Ponder This Challenge - October 2023 - Non-palindromic numbers with palindromic squares

This riddle was suggested by our previous puzzlemaster, Oded Margalit. Thanks Oded!

We are entering the 26th year of Ponder This and this is the 307th riddle. Both 26 and 307 are not palindromes, but their squares 676 and 94249 are.

Your goal: Find a number $n$ that is not a palindrome, such that $n^2$ is a palindrome and $n \equiv 845 (\text{mod } 1201)$

A Bonus "*" will be given for finding that number, such that in addition to the previous constraints, we also have $n \equiv 2256 (\text{mod } 3565)$

An open question that we are pondering about: Does there exist an $n$ that is not a palindrome but $n^2$ is, such that all the digits of $n$ (in base 10) are odd?

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  October 2023 Solution:

  When engineering the problem we did not consider negative numbers and forgot to explicitly forbid them, making solutions such as -12866669 possible. We accepted those solutions since they are nice tricks (which doesn't help with the bonus part).

  Here is a brief description of one method of solving the problem, given by Karl Mahlburg:
  Regular version:
  n = 100000000000000100009999999999999990000000000000010001000000000000001
  * version:
  n = 10^{2*s3} + 1 + 10^{s3+s2} + 10^{s3-s2} + 10^{s3+s1} + 10^{s3-s1} - 10^s3,
  where [s1, s2, s3] = [415, 3819, 4234]
  (A 8469-digit number!)

  The regular version is a similar construction as the * version, just with S = [15, 19, 34]. The next smallest case that satisfies the problem is [24, 43, 67].

  I found this construction in some of the links from the OEIS page for palindromic squares

  Most relevantly, the Asymmetric (A) family from here

  This shows that any S = [s1, s2, s1 + s2] will give an asymmetric integer w/ palindromic square, and a brief search starts returning examples that satisfy the congruence mod 1201.

  The congruence mod 3565 = 5*23*31 is not much harder, since mod 5 is satisfied automatically, and 23 and 31 are small primes, so Chinese Remainder Theorem works out pretty easily (note my * version is a lift of the regular solution, since 10 has order 200 mod 1201).

