Monday Puzzle: Some Light Number Music

This is a view of the solar system showing an unidentified object whose orbit is shown in blue. You have to identify the object by solving today’s simple arithmetic puzzles. This is a view of the solar system showing an unlabeled object whose orbit is shown in blue. You have to identify the object by solving today’s simple arithmetic puzzles.

After the heavy metal physics arguments about the papaya battery which are still ongoing, I thought we should relax with some lighter number music. Today’s problem requires nothing more than simple arithmetic and in one case, finding the square root. Today’s puzzle blog is based on problems devised by Dr. Aziz Inan, whom we met previously in Backdating.

Each of the first five problems below has a single target number, whose properties are described. In case of the first question, several numbers may satisfy the condition, but only one of them is the target number. You have to try and find the five unique numbers based on the clues contained in the other questions as well, since the numbers all hang together as described in the conditions.

1. The first number is a four-digit number such that when you divide it by ten you get a palindromic three digit number. When you add this three-digit number to the first number, you get a palindromic four digit number. [Extra question: How many such numbers are there?]

2. The second number is a four-digit number which is a perfect square. The square root can be obtained by subtracting the first number above from this second number and adding 4. [How many pairs of numbers satisfy the conditions in question 1 and 2?]

3. The third number can be obtained by adding the second number to its square root and doubling the result.

4. The fourth number is obtained thus: Take the third number and swap its first two digits. Next, add thirteen to it and reverse the result. Now multiply the reversed number by 13 and introduce a zero at each end.

5. The fifth number is a four digit number with the following property: Split it down the middle and multiply together the two two-digit numbers so obtained. When you add this product to the first number above, you obtain the fifth number again. This is the second lowest number with this property.

6. The numbers above are related to a famous person and to the heavenly body in the solar system whose orbit is shown in the figure above. Can you figure out who and what they are? Why has this puzzle been posted today, and what is the significance of each of these numbers?

For our word challenge today, let’s construct sentences or paragraphs that conceal the names of famous people, as follows:

To say that artists and musicians are more loony than others is a myopic association. This is not research conducted at first hand, but incorporates second-hand elements such as hearsay.

The above sentences conceal a famous artist and a famous musician. Highlight the empty space below this sentence if you can’t find them.
Picasso, Handel

As usual, you can submit answers as comments here. Don’t be shy about responding to individual questions. Comments posted today will not be released for a day or two. We’ll award a copy of Martin Gardner’s book, “My Best Mathematical and Logic Puzzles,” to someone who comes up with an especially interesting answer to today’s puzzles or to someone who proposes a sufficiently intriguing puzzle for Lab readers to solve. Thanks again to Dr. Aziz Inan for today’s puzzle. Dr. Inan will receive a second Gardner book. (To submit a new puzzle, send an e-mail message with “NEW LAB PUZZLE” in the subject line to Please include the solution to the puzzle and indicate whether or not the puzzle is original.)

Comments that contain interesting or well-explained answers will be highlighted. Unless otherwise noted, highlighted comments will have correct answers.