact we do, and we catch them every once in a while. But if we got a bad one, it would bring down all the sites you go to. You wouldn't be able to go to those sites anymore. Uh, and everybody would notice. It would be sort of embarrassing. Uh, now, this really does matter.


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[Assistant] You know, and this is going to sound a little scary, but someday, probably 30, 20 years from now, somewhere in there, it's possible that all of us


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[Assistant] Our lives may depend on the software that some of you write.


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[Assistant] In fact, to bring this home, how scary it is, look at the person sitting next to you.


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[Assistant] And imagine that in 25 years, your life depends on whether their code does what it's supposed to do. Little scary, you know. That's why we are very motivated


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[Assistant] To help you learn how to make rock solid arguments.


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[Assistant] Alright, so you don't have code that fries one of your classmates someday or puts them in a bad, plain situation.


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[Assistant] Now unfortunately, writing rock solid proofs is a very hard thing to do. Even the world's best mathematicians mess them up on a regular basis. In fact, it's estimated that one third of all published proofs have bugs, have flaws that render the proof incorrect.


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[Assistant] The trouble often arises because we get lazy.


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[Assistant] You know, we don't write down all the details or all the steps, cause wow, it's just clear, let's just move on. Now, this can be okay, not such a good practice, but it dramatically increases the chances of making a mistake.


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[Assistant] And there's some very famous examples in the math literature from the world's most famous mathematician.


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[Assistant] You know, Gauss, and we'll, we'll talk about Gauss later during the term. He's one of the most famous mathematicians ever. He wrote his PhD thesis in 1799.


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[Assistant] And it's usually referred to as the first rigorous proof of the fundamental theorem of algebra. And that says that every polynomial has a zero over the complex numbers. Something probably a lot of you learned in high school.


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[Assistant] You get a polynomial, you can find roots of the polynomial over the complex numbers. But his thesis contains the following quote.


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[Assistant] If a branch of an algebraic curve enters a bounded region, it must necessarily leave again. Nobody to my knowledge has ever doubted this fact.


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[Assistant] Warning signs, buzzers should be going off in your brain. But if anybody desires it, then on another occasion I intend to give a demonstration which will leave no doubt. So he's using something that he believes to be true. He sort of thinks it's clear that everybody knows it.


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[Assistant] But he writes this down, and when you're writing that down, you know there's a problem.


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[Assistant] And in fact, there's another Fields Medalist, we talked about Fields Medals last week. Steven Smale writes,


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[Assistant] That this was an immense gap in the proof that was not filled until 1920, more than a hundred years later. So Gauss never could give the proof, and nobody did for over a century more. Remember the Poincaré conjecture from last week? You know we had Colbert talking about that?


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[Assistant] In 1900, Poincare claimed it was a simple fact. Four years later, he decided it wasn't so simple, and he demoted his claim to the status of a conjecture. And of course, this became the famous Poincare conjecture, which took another century to solve.


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[Assistant] Alright, so when you, you know, think you see yourself doing this, famous mathematicians do it too and it catches them.


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[Assistant] Uh, so you wanna not try to do that. Now, in fact, just to really bring home the point, we've made a, uhm, top ten list of proof techniques you should not use in 6042. So everybody go to the handout here.


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[Assistant] There's a lot of them on the back too, but on the front page there's the top ten. So we're gonna go through these.


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[Assistant] These are the ones we've most observed in 6042 over the years. Number 10. Proof by throwing in the kitchen sink.


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[Assistant] The author writes down every theorem known to mankind and then adds a few more for good measure.


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[Assistant] Now this is good because when you're questioned later, you're trying to get a better score on your exam, uh, you say, look, the proof contains all the key facts. They're all here.


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[Assistant] Uh, and so that does help. You get extra credit for all the facts in there. We have literally seen


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[Assistant] Students copy over their crib sheet if they got enough time for a problem they're not knowing how to solve. Just to get all the facts in there. Number nine, proof by example.


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[Assistant] The author gives the case N equals 2 and suggests it contains most of the ideas of the general proof. In fact, a student before class was telling me on one of the homework problems, he talked about it to his non-math friends, and they gave examples. Not the proof. Number eight, proof by vigorous hand waving.


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[Assistant] One of my favorites, I get up here and wave my hands and it must be true.


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[Assistant] Okay, number seven, proof by cumbersome notation.


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[Assistant] Here the reader gets hopelessly confused, gives up and says okay.


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[Assistant] In fact, I once had a grad student and we called him the encryptor because he could take the simplest proof


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[Assistant] And so encrypted in notation in God knows what, you can never understand the proof. And you figure out if it was really right or wrong. That goes well with number six, proof by exhaustion. So does throwing in the kitchen sink.


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[Assistant] Number five, proof by omission. You will see that a lot, also by faculty. The reader may easily supply the details.


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[Assistant] The other cases are analogous. Trivial. Experts use that all the time. You can find a lot of my papers online. The proof is trivial. Very bad. Every once in a while it turned out not to be so trivial.


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[Assistant] Number four, proof by picture. We've seen those. Saw one on Tuesday. Number three, proof by vehement


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[Assistant] Assertion, sort of like the hand waving. The more forceful you are in your argument, the higher you raise your voice, the more intimidating you become. Doesn't make it more true. Number two, proof by appeal to intuition. Now you say, any moron knows that. Well, you're sort of reluctant to now question it.


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[Assistant] And then number one, proof by reference to eminent authority. I saw Fermat on the elevator and he said he had a proof. I don't think so.


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[Assistant] Fairmont's been dead for over 300 years, and he was not so reliable in the first place.


