Using your Head is Permitted

July 2013 riddle

This month's riddle comes from an idea by Graham Farr. More credits and references for it will be given on the solution page.

We begin with three definitions. The first two are for familiar mathematical objects, the third is new.

We call a number a palindrome if its digits are the same when read forward-to-backward as when read back-to-front. For example, "131" and "2552" are both palindromes, but "253" is not. Also, "10" is not: you can't add leading zeroes to make a palindrome.

We call a number a square if it equals another number multiplied by itself, so "9" and "16" are both squares, but "35" is not. The number that is to be multiplied by itself to generate the original number is called its root, so "3" is the root of "9" and "4" is the root of "16". (In the present context, "number" refers to the naturals, so "square root of 35" is not a number.)

We call a number a PSP if it satisfies the following 3 conditions:

  1. It is a palindrome.
  2. It is a square.
  3. Its root is a palindrome.

This month's question: as a function of N, how many PSP numbers are there, that are at most N digits in length?

Provide your answer as a closed-form formula. (As usual, with proof of correctness.)

List of solvers:

Joseph DeVincentis (2 July 21:55)
Lewei Weng (3 July 15:42)
Albert Su (3 July 17:15)
Oded Margalit (7 July 01:25)
Andreas Razen (10 July 05:32)
Dan Dima (11 July 22:01)
Rohan Joshi (13 July 07:40)
Graham Hemsley (14 July 18:03)
Daniel Bitin (15 July 00:56)
Jan Fricke (16 July 01:29)
Lorenz Reichel (18 July 23:12)
Yuanming Yu (19 July 02:15)
Andreas Stiller (28 July 11:02)
Radu-Alexandru Todor (31 July 19:35)
Harald Bögeholz (1 August 03:40)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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