Solvers

• *Lazar Ilic (1/10/2023 11:47 AM IDT)
• Paul Revenant (1/10/2023 8:21 PM IDT)
• *Ariel Landau (1/10/2023 9:32 PM IDT)
• Gary M. Gerken (2/10/2023 3:24 AM IDT)
• *Bertram Felgenhauer (2/10/2023 6:35 AM IDT)
• Nicolas Paris (2/10/2023 8:15 AM IDT)
• Giorgos Kalogeropoulos (2/10/2023 1:18 PM IDT)
• Richard Gosiorovsky (2/10/2023 1:32 PM IDT)
• *Tim Walters (2/10/2023 5:12 PM IDT)
• *Paul Revenant (2/10/2023 5:50 PM IDT)
• Hakan Summakoğlu (2/10/2023 11:01 PM IDT)
• *Bert Dobbelaere (2/10/2023 11:18 PM IDT)
• Michał Franczak (3/10/2023 1:17 AM IDT)
• *Sanandan Swaminathan (3/10/2023 4:17 AM IDT)
• Kipp Johnson (3/10/2023 5:26 AM IDT)
• *Dan Ismailescu (3/10/2023 5:28 AM IDT)
• Lorenzo Gianferrari Pini (3/10/2023 11:03 AM IDT)
• *Harald Bögeholz (3/10/2023 1:00 PM IDT)
• *Yan-Wu He (3/10/2023 3:21 PM IDT)
• Michael Liepelt (3/10/2023 3:58 PM IDT)
• Andreas Knüpfer (3/10/2023 4:41 PM IDT)
• Kipp Johnson (3/10/2023 5:45 PM IDT)
• Soham Sud (3/10/2023 5:54 PM IDT)
• Stéphane Higueret (3/10/2023 6:45 PM IDT)
• Daniel Chong Jyh Tar (4/10/2023 4:31 PM IDT)
• John Kelly (4/10/2023 5:14 PM IDT)
• Daniel Stadtmauer (4/10/2023 10:52 PM IDT)
• Paul Lupascu (5/10/2023 6:58 AM IDT)
• Lorenzo Gianferrari Pini (5/10/2023 12:51 PM IDT)
• *Alper Halbutogullari (5/10/2023 2:41 PM IDT)
• *Dieter Beckerle (5/10/2023 3:27 PM IDT)
• *Latchezar Christov (5/10/2023 5:28 PM IDT)
• Alex Fleischer (5/10/2023 7:06 PM IDT)
• Kang Jin Cho (5/10/2023 8:22 PM IDT)
• *Karl Mahlburg (6/10/2023 2:43 AM IDT)
• Eytan Rosenstock (6/10/2023 2:47 AM IDT)
• *Franciraldo Cavalcante (6/10/2023 7:33 AM IDT)
• Reda Kebbaj (6/10/2023 8:17 AM IDT)
• Quentin Higueret (6/10/2023 8:34 AM IDT)
• *Guglielmo Sanchini1 (6/10/2023 10:13 AM IDT)
• Cameron Ritson (6/10/2023 1:45 PM IDT)
• *Vladimir Volevich (6/10/2023 4:04 PM IDT)
• *Marco Bellocchi (7/10/2023 1:59 AM IDT)
• Andreas Höglund (7/10/2023 2:54 AM IDT)
• *Andrew Gauld (7/10/2023 9:43 AM IDT)
• Fabio Michele Negroni (7/10/2023 1:21 PM IDT)
• Peyo (7/10/2023 8:03 PM IDT)
• Adrian Neacsu (7/10/2023 9:14 PM IDT)
• *Aaron Kim (7/10/2023 9:49 PM IDT)
• Dan Dima (8/10/2023 3:01 PM IDT)
• Clive Tong (8/10/2023 3:45 PM IDT)
• *Dan Dima (8/10/2023 4:32 PM IDT)
• *Jeff Harris (8/10/2023 7:56 PM IDT)
• Dominik Reichl (8/10/2023 8:37 PM IDT)
• *David F.H. Dunkley (9/10/2023 12:40 AM IDT)
• *Thomas Egense (9/10/2023 9:54 AM IDT)
• *Bertram Felgenhauer (9/10/2023 12:56 PM IDT)
• Matt Cristina (9/10/2023 7:05 PM IDT)
• *Maksym Voznyy (10/10/2023 12:56 AM IDT)
• *Dillon Davis (10/10/2023 1:39 AM IDT)
• *Alfred Reich (10/10/2023 9:34 AM IDT)
• *Martin Thorne (10/10/2023 11:12 PM IDT)
• Daniel Bitin (11/10/2023 12:01 PM IDT)
• Evert van Dijken (11/10/2023 8:42 PM IDT)
• *Govind Jujare (12/10/2023 1:46 AM IDT)
• Abhinav Bana (12/10/2023 8:14 AM IDT)
• *Lyes Belhoul (12/10/2023 8:31 PM IDT)
• *Hugo Pfoertner (12/10/2023 10:54 PM IDT)
• *Wolfgang Kais (15/10/2023 4:04 AM IDT)
• Florian Sikora (15/10/2023 2:55 PM IDT)
• *Alain Michiels (15/10/2023 4:03 PM IDT)
• Amos Guler (16/10/2023 8:01 PM IDT)
• Sri Mallikarjun J (17/10/2023 5:27 PM IDT)
• *Lorenz Reichel (17/10/2023 11:51 PM IDT)
• *Rong Fan (18/10/2023 12:09 AM IDT)
• *Julien Vincent Pradier (18/10/2023 12:16 AM IDT)
• *Kabir Menghrajani (19/10/2023 6:21 AM IDT)
• *Arthur Vause (19/10/2023 2:26 PM IDT)
• Evan Semet (19/10/2023 5:20 PM IDT)
• *David Greer (19/10/2023 6:29 PM IDT)
• John Tromp (20/10/2023 2:09 PM IDT)
• Guy Daniel Hadas (20/10/2023 6:16 PM IDT)
• Blaine Hill (22/10/2023 11:38 PM IDT)
• Isaac Greenleaf (23/10/2023 11:36 PM IDT)
• *Nyles Heise (24/10/2023 5:17 AM IDT)
• Armin Krauss (25/10/2023 12:12 PM IDT)
• Mark Beyleveld (25/10/2023 11:07 PM IDT)
• *Hans-Peter Schrei (26/10/2023 8:26 PM IDT)
• Maya Farber Brodsky (27/10/2023 1:20 PM IDT)
• Samuel Grondahl (27/10/2023 5:45 PM IDT)
• *Chris Shannon (28/10/2023 2:43 AM IDT)
• *Thomas McClave (29/10/2023 12:01 AM IDT)
• Christoph Baumgarten (30/10/2023 11:23 AM IDT)
• Victor Rasmussen (31/10/2023 1:41 AM IDT)
• *Oscar Volpatti (1/11/2023 4:13 AM IDT)
• Daniel Copeland (1/11/2023 6:48 AM IDT)
• *Radu-Alexandru Todor (2/11/2023 4:00 AM IDT)
• Shouky Dan & Tamir Ganor (2/11/2023 4:03 PM IDT)