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[Assistant] In fact, he has one of the most famous assertions that turned into a conjecture called Fermat's Last Theorem. And it was not really a theorem by the time he died.


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[Assistant] How many people have heard of Fermat's last theorem? Raise your hand. Yeaah, pretty famous. It's the one theorem he proved that, actually he didn't prove.


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[Assistant] Says, for all n bigger than 2, there does not exist


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[Assistant] And x, y, and z in the natural numbers.


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[Assistant] Uh, plus the positive natural numbers such that x to the n plus y to the n equals z to the n. Now of course for n equals 2,


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[Assistant] That's just the Pythagorean theorem. You can find like 3, 4, and 5 when you square them. Satisfy that. But it says this does not work for any higher value, any higher power than 2.


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[Assistant] Now, in 1637, Fermat wrote in the margin of a book,


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[Assistant] That he had discovered a proof of this result, but the proof was too long to fit in the margin. And so he wasn't going to supply the details there.


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[Assistant] And he never did supply the details and it took 350 years.


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[Assistant] And hundreds and hundreds of pages, a fellow named Andrew Wiles took him personally over ten years. He more or less locked himself in his room for ten years.


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[Assistant] Uh, produced hundreds of pages and he finally did prove Fermat's Last Theorem.


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[Assistant] Uh, so we, Fermat was right about one thing. It would not fit in the margin. At least, there's no proof that we know of that would fit in the margin of a book. Any questions on proof technique? We're gonna pound a lot on this, uh, over the next few weeks.


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[Assistant] Alright, next we're gonna look at a class of puzzles.


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[Assistant] That was very popular in the late 1800s.


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[Assistant] Now in these puzzles you have a grid of letters or numbers and you gotta slide the letters and numbers around to put them in order. And so as an example,


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[Assistant] Here's the problem on a 3x3 grid or an 8 puzzle. You want to find a sequence of moves.


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[Assistant] This configuration, I'll do a 3x3 case.


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[Assistant] I got A here, B here, C, D, D, F, and I put G and H out of order. And that square is blank, and I can move


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[Assistant] Tiles or letters into it. Want to start here and I eventually want to get to


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[Assistant] The configuration with the letters are all in alphabetical order.


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[Assistant] Now, a legal move means you slide a letter into an adjacent blank square.


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[Assistant] You slide these things that could go down, that could go across. How many people played the game like this before?


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[Assistant] Good, good, good, a lot of you. Okay, so I need some volunteers who are good at solving puzzles that can do this. They've played games like this, and they're pretty quick at it.


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[Assistant] Alright. So I've got it here.


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[Assistant] Alright, who'd like to come down? There's prizes if you can solve this. What we're gonna do is solve this and see if you can solve it in three minutes. But I need three students to come on down and do this.


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[Assistant] Who'd like to volunteer? Who thinks they can handle puzzles under pressure? They have the G and H out of order. See, you know, it's not so hard. You move the F down there. I don't know. You know, play around like that. So I need some volunteers. Who'd like to come down and do this? You can win candy. I got candy here. You know? There's even a prize if you don't make it in three minutes. Who wants some candy? Nobody?


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[Assistant] What's that? There is, come on down. Alright, now who's gonna help him out?


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[Assistant] I need a couple more people here just to mess him up, cause by himself, well there we go, good. Anybody else like to come on down? Cause if we get three, it really gets hard. Cause you fight over what moves to make.


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[Assistant] Now, uh, what I'd like you all to do is sort of scream out possibilities for them. Like you figured out there, if you scream out, that'll really screw them up. Okay, so, uh, now you can't pick them up, you just gotta slide them around, and you can go, and we'll go to 256, and see if you can solve it. If you can solve it, you guys get, get candy.


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[Assistant] Alright, you got about two minutes. Oh, they got the G and H in order. Uh, everything else is pretty well screwed up though.


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[Assistant] I think they need help. We've got about a minute to go. You can't see anything. Well, I don't know if it would help.


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[Assistant] Alright, well, now you guys are good sports, so you do get a prize.


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[Assistant] Now, I was a little cruel to these guys. Can anybody, for a lot of reasons, it's impossible? You think?


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[Assistant] Yeah, it is impossible. Uhm, this is impossible to do. Let's see if we can prove it's impossible.


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[Assistant] Uh, cause it doesn't seem so hard necessarily. Uh, but let's, let's state that as a theorem.


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[Assistant] Now, to prove this, we're going to use what's called an invariant.


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[Assistant] It's a very powerful and commonly used concept in computer science, very closely tied to induction as we'll see. In order to show that your system can never reach a particular special state,


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[Assistant] It is sufficient to show there's some property called the invariant that holds at the initial state and that is preserved by every legal move. And


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[Assistant] Is not present, does not hold in that special state.


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[Assistant] The idea is if you get this magic property called the invariant and it holds at the start and it holds across every step, then the only states you can reach have to have this special, have to have the property, the invariant.


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[Assistant] And if the special state doesn't have that property, you can't reach it.


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[Assistant] There is gonna be some property we're gonna look for that held at the beginning here. In the beginning had this set up.


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[Assistant] That holds, it's preserved by every move, every legal move, but does not hold in this state. And therefore you could never reach this state legally. So they were doomed.


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[Assistant] Uhm, and that's always the trick in analyzing systems or algorithms is what's that key property? The invariant property.


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[Assistant] So to figure out the invariant, we just sort of have to look at what happens during a move in this system, a transition. Well there's two kinds of moves. There's a row move.


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[Assistant] And I'm gonna move the G in its row to the blank square. So it would become this state.


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[Assistant] So G moves rightward into the blank, and the rest stays the same.


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[Assistant] Did the relative order of the items change?


