My first 6×6 magic square

There are two kinds of even-ordered magic squares. Singly-even and doubly-even. Doubly-even magic squares have its dimensions divisible by both 2 and 4. Thus, they have dimensions that are multiples of 4. Singly even magic squares are only divisible by 2 and not 4. They include 2, 6, 10, 14, 18, 22, and so on.

For my first ever even ordered magic squares, I for some reason chose the most difficult squares to construct, and these are squares of singly even order. In fact, I have to be so careful to watch out for all possible things that can go wrong, that I have to set up a spreadsheet to verify the details for me that can be non-trivial to spot. Such as:

  • duplicate numbers, which can take a while to spot
  • missing numbers, which go hand-in-hand with the first problem (the numbers in this square should be from 1 to 36, with no skips or repeats)
  • duplicate lines in one of the preliminary squares used in the sum
  • sums not adding to 111, the magic number for this kind of square

I can do odd-ordered squares in my sleep, but the construction of singly-even ordered squares are not as straightforward, since they appear to have random and non-random elements in them. In fact, the best constructions I have heard of (namely the method devised by Philippe de la Hire (1640-1719)) makes most of the square very deterministic, and much less random than we saw for 5×5 and 7×7.

Not all of the “random” elements in 6×6 squares work, and it looks as if one has to just try out different combinations. Following instructions may always lead to squares with the magic number in all of the required rows, columns and diagonals, but it still may suffer from the pitfalls mentioned above.

It took me several weeks of trial and error before I could make one that I could share. For the first while I wasn’t looking for duplicate lines in the preliminary squares, but after that plus a few other modifications to the method, pure magic squares were being produced.

I will leave the construction of the magic square in the next blog article, but here is a square I made using these methods:

1 5 33 34 32 6
25 8 10 27 11 30
18 20 15 16 23 19
24 17 21 22 14 13
12 26 28 9 29 7
31 35 4 3 2 36

To be continued …

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I am Paul King, a math and science teacher. I help maintain the MIT FAQ Archive along with Nick Bolach. I am also the maintainer of the FAQ for