This week the #puzzle is: Happy (Almost) New Year from The Fiddler! #MagicSquare #primes

A magic square is a square array of distinct natural numbers, where each row, each column, and both long diagonals sum to the same “magic number.” ,,,

A prime magic square is a magic square consisting of only prime numbers. Is it possible to construct a 4-by-4 prime magic square with a magic number of 2026? If so, give an example; if not, why not?

And for extra credit:

Find all values of N for which it is possible to construct an N-by-N prime magic square with a magic number of 2026. (Remember, the numbers in a magic square must all be distinct!)

Happy (Almost) New Year from The Fiddler!

Highlight to reveal (possibly incorrect) solution:

Examples of prime magic squares OEIS sequence Magic Squares (4 x 4), Analytic Solution, Pan Magic Squares Prime Magic Squares (4 x 4), Simple Magic Squares (4 x 4) Magic Square Generator

Program

Example of a magic square with magic sum 2026:
Example of a prime magic square with magic sum 240:

506	509	512	499		47	7	79	107
511	500	505	510		37	101	31	71
501	514	507	504		73	19	89	59
508	503	502	513		83	113	41	3

Based on the methods found on entertainmentmathematics.nl, I code my own “find a prime pan magic square, 4×4”. For a magic sum of 240, it correctly finds a solution (see example above). For 2026, it does not. There is no magic square of this kind. However, here’s a few others:

43	131	877	977		193	181	809	857
857	997	23	151		653	1013	37	337
137	37	971	883		211	163	827	839
991	863	157	17		983	683	367	7

And for extra credit:

Thoughts:

• N odd won’t work, as the sum of (odd) primes has to be even.
• N = 2 won’t work, as no magic square of any kind exists for N = 2.
• We’ve already shown that N = 4 doesn’t work.
• There are 305 primes below 2026. N = 18 and above won’t work, because 18² = 324.
• As for the rest…?