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[Assistant] No, in fact, of course to be precise, I better define what relative order means, or the natural order. So, by the natural order,


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[Assistant] G moved from five to six, but didn't change its order with respect to any of the other guys.


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[Assistant] Cause that'll sort of, that'll be very useful. Number one. A row move.


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[Assistant] Of the items. Of the letters.


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[Assistant] Hmmm. Now you can't let me get away with that, right? Cause I'm not gonna let you get away with that. On homework. That's not so good. Uh, so we gotta be a little more careful in the proof here. Otherwise we'll just head down the path to trouble.


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[Assistant] So let's sort of specify what does a row move mean. Just get some little bit of math and specification around it. Well, in a row move,


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[Assistant] Okay, into cell I minus one or I plus one.


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[Assistant] The order of the items is preserved.


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[Assistant] Because if you're going with i to i plus one, everything else is an i minus one or less, i plus two or more. Relative order does not change in this case.


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[Assistant] I, you know that I've thought about it a little bit, I've really quantified that i goes to i minus one or i plus one, nothing else moves.


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[Assistant] Therefore, we're done. Now I could have added some more sentences here like I talk about, but this is probably enough in this case. For this guy.


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[Assistant] If we only had row moves, there's not much that happens. The order never changes with row moves. So we never get there, but column moves are more interesting.


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[Assistant] Did the ordering change? For this move? Yeaah. Uhm, how so?


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[Assistant] G changed by 3, moved 3 in the order, and what letters did it change relative order with?


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[Assistant] Yeaah, the two preceding it. G used to be after H and E, now it's before H and E. Alright, so it


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[Assistant] Changed relative order with two items and changed its position by three, moved up three in the ordering and so it changed order with the two in between. Let's look at another one.


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[Assistant] Cause we're really doing this, these examples to hunt for an invariant, is what we're trying to do here.


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[Assistant] Something that'll let us prove we can never get to the desired state. Alright, let's look at this one.


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[Assistant] Relative order changes, which pairs got changed in their relative order?


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[Assistant] B used to be before C and D, when it moves down here, it goes after C and D.


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[Assistant] Any guess about what Lemma 2 is going to be? What happens when we make a column move?


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[Assistant] Good. So a column move changes the order of a guy with the previous two or the next two, which means the relative order of two pairs change. So let's take that as Lemma 2.


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[Assistant] Yeah, when we say don't do proof by example, we don't mean don't try examples, because by trying examples you find out what you're trying to prove. Alright, so that's good to do. A column move


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[Assistant] We move an item in cell i, for some i.


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[Assistant] Let's draw out the natural order. I got one, two, three, four, five, six, seven, eight, nine. Column moves could be here. One to four and four to one.


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[Assistant] Two and five, three and six, four and seven, five and eight, six and nine. All twelve possible column moves this works for. They're always separated by three.


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[Assistant] And when an item moves three positions,


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[Assistant] It's i minus 1, i minus 2, or i plus 1, i plus 2. Those are the guys it changes order with.


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[Assistant] In a row move, order does not change. In a column move, two pairs of letters get flipped. That's all you can do with this puzzle now. So can anybody think of something we can work with to get an invariant here? Something that won't change? What we should be focusing on?


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[Assistant] Yeah, that's good. So really to focus in on there is if you have an even number of things out of order, start.


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[Assistant] You're gonna have an even number forever because you can only change two at a time.


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[Assistant] That's a great idea. Let's, let's really specify that now. And, you know, define that.


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[Assistant] So for example, let's see, how many inversions do we have in this case?


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[Assistant] Yeah? Three, good. And what are they?


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[Assistant] And what's the last one? EG is an inversion. Yes. So the answer is there's three inversions in this puzzle.


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[Assistant] And if I keep doing row and column moves to this, what do I know about the parity of the number of inversions?


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[Assistant] That's sort of interesting. Okay.


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[Assistant] Now, how many inverted pairs are there in the start state, way back over here? How many inverted pairs here? One inversion.


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[Assistant] Alright, and can you see where we're gonna head here now?


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[Assistant] If I started with one, that's odd.


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[Assistant] And every time I do a row move or column move, it's gonna stay odd, and I can never get here.


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[Assistant] Is that clear, what we're trying to do? So let's keep doing it.


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[Assistant] Alright, so now we'll use Lemma 1 and Lemma 2 to create this notion that we


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[Assistant] Only change the number of reversions by an even number. In Lemma 3.


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[Assistant] Alright, this proof is pretty easy. There's a couple of cases.


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[Assistant] In a row move, what happens? How many, how does the number of inversions change in a row move? Stays the same. No changes. And that's by Lemma 1.


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[Assistant] Now in a column move, there's three cases.


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[Assistant] But we know two pairs change order, that we know. That's limit two.


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[Assistant] So case A is that both of the inverted pairs


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[Assistant] We're in order, originally, before the column moved.


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[Assistant] What happens to the number of inversions in this case as a result of the column move?


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[Assistant] Is that both pairs were inverted before I made the column move. What happens to the number of inversions in that case?


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[Assistant] Decreases by two, they were in order, so they were, they were inverted. Now when I flop them they become...


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[Assistant] Uh, in order, so inversions drops by two. And case C is, there's one of each. One of the pairs was inverted.


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[Assistant] The other wasn't. What happens in the number of inversions in that case? Yeaah, stays the same.


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[Assistant] Alright, and we're done. That's what the lemma says. I can go up by two, down by two, or stay the same.


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[Assistant] Alright, now, a simple corollary of this.


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[Assistant] Is that during a move, the parity, i.e. even or odd,


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[Assistant] Uh, the number of inversions stays the same. It can't change at all.


