Ponder This Challenge - February 2025 - Prime number magic square

Given a number $A$, we wish to construct an $N\times N$ square of digits (0,1,2,3,4,5,6,7,8,9) such that

1. The sum of each row, column, and diagonal is $A$.

2. The number appearing in each row (left-to-right), column (top-down), and diagonals (top left to bottom right and top right to bottom left) is an $N$-digit prime number (no leading zeros).

Note that the same digit can appear multiple times in the square, and the same prime number can appear multiple times in the square.

The cost of a square is computed in the following manner: We list all the prime numbers appearing in the rows, columns, and diagonals of the square. We remove any duplicates from the list. For each digit $d$, we count the total number of times it appears in the primes in the list. Each time incurs a cost: the first time the cost is 0, the second time it's 1, the third time it's 2, and so on. Note that the same digit in the square appears in more than one number and hence might be counted more than once; we count the occurrences of the digits in the prime numbers, not in the square itself.

For example, for $N=4$ we have the following square:

in which $A=22$ and which contains the unique primes 7717, 9391, 5557, 7591, 7537, 1597, and 1777. The square has a total of 5 occurrences of the digit 1, 2 of the digit 3, 6 of the digit 5, 11 of the digit 7, and 4 of the digit 9, resulting in a cost of 87 in total. This is the $4\times 4$ square with minimum cost. Similarly, the $4\times 4$ square of maximum cost is

which has a cost of 143.

Your goal: For $N=5$, find the squares of minimal cost and maximal cost (without limiting A to A=22). Provide your squares as written above, with the score of each square written above it.

A bonus "*" will be given for finding squares of minimal and maximal costs for $N=6$.

Solution

• Click here to view the solution

  February 2025 Solution:

  For the N=5 case, the minimal solution is 61, given by two boards, one of which is

  2 8 4 6 3
  8 9 0 5 1
  4 0 7 3 9
  6 5 3 2 7
  3 1 9 7 3
  

  And the maximal is 488 given by

  1 7 3 3 3
  4 1 8 1 3
  3 3 3 1 7
  2 3 2 9 1
  7 3 1 3 3
  

  For N=6 the minimum is 158, given by

  2 7 8 2 2 7
  7 1 5 1 5 9
  8 9 4 4 0 3
  2 1 4 9 9 3
  2 1 4 9 9 3
  7 9 3 3 3 3
  

  And the maximum is 1310, given by

  3 1 3 7 7 7
  3 7 3 3 3 9
  5 3 9 5 3 3
  9 7 3 3 3 3
  5 7 7 3 3 3
  3 3 3 7 9 3
  

  In both cases, the solutions require extensive search which can highly depend on various possible optimizations.

