orbits are strictly Keplerian no more; compared to a linear ephemeris, some transits occur earlier and some later. That is what the TTVs stand for.

The promise of the TTV method was that if there is, say, transiting Neptune closer in and an Earth-size planet farther out, then the inner planet’s TTVs should reveal the existence of the outer planet. No other observations needed. Awesome, isn’t it!? That was the theme that the first paper, by Holman & Murray (2005), highlighted. The second paper, by Agol et al. (2005), was broader in scope. It discussed TTVs for all sorts of planetary configurations and gave an approximate scaling of the TTV amplitude for each.

So far so good, I recall thinking, but how about the heart of the matter, which is the [*inverse*]{} problem. The inverse problem arises when someone attempts to figure out the planetary parameters, such as the mass and orbit, from TTVs. Does the inverse problem have a unique solution? What if the companion planet is not transiting?

My goal in 2007/2008 was somewhat naive, I thought it could be possible to fully resolve the inverse TTV problem by employing computer algebra. For that I considered a case 