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[Assistant] A parody, and a parody means even odd, sometimes it's called zero one.


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[Assistant] But the evenness or the oddness of the number of inversions


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[Assistant] Oops. I mean one pair is inverted, one pair is not.


[User] Please transcribe this text

[Assistant] You know what I mean? Two pairs change order in a column move. Right? Let's go back and look at that. So I'm doing a column move, for example, G goes up,


[User] Please transcribe this text

[Assistant] GH changes order, GE changes order.


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[Assistant] G changes order with these two guys, so there's two pairs that got flipped.


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[Assistant] Alright, so in a column move, two pairs reverse their order. If they were in order, they become inverted. If they were inverted, they become in order.


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[Assistant] Yes, a letter can be in, uh, letter E is in seven pairs. Cause there's seven other letters.


[User] Please transcribe this text

[Assistant] And in fact we're looking at a space of, what, uhm,


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[Assistant] 8 times 7 divided by 2, 28 pairs of letters. So there could be 28 inversions if they're all out of order. Any other questions?


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[Assistant] Okay, well we're almost, we're getting there, we're almost done. I gotta prove this corollary that during a move, the evenness or oddness of the number of inversions does not change.


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[Assistant] And that's because adding or subtracting 2 from a number does not change its parity. Stays odd or stays even.


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[Assistant] Anybody tell me what the invariant is gonna be? What's the invariant gonna be in this system?


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[Assistant] Yeaah, the parity of the number of inversions is odd. Because it starts that way, it's preserved, and it won't hold in the desired end state that we're looking at. So we can state that as follows.


[User] Please transcribe this text

[Assistant] In every state or configuration reachable,


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[Assistant] Alright, and the proof will be by induction. And invariant proofs are always by induction.


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[Assistant] And the inductive hypothesis, and this is very typical in a proof by using invariance, is, so p of n is after any sequence


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[Assistant] Well, in fact, just the rest of this is what it is.


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[Assistant] P of n, after any sequence of n moves, so n is the number of moves you took to get there, from the start state, the parity of the number of inversions is odd. That is the inductive hypothesis.


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[Assistant] This is the invariant and they become one and the same. And the parameter n is the number of moves you've taken.


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[Assistant] And that's how you set up a proof using an invariant. Always looks just like this.


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[Assistant] Okay, so we're doing a proof by induction. We've got the, uh, inductive hypothesis. What's the next step?


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[Assistant] And in these circumstances with the proof by invariant, the base case is always zero. We haven't made any moves yet.


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[Assistant] Now in this case, if we haven't made moves, the number of inversions in the start state now is one. That means the parity is odd.


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[Assistant] And the hypothesis is satisfied. P of zero is true. Because in any sequence of zero moves,


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[Assistant] After this, i.e., in this state, you got one inversion, that's odd.


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[Assistant] And here we need to show for any n bigger than or equal to zero, we need to show


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[Assistant] P of n implies P of n plus 1 is true. Alright, the standard thing.


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[Assistant] That's talking about where we are after a sequence of n plus 1 moves in the puzzle, in the start state.


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[Assistant] And call these moves labeled by M1 all the way to M, N plus 1. Those are the moves we made.


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[Assistant] Now, by the inductive hypothesis, because of PN, we know that after the first N moves, the parity is still odd. That's what PN says.


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[Assistant] Alright, that's pn. After any sequence of n moves from here, the parity of the number of inversions is odd.


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[Assistant] So we could say by the inductive hypothesis, or we could say by PN.


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[Assistant] Now, by corollary 1, do we still have corollary 1?


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[Assistant] Parallel, yeaah, good. We got Correlate 1. During any move, the parity doesn't change. That says from


[User] Please transcribe this text

[Assistant] The nth, after the nth move to after the n plus first move, one extra move doesn't change the parity. So it's still odd.


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[Assistant] The parity, uh, the number of inversions,


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[Assistant] Therefore, this implies that the parity


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[Assistant] After all n plus 1 moves, these guys.


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[Assistant] Start at odd, it stayed odd after the first n, it stayed odd during the n plus first, so it's odd. And that's exactly the statement of Pn plus 1. So we've completed the induction. We've shown that Pn implies Pn plus 1.


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[Assistant] Alright, so now we've, any questions so far on this? Yeaah.


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[Assistant] You know, really, there's not a lot of difference. Corollary usually is something that's a simple consequence of something else. And,


[User] Please transcribe this text

[Assistant] The corollary was, had a pretty short proof. Usually when you see corollary, there's not a ten page proof there.


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[Assistant] Uh, lemma is something that you're gonna use as a tool for a bigger thing. Often a theorem or another lemma.


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[Assistant] In fact, now we're gonna get to the theorem, which is sort of the final thing, the thing we're really after. And that's sort of the big deal.


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[Assistant] And now the proof will be simple because we've done all these lemmas and corollaries. Any other questions?


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[Assistant] Uh, in fact did I write the theorem? Yeah, the theorem's up here, right?


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[Assistant] There is no sequence of legal moves to invert G and H. In other words, the students were screwed. They couldn't do it, uh, fairly. So let's prove the theorem now.


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[Assistant] In the desired state, i.e. in order. The target state.


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[Assistant] Is even. Nothing's out of order. There's zero inversions. By Lemma 4,


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[Assistant] Using legal moves. So we're done. So the proof of the theorem, by now, is short.


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[Assistant] Because we did four lemmas, a corollary, and lots of arguments.


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[Assistant] Uh, could you have solved the puzzle if I allow the blank to be somewhere else at the end?


[User] Please transcribe this text

[Assistant] Doesn't impact the parity at all, so you can't do that.