Solvers

• *Yi Jiang (2/2/2025 4:33 PM IDT)
• Lazar Ilic (2/2/2025 7:27 PM IDT)
• King Pig (2/2/2025 7:30 PM IDT)
• Gary M. Gerken (2/2/2025 9:45 PM IDT)
• *Ankit Aggarwal (2/2/2025 10:07 PM IDT)
• *Harald Bögeholz (2/2/2025 10:22 PM IDT)
• *Sanandan Swaminathan (2/2/2025 10:22 PM IDT)
• *Daniel Chong Jyh Tar (3/2/2025 1:32 AM IDT)
• Károly Csalogány (3/2/2025 4:44 AM IDT)
• *Bertram Felgenhauer (3/2/2025 5:37 AM IDT)
• *Alper Halbutogullari (3/2/2025 10:26 AM IDT)
• Alex Fleischer (3/2/2025 1:04 PM IDT)
• Sam Fischer (3/2/2025 5:01 PM IDT)
• Ralf Jonas (3/2/2025 10:00 PM IDT)
• Noam Zaks (3/2/2025 10:12 PM IDT)
• *Karl-Heinz Hofmann (4/2/2025 8:45 AM IDT)
• Vassili Chesterkine (4/2/2025 1:03 PM IDT)
• Lorenz Reichel (4/2/2025 3:26 PM IDT)
• Reiner Martin (4/2/2025 3:38 PM IDT)
• Robin Guilliou (4/2/2025 5:02 PM IDT)
• Dan Ismailescu (4/2/2025 7:08 PM IDT)
• *Asaf Zimmerman (4/2/2025 7:52 PM IDT)
• Shirish Chinchalkar (4/2/2025 8:52 PM IDT)
• *Daniel Bitin (5/2/2025 10:30 AM IDT)
• Juergen Koehl (5/2/2025 11:33 AM IDT)
• Jiri Spitalsky (6/2/2025 8:53 AM IDT)
• *Vladimir Volevich (6/2/2025 9:55 AM IDT)
• Mathias Schenker (6/2/2025 12:32 PM IDT)
• Fakih Karademir (7/2/2025 1:13 PM IDT)
• Shouky Dan & Tamir Ganor (7/2/2025 1:31 PM IDT)
• Stéphane Higueret (7/2/2025 6:50 PM IDT)
• *Rémi Pérenne (7/2/2025 9:11 PM IDT)
• Dominik Reichl (8/2/2025 12:37 AM IDT)
• Sanjay Rao (8/2/2025 9:42 AM IDT)
• Michael Liepelt (8/2/2025 9:57 AM IDT)
• Rudy Cortembert (8/2/2025 11:43 AM IDT)
• Alex Izvalov (8/2/2025 2:47 PM IDT)
• Joaquim Carrapa (8/2/2025 3:50 PM IDT)
• Dieter Beckerle (8/2/2025 5:12 PM IDT)
• Fabio Michele Negroni (9/2/2025 11:58 AM IDT)
• Todd Will (9/2/2025 5:03 PM IDT)
• *Florian Sikora (9/2/2025 10:20 PM IDT)
• Emmanuel Lazard (9/2/2025 10:22 PM IDT)
• Arthur Vause (10/2/2025 6:45 PM IDT)
• Hakan Summakoğlu (10/2/2025 6:57 PM IDT)
• Amos Guler (10/2/2025 8:01 PM IDT)
• José Eduardo Gaboardi de Carvalho (11/2/2025 2:07 AM IDT)
• John Spurgeon (11/2/2025 3:53 AM IDT)
• Abhinav Bana (11/2/2025 6:34 AM IDT)
• *Klaus Nagel (11/2/2025 7:24 PM IDT)
• *Hugo Pfoertner (11/2/2025 8:27 PM IDT)
• Merouane Benadda (12/2/2025 12:13 AM IDT)
• *Guangxi Liu (12/2/2025 3:49 AM IDT)
• Paul Lupascu (12/2/2025 12:12 PM IDT)
• Sri Mallikarjun J (12/2/2025 4:39 PM IDT)
• Loukas Sidiropoulos (12/2/2025 5:05 PM IDT)
• Franciraldo Cavalcante (12/2/2025 5:21 PM IDT)
• Stephen Ebert (12/2/2025 10:15 PM IDT)
• Zach Liu (13/2/2025 3:20 AM IDT)
• Mohit Garg (13/2/2025 4:17 AM IDT)
• Piyush Agrawal (13/2/2025 6:41 AM IDT)
• Alfred Reich (13/2/2025 5:03 PM IDT)
• Lavanya Kannan (13/2/2025 7:02 PM IDT)
• William Han (14/2/2025 2:03 AM IDT)
• Andrea Andenna (14/2/2025 5:48 PM IDT)
• Sumanth Nethi (14/2/2025 10:36 PM IDT)
• Björn Hendriks (15/2/2025 7:20 AM IDT)
• *Bert Dobbelaere (15/2/2025 11:58 AM IDT)
• Jason Kim (15/2/2025 7:56 PM IDT)
• Gyozo Nagy (15/2/2025 8:17 PM IDT)
• Jose Coelho (16/2/2025 12:03 AM IDT)
• John Tromp (16/2/2025 4:57 PM IDT)
• Joydeep Pal (16/2/2025 8:25 PM IDT)
• Guglielmo Sanchini (17/2/2025 3:25 PM IDT)
• *Evert van Dijken (17/2/2025 5:08 PM IDT)
• *Ward Beullens (17/2/2025 6:54 PM IDT)
• Clive Tong (17/2/2025 9:50 PM IDT)
• *Matt Rearick (18/2/2025 8:42 AM IDT)
• *Eran Vered (18/2/2025 6:51 PM IDT)
• Joydeep Paul (19/2/2025 5:43 AM IDT)
• Evan Semet (19/2/2025 10:22 AM IDT)
• *Andreas Stiller (20/2/2025 12:13 PM IDT)
• Nadir S. (20/2/2025 9:52 PM IDT)
• Kang Jin Cho (21/2/2025 12:57 AM IDT)
• *Chris Shannon (22/2/2025 5:32 AM IDT)
• Blaine Hill (22/2/2025 9:11 AM IDT)
• Peter Moser (22/2/2025 3:58 PM IDT)
• Michael Vahle (22/2/2025 4:47 PM IDT)
• Dan Dima (22/2/2025 5:28 PM IDT)
• Balkrishna Ketkar (23/2/2025 8:29 AM IDT)
• *Latchezar Christov (23/2/2025 4:27 PM IDT)
• Radu-Alexandru Todor (23/2/2025 7:15 PM IDT)
• Lorenzo Gianferrari Pini (23/2/2025 7:47 PM IDT)
• Ido Meisner (23/2/2025 11:39 PM IDT)
• Sachal Mahajan (24/2/2025 12:36 AM IDT)
• Nyles Heise (24/2/2025 3:11 PM IDT)
• Thomas Egense (24/2/2025 6:32 PM IDT)
• *David F.H. Dunkley (25/2/2025 11:27 AM IDT)
• Michael J. Swart (25/2/2025 7:27 PM IDT)
• Micheal Malki (25/2/2025 8:16 PM IDT)
• *David Greer (27/2/2025 1:10 AM IDT)
• Mark Mammel (27/2/2025 4:14 PM IDT)
• Paul Revenant (28/2/2025 11:42 AM IDT)
• Christopher Bender (28/2/2025 8:10 PM IDT)
• Stefan Wirth (28/2/2025 8:53 PM IDT)
• Tim Walters (28/2/2025 10:23 PM IDT)
• Wenjie Yang (1/3/2025 4:32 AM IDT)
• Dezhi Zhao (1/3/2025 6:09 AM IDT)
• Marco Bellocchi (1/3/2025 1:35 PM IDT)
• *Motty Porat (1/3/2025 9:21 PM IDT)
• *Oscar Volpatti (2/3/2025 5:16 AM IDT)
• Avi Levin (2/3/2025 5:36 PM IDT)
• Karl Mahlburg (3/3/2025 2:12 AM IDT)
• *Li Li (4/3/2025 4:48 PM IDT)
• Erik Wünstel (4/3/2025 10:46 PM IDT)
• Matt Cristina (5/3/2025 3:36 AM IDT)
• Maths ForFun (5/3/2025 6:25 AM IDT)