Almost Magic

April 2022 : Puzzle

For the purposes of this puzzle, define a magic square to be a 3-by-3 grid of positive integers for which the rows, columns, and two diagonals all have the same sum. An example is shown top left; the 8 sums all equal 39.

An almost magic square is, well, almost a magic square. It differs from a magic square in that the 8 sums may differ from each other by at most 1. An example is shown top right; the sums are all 138 or 1391.

For this puzzle, place distinct positive integers into the empty grid above such that each of four bold-outlined 3-by-3 regions is an almost magic square. Your goal is to do so in a way that minimizes the overall sum of the integers you use.

(Cutoff updated April 4th.) If you find an entry with a sum lower than 1111, send it in! To submit an entry, submit a comma-separated list of the 28 integers in your diagram in top-to-bottom order. For the example above, that list would be “50, 72, 16, 12, 46, 80, 75, 53, 77, 76, 20, 43, 69, 96, 1, 58, 114, 85, 64, 59, 95, 40, 39, 88, 137, 111, 90, 62”.

[1] Not that it matters for this puzzle, but a magic square is, technically, also an almost magic square.