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[Assistant] Is you're looking for a property that holds the beginning, is preserved by every step, but is not present in the target state.


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[Assistant] Alright, now, actually this puzzle was enormously popular, uh, in the late 1880s. In fact, there was a thousand dollar prize offered.


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[Assistant] For anybody who could solve the larger version. Back then it was the 15 puzzle, which you can still buy today. Same problem on a 4x4 grid. Exactly the same problem. A pair is out of order, you gotta get them in order. You can't do it. Uh, now the proof you can't do it,


[User] Please transcribe this text

[Assistant] Is a little harder. Same idea. One extra idea in it. Uhm, one extra lemma. And that's for homework.


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[Assistant] Alright, so you'll go through and see exactly the structure with one extra little trick in it. Uh, but,


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[Assistant] Back then, I guess most people didn't, well, of course most people didn't realize it was undoable so they could offer this prize safely. You know, a thousand dollars then, probably worth a quarter million dollars today.


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[Assistant] Ah, let's see, now the moves are, we have the first move is M1, the second is M2. Now when I talk about zero moves, that means no moves have taken place, I'm still in the start state. So a move is a transition between states. So you could have state zero, that's the start state. State one would be the move, state after move one.


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[Assistant] I got that right here, so the inductive step I'm showing that, yeaah, P0, I need to show P0 implies P1, P1 implies P2, and so forth.


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[Assistant] And pn is the parity of inversions after n moves.


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[Assistant] So you could have state S0. We didn't label the states here, but you could have state S0.


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[Assistant] Ah, well that's a good question. Okay, so really there's two cases. We'd have to sort of


[User] Please transcribe this text

[Assistant] To be really precise, we should have argued there's only two types of moves.


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[Assistant] A row move and a column move. And in fact, if you look at what can move into a blank square, there's four guys that can do it. The guy above, the guy to the right, the guy to the left, and the guy below. Two of those are row moves, two are column moves. So that's right, to really pin down the details we should have checked those are the only two possible moves.


[User] Please transcribe this text

[Assistant] And if there was a third kind of move that I hadn't considered, this proof would be bogus. That's a good question. Any other questions?


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[Assistant] I think probably you see why the invariants are important, but, you know, say that someday you're building software to run a nuclear reactor. Now there's a certain state you'd really like to avoid. Meltdown.


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[Assistant] And you'd like to be able to prove that in any sequence of moves that happen in your reactor program, you never reach the meltdown state. Well, you'd use invariance to do that. Or if you're building software for an airplane, you'd like to never reach the crash state.


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[Assistant] Or if you're building a radiation device, you never want to get in a state that fries the patient.


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[Assistant] Uh, so these things actually, this is an important notion.


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[Assistant] Alright for the rest of the day we're going to talk about a different kind of induction called strong induction.


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[Assistant] It's very similar to ordinary induction, but it's a little easier to use when solving certain problems.


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[Assistant] Now like regular induction, strong induction can be expressed with an axiom.


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[Assistant] Alright, now the only difference between strong induction and ordinary induction that we did last time is this part.


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[Assistant] In ordinary induction, you're showing that Pn implies Pn plus 1. In strong induction, you're showing that all these facts put together imply Pn plus 1.


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[Assistant] Now to show an implication, remember you get to assume that all these things are true.


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[Assistant] So you know how in ordinary induction we say assume Pn is true and you're going to prove Pn plus 1?


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[Assistant] In strong induction you get to say, assume pn is true, pn-1 is true, dot dot dot, p1 is true and p0 is true. You get to assume a lot more.


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[Assistant] So it's a stronger proof technique, because it allows you to do more, or so you would think. Now in fact,


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[Assistant] Any proof you can do with strong induction you can do with ordinary induction. It just might be harder. So you can't prove anymore with it, but it makes your proofs much easier and we're gonna do an example in a minute where the proof is much easier.


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[Assistant] By getting to assume all of these things are true in order to prove Pn plus 1.


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[Assistant] Are there any questions about the difference here with strong induction? That now you get to assume all these are true as part of the inductive step?


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[Assistant] Alright, our first example with strong induction is gonna be a simple game. We got a stack of eight blocks here. Now in this game, what you're gonna do is divide the stack into two sub stacks. So I just took eight and split it into three and five.


[User] Please transcribe this text

[Assistant] For that move, you get fifteen points. Three times five. Let's write that down.


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[Assistant] And in the first move we went 8 split to 5 and 3 and that's worth 15 points.


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[Assistant] I'm gonna split the stack of five into four and one, and for that you get four points. Four times one.


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[Assistant] And I keep on going until I have eight stacks of height one.


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[Assistant] And the last move I'll be splitting a two into two ones, I'll give me one point. Then I add up all the points, and that's your score.


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[Assistant] And the goal is to get the most number of points when you add up the entire score. Now, just so this game is clear, we're gonna play it. And I'm gonna raise the stakes here.


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[Assistant] What we're gonna do is we're gonna have a competition between three members of the class and the TAs.


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[Assistant] Now, if the class team wins, they get more points, we're gonna give everybody in the class candy. And I got lots of candy here. If the TAs beat you, they get the candy.


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[Assistant] I will come to that later. But you want to win this thing. Alright, so I need some volunteers for the class who think they can do well. Yeah? Come on down. Alright, you two guys, come on down. Alright, TAs, come on up.


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[Assistant] Now I want you to make some noise if you think your three class reps here can beat the TAs.


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[Assistant] Alright, how many people? Make some noise if you think the TAs are going to win.


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[Assistant] Alright, so uhm, let's see, I think we're gonna let the class go first. This is your stack. Now think about what move you want to make first, cause that matters. And uhm, you guys can give them some advice if you want, as to the move they should make first. Then I'm gonna keep score over here.


