Another crack at 6×6 magic squares

Even-ordered magic squares are not difficult just because they are even, in my opinion. They are difficult to design because their order is composite. My experience has shown that by far the easiest to design are magic squares whose order is a prime number like 5, 7, 11, or 13. I have run into similar problems with 9×9, 15×15, as well as 6×6 and 8×8. The 6×6 seems to have the reputation for being the most difficult to make magic, although I have stumbled on one system that produced them, and wrote about it a few years ago, about how I applied that method to a spreadsheet. That method, however, led only to 64 possibilities.

Spreadsheets are a great way of checking your progress as you are building such squares, especially when you are trying to build a square using, say, a method you made up on your own, such as applying a Knight’s tour (which works OK with an order-8 square) to an order-6 square. This would require some facility with using spreadsheet formulae and other features which improve efficiency. Using your own method is very much based on trial and error, and you have to make a rule as to whether you will be wrapping the Knight’s moves (if you decide to use a Knight’s tour) to the opposite side of the board, or will you be keeping your moves within the board limits, changing direction of the “L’s” in your movements (this seems to lead to dead ends as you find you have no destinations left which follow an “L”, and consequently, squares which are not really magic). At any rate, the best squares follow some kind of rule which you need to stick to once you make it.

The best ones I have been able to make with a knights tour are: 1) when your L’s are all in the same “direction”‘ 2) when you sum up two squares. The problem is, all of the ones I have made with these methods so far either end up