The magic sum of a 4 by 4 square

Question

A 4 by 4 magic square contains each of the numbers 1 to 16 exactly once, with every row, every column and both main diagonals adding to the same total. 1. What is that total? 2. Find the formula for the magic sum of an n by n square using 1 to n squared, and evaluate it for n = 5 and n = 10. 3. Give an example of a 4 by 4 magic square.

Accepted Answer

The magic sum is 34. In general it is n(n squared + 1)/2. The 4 by 4 case. The numbers 1 to 16 add to 16 × 17 / 2 = 136. Four rows share that total equally, so each row sums to 136 / 4 = 34, and so must each column and diagonal. The formula. The numbers 1 to n squared add to n squared (n squared + 1)/2. Dividing by the n rows gives the magic sum n(n squared + 1)/2: 15 for n = 3, 34 for n = 4, 65 for n = 5, and 505 for n = 10. An example. Dürer's square, from his 1514 engraving "Melencolia I": Every row, column and diagonal adds to 34, and so do the four corners, the central 2 by 2 block, each…

The magic sum of a 4 by 4 square

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