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[Assistant] Oh, okay. They went for the maximum move. That's good. They realized four times four is sixteen. Very good. So let's write that up here.


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[Assistant] That's the maximum number of points for the first move, I think. Good job.


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[Assistant] Ah, and they've got a move. That's a pretty pathetic first move for the TAs. Looking good for the candy for you guys. They got seven points for the TAs. Class, what would you like to do for your next move?


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[Assistant] Ah, no, no. One, one at a time. No, no, just one. One stack gets split. Either one.


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[Assistant] The TAs are convinced they're gonna lose. Three and one, okay, good, alright. Interesting. So we got three and one, that gives you three points. Alright, TAs, we need to move.


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[Assistant] Alright, so they've gone six and one, six points.


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[Assistant] You think you're dead already, huh? So you've gotten three and one now, again. For three. So you're up to twenty-two. You're way ahead. Twenty-two to thirteen. Alright, TAs. They look a little disorganized to me. What do you think?


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[Assistant] Another one. You can tell where they're headed here. Five and one. They got five points. They're up to eighteen. They're four behind you. Alright class. Alright. You got two points. TA's.


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[Assistant] Oh, big change in strategy there, yeaah. Alright. Why, all that thinking and that's what we got. Okay. Four points. Class.


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[Assistant] Alright, they pick up two more here. The three just got split into two and one.


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[Assistant] For two points, how you doing? You got 19, 22, 24, up to 26. 13, 18, 22. You're still doing good. TA's are behind. Let's see what they got up their sleeve here.


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[Assistant] Alright, they definitely have a strategy going here. That's three points. Uh, you're, ooh, there's not much option left for you guys. Yeah?


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[Assistant] Alright, that gave you a point. Two went to one on one, one point.


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[Assistant] Not a lot of choice for you guys either, I guess. There's only one way to split that stack.


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[Assistant] Alright, two points, and your final move for the class, and all the candy, you get one more point.


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[Assistant] Let's see what your total is. You got twenty-two, twenty-four, twenty-six, twenty-seven, twenty-eight. And the TAs are gonna get, what last move for the TAs?


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[Assistant] Alright, and you got 2, 1, 1, 2, that's interesting, 1, 2, 3, 4, 5, 6, 7, what else, 7 times 8 over 2, uh, 15, no, that's not right. Is that right? Oh, 28! We got a tie! Ah, jeez. Alright, I tell you what, I'll give you guys one more chance to start over. You've seen that, you gotta beat, you gotta beat them.


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[Assistant] You gotta beat them though. You got 28. You already tied them. They came from behind and tied you. You gotta beat them. You got off to a good start, but then they caught up.


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[Assistant] Oh, now your class reps had an interesting thought here. They asked, is it possible that any strategy gives 28?


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[Assistant] And you can't do worse than 28 either, because every strategy gives 28.


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[Assistant] So I guess we're gonna have to throw it to the class to vote who should get the candy.


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[Assistant] So if you think your reps really should win this and you should get the candy, make some noise.


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[Assistant] Alright, so you get the candy, so why don't we, here, help pass out the candy here. We got plenty of bags and baskets. Take it all up on both sides and pass it out.


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[Assistant] We should have enough for everybody unless somebody got really hungry down here in front.


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[Assistant] Okay, so while you're doing that, let's try to prove this theorem.


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[Assistant] So the theorem is that all strategies for the end block game


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[Assistant] And the score we'll call it S of N for the N block game. So for example,


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[Assistant] Strong induction. So you want to write that down? Proof by strong induction.


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[Assistant] What's the next step in a proof by any kind of induction? Ah, base case, not yet. The predicate. What's the predicate here? Alright, so, oh, the inductive hypothesis, the predicate. Any guesses as to what p of n is going to be?


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[Assistant] Uh, yeaah, but that's part of it, but...


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[Assistant] We're gonna use this as a predicate to start with. That, you've got, you've gone a step ahead there.


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[Assistant] So I'm going to start with this as being the predicate, because I'm trying to show they're all the same, and I haven't figured out what the score is going to be, but I think you're already on a good track here, what the score is going to be. Now we do the base case.


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[Assistant] So we take n equals one, the one block game. S of one is zero.


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[Assistant] We never took a move. We never got a score. There was only one block. We ended before we started. So that's okay.


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[Assistant] And it could be any split. We might have k on one side, then n plus 1 minus k on the other side. For any k between 1 and n.


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[Assistant] And let's figure out our score for that scenario.


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[Assistant] Well, we get the product of these numbers for the first step.


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[Assistant] And then as we recurse and we split up that block of k blocks.


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[Assistant] How many points do we get for splitting that all the way down? P of K.


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[Assistant] Because that's the induction, we can assume that's part of the induction hypothesis.


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[Assistant] And how much points do we get for splitting this stack all the way down?


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[Assistant] And do you see why strong induction's coming in handy here? For the TA strategy, K would be 1. We know P of 1 is 0. And you would just need ordinary induction. This would be P of N.


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[Assistant] But for a general strategy, could be any split. So we need to have all these assumed here. Makes it much easier when using strong induction.


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[Assistant] Alright, so this is our total score for the game, for n plus 1 blocks. And we're trying to show it doesn't depend, it just depends on n. And that it doesn't depend on k.


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[Assistant] Well, it looks like it to me, right? I got K in every single term.


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[Assistant] You see why I'm stuck? Because I'd like to be able to say that the score I get for the n plus one block game, this is s n plus one.


