A magic square is a square of numbers in which all rows, all columns and both diagonals add to the same number, or magic total. The simplest magic square using distinct numbers is this:

6 1 8 7 5 3 2 9 4

It’s easy to prove that the magic total of a 3×3 magic square must be three times the central number. Accordingly, if the central number is 37, the magic total must be 111. There are lots of ways to create a magic square with 37 at its heart, but this is my favourite:

43 | 01 | 67 61 | 37 | 13 07 | 73 | 31

The square is special because all the numbers are prime, or divisible by only themselves and 1 (though 1 itself is not usually defined as prime in modern mathematics). I like the 37-square even more now that I’ve discovered it can be found inside another all-prime magic square:

0619 = 0006[37] | 0097 = 00000010 | 1123 = [11][56] 1117 = [11][50] | 0613 = 0006[31] | 0109 = 0001[12] 0103 = 00000016 | 1129 = [11][62] | 0607 = 0006[25] Magic total = 1839

The square is shown in both base-10 and base-97. If the digit-sums of the base-97 square are calculated, this is the result (e.g., the digit-sum of 6[37]_{[b=97]} = 6 + 37 = 43):

43 | 01 | 67 61 | 37 | 13 07 | 73 | 31

This makes me wonder whether the 613-square might nest in another all-prime square, and so on, perhaps *ad infinitum* [Update: yes, the 613-square is a nestling]. There are certainly many nested all-prime squares. Here is square-631 in base-187:

661 = 003[100] | 379 = 00000025 | 853 = 004[105] 823 = 004[075] | 631 = 003[070] | 439 = 002[065] 409 = 002[035] | 883 = 004[135] | 601 = 003[040] Magic total = 1893 Digit-sums: 103 | 007 | 109 079 | 073 | 067 037 | 139 | 043 Magic total = 219

There are also all-prime magic squares that have two kinds of nestlings inside them: digit-sum magic squares and digit-product magic squares. The digit-product of a number is calculated by multiplying its digits (except 0): digit-product(37) = 3 x 7 = 21, digit-product(103) = 1 x 3 = 3, and so on. In base-331, this all-prime magic square yields both a digit-sum square and a digit-product square:

503 = 1[172] | 359 = 1[028] | 521 = 1[190] 479 = 1[148] | 461 = 1[130] | 443 = 1[112] 401 = 1[070] | 563 = 1[232] | 419 = 1[088] Magic total = 1383 Digit-sums: 173 | 029 | 191 149 | 131 | 113 071 | 233 | 089 Magic total = 393 Digit-products: 172 | 028 | 190 148 | 130 | 112 070 | 232 | 088 Magic total = 390

Here are two more twin-bearing all-prime magic squares:

Square-719 in base-451: 761 = 1[310] | 557 = 1[106] | 839 = 1[388] 797 = 1[346] | 719 = 1[268] | 641 = 1[190] 599 = 1[148] | 881 = 1[430] | 677 = 1[226] Magic total = 2157 Digit-sums: 311 | 107 | 389 347 | 269 | 191 149 | 431 | 227 Magic total = 807 Digit-products: 310 | 106 | 388 346 | 268 | 190 148 | 430 | 226 Magic total = 804

Square-853 in base-344:

883 = 2[195] | 709 = 2[021] | 967 = 2[279] 937 = 2[249] | 853 = 2[165] | 769 = 2[081] 739 = 2[051] | 997 = 2[309] | 823 = 2[135] Magic total = 2559 Digit-sums: 197 | 023 | 281 251 | 167 | 083 053 | 311 | 137 Magic total = 501 Digit-products: 390 | 042 | 558 498 | 330 | 162 102 | 618 | 270 Magic total = 990

Proviously Post-Posted (please peruse):