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[Assistant] Is the same for any sequence of moves. Doesn't even depend on K. But I can't do that. What do you do when you're stuck with an induction proof? What's one of the tricks? Make it stronger. Which means I get a stronger induction hypothesis. Now somebody gave me one up there before. How could I strengthen my induction hypothesis here?


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[Assistant] Her SN. That would make it stronger. I tell you what it is. What's a good guess for the formula here?


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[Assistant] What, 8 times 7 over 2 is 28, it works here. Let's check another one. What's, what's s of 2?


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[Assistant] One. Well, two times one over two, it works. What's s of three?


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[Assistant] Two of the first move, then one more. Three. Three times two over two is three. Ooh, looks good.


[User] Please transcribe this text

[Assistant] So now I've got a stronger induction hypothesis. Not only is it always the same, it is that number is your score. Let's see if that works. Let's plug that in now.


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[Assistant] So, base case, let's check it out. Is 1 times 1 minus 1 over 2 equal to 0? Yes. Alright, let's plug it in here now.


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[Assistant] Well, p of k would be now k, k minus 1 over 2.


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[Assistant] Alright, let's, I gotta add all these things up and see if it equals that expression for n plus 1. Let's do that.


[User] Please transcribe this text

[Assistant] Alright, I get 2kn, plus 2k, minus 2k squared, I'm putting all this over 2. One giant mess over 2 here. Plus k squared, minus k, yikes, plus n squared,


[User] Please transcribe this text

[Assistant] Well, let's write it just as n plus 1 times n.


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[Assistant] Minus Km. All over 2. And now let's cancel.


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[Assistant] Two KNs here. Cancel KNKN.


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[Assistant] Uh, 2k squared, minus 2k squared cancels k squared. Oop, and I forgot. Plus a k, plus a k squared. From here and here.


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[Assistant] So I'm left with n plus 1 times n over 2. That is s n plus 1.


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[Assistant] Is for n blocks, n times n minus 1 over 2.


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[Assistant] It's pretty good for strong induction, unless you do some pretty powerful things. Okay, very good. That's it for today.


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[Assistant] Defend Objective Golf. Defend Objective Hotel. Defend the bomb. Defend the carrier. Defend the device. Defend the flag. Defend the HVI. Defend the package. Defending here! Diffuse Objective Bravo. Diffuse Objective Charlie. Diffuse Objective Delta. Diffuse Objective Echo. Diffuse Objective Foxtrot.


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[Assistant] Diffuse objective golf. Diffuse objective hotel. Diffuse a charge. Diffuse a device.


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[Assistant] Diffuse the objective. Diffuse the objective. Diffuse the package. Diffuse the target. Deployable cover here. Destroy target Alpha. Destroy target Bravo. Destroy target Charlie. Destroy target Delta. Destroy target Echo. Destroy target Foxtrot. Destroy target Golf. Destroy target Hotel. Destroy the bomb.


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[Assistant] Destroy the flag. Destroy the HVT. Destroy the package. Destroy the target. Destroy the target. Dibs on gunner. Dibs on shotgun. Dibs on the satchel. Don't make this a habit, yeah?


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[Assistant] Doors locked. Down low. Down range. Drill charge here. Dropping ammo here. Dropping some armor. East. Easy pickings here. EKIA. Elevator here. Eliminate the device. Eliminate the HVT. Eliminate the package. Eliminate the target. Elimination target marked. Encrypting data.


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[Assistant] It's UAV marked. Enemy airdrop marked. Enemy air strike is active. Enemy air support active in the AO. Enemy air support marked.


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[Assistant] Enemy tear gas here. Enemy thermite marked. Enemy transport helo marked. Enemy trophy system active. Enemy truck marked. Enemy UAV marked. Enemy vehicle. Enemy visual. Enemy VTOL jet marked. Enemy Wilson marked. Engage the HVT. Engaging on my mark. Extraction balloon here.


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[Assistant] Enemy recon overhead! Eyes up! Someone's bent here! Eyes up! The sector's hot! Failed upgrade here. Blanket this way! Flashbang grenade here! Found cash here! Found a bomb drone. Found a care package. Found a chopper gunner.


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[Assistant] Found a counter UAV here. Found a mortar strike. Found a stash. Found a stealth bomber. Found a UAV. Found a VTOL jet. Found a Wilson. Found an advanced UAV. Found an emergency airdrop.


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[Assistant] Friendly airdrop here! Friendly airstrike here! Friendly air supports overhead! Friendly C4 here! Friendly care package marked! Friendly chopper gunner overhead! Friendly cluster mine here! Friendly cruise missile inbound!


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[Assistant] Friendly decoy mine here. Friendly emergency airdrop here. Friendly gunship in the airspace. Friendly killstreak here. Friendly lethal here. Friendly recon overhead. Friendly stealth bomber in the sky. Friendly suppression mine here. Friendly tactical insertion here. Friendly tacticals here. Friendly UAV orbiting above. Friendly VTOL active. Geiger counter here.


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[Assistant] Get to Echo. Get to Foxtrot. Get to Golf. Get to Hotel. Get to me, I'm down. Get to the bomb. Get to the device. Get to the LZ. Get to the package. Good loot here. Good shit soldier. Got a boat here. Got a boss here. Got a closed door. Got a cruise missile here. Got a field upgrade.


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[Assistant] Got eyes on an enemy stronghold. Got eyes on an enemy vehicle.


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[Assistant] Got eyes on the objective. Got eyes on the package. Got lethals. Got loot here. Got O2 here. Got one sighted here. Got recon here. Got rockets here. Got some cash. Got visual on a stronghold. Got a friendly jug crate here. Got a safe here. Hang tight, coming your way.


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[Assistant] I'll have that weapon. Hey! Enemy field upgrade here! Hey!


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[Assistant] We need tactical gear here! Hey, need a field upgrade! Hey!


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[Assistant] On me. Hey, we can push this. Hit a container here. Hold on the door. Hold in what I got here. Hook me up with cash. Hunt contract here. HVT in my sights. HVT located. Let's go. I can use the armor. I can use that cash. I got a key card. I got a melee weapon. I got eyes here. I got you.


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[Assistant] Custom weapon! I have a launcher! I have a marksman rifle! I have a pistol! I have a riot shield! I have a shotgun!


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[Assistant] I have a sniper rifle. I have an assault rifle. I have an LMG. I have an SMG. I have medical. I've got some high caliber rounds. I've got some mid caliber rounds. I've got some small caliber rounds. I've got some tacticals. I have some valuables. I need a ride. I need a weapon. I need launcher ammo. I need those rounds. I want that buzzsaw.


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[Assistant] I'll drive, I know the way. I'll fly the bird. I'll get you back. Hold tight. I'll go for a ride. I'll take that valuable.


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[Assistant] I'll take point. I'll take that. I'll take that dog tag. I'll take that medical. I'll take that oxygen. I'll take that self-revive. I'll take that weapon. I'll take the field upgrade. I'll take the key. I'll take the key card. I'll take the kill streak tablet. I'll take those nods.


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[Assistant] I'm gonna enjoy hunting your team. I'm outta here. Intel contract here. Intel here. I've got a kill streak tablet. I've got rockets. I've got room. I've got shotgun shells. I've got some armor. I've got some enemy dog tags. Ugh. Jammed.


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[Assistant] Let me handle the big gun. Let me drive. Lethal's here. Let's give him an encore, huh? Let's grab that recon contract. Let's grab that scavenger contract. Let's have that hunt contract. Let's not do that again.


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[Assistant] Located a cluster mine. Located a gunship here. Located a sentry here. Located an airstrike. Looking for a kill streak. Loot here. Looting here. Marked a tank. Marked an SMG. Marked our deployable cover. Marked the enemy target area.


[User] Please transcribe this text

[Assistant] Marking Lethals. Marking LZ. Marking Self-Revive. Marking Shotgun Shells. Marking Sniper Rifle. Marking Target Area. Marksman Rifle here. Maybe I'll see ya, maybe I won't.


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[Assistant] Roger, visual on the LZ! Roger, visual on the radio tower. Roger, visual on the stronghold.


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[Assistant] Safe contract here. SAM site here. Secure. Secure Alpha. Secure Bravo. Secure Charlie. Secure Delta. Secure Echo. Secure Foxtrot. Secure Golf. Secure the device. Secure the explosive. Secure the flag. Secure the juggernaut. Secure the objective. Secure the package. SAM text here.


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[Assistant] Quiet! Shit yeah! Shopstick here! Shotgun here! Small cali-


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[Assistant] Switching to AP rounds! TAC cable here. TAC insert here. Tactical camera here. Tacticals here. Tacticals marked. Take out the device! Target of interest here! Tear gas grenade here! Tell me!


[User] Please transcribe this text

[Assistant] That's our thermite. That's a friendly airstrike. That's a friendly counter UAV. That's a friendly Overwatch helo. That's better off with me. That's friendly smoke. That's our jug. That's our proximity mine. That's the way to get it done.


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[Assistant] The enemy deployed cover here. There's a valve here. There's an upgrade here. Thermite here. This one's got a riot shield. This vehicle's been destroyed. Threats in the area. Throwing knife here. Time to finish this one off. Time to get out of here. To your east! To your north! To your northeast! To your northwest! To your south!


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[Assistant] To your southeast! To your southwest! To your west! Transport helo down! Trap spotted! Trophy system here! UAV tower here! Upload station here! Vehicle marked! Visual on an enemy tactical camera! Visual on tags! Visual on the device!


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[Assistant] Visual on the X-fil-bird. Visual on the explosive. Visual on the flag. Watch it, that vehicle's foo-bar.


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[Assistant] Aways out! Active friendly recon up here! Advanced infantry here! Advanced infantry sighted! Affirmative.


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[Assistant] Affirm. Airstrike tablet here. Air support kill streak here. Alright. Alright. I'll finish the fight for ya. Ammo box here. Ammo here! AP rounds here. APC here. Appreciate ya. Armor box here.


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[Assistant] Attack target Alpha. Attack target Bravo. Attack target Delta. Attack target Echo. Attack target Foxtrot. Attack target Golf. Attack target Hotel. Attacking at this location. ATV marked. Back off before it blows. Bandage here. Battle rage here. Be seeing ya. Below you. Bike marked. Bird marked.


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[Assistant] Black site marked. Bus saw here. Calling dibs on that. Cancel last. Car marked. Careful!


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[Assistant] Copy, let's grab the gun safe contract. Copy, let's put him down. Copy, let's secure that intel. Copy on that drop zone. Copy that. Copy that, eyes on the enemy. Copy, visual on the hostage.


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[Assistant] Copy, visual on the safe. Copy, visual on the SAM site. Copy, visual on the target. Copy, visual on the UAV tower.


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[Assistant] I'm not blowing up the plant. Just harmlessly stopping it from doing more damage to Mother Earth.


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[Assistant] You know, Dad, just because your mission's complete, it doesn't mean you have to go back to pork eating and beard drinking.


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[Assistant] Hello, boy. At times like this, there's not much you can say. Especially if you don't know what happened.


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[Assistant] As I always say, don't get mad, get dinner. Then get even. With Skinner.


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[Assistant] Good game everyone! I never dreamed a night where you discovered I was cheating and beat the crap out of me could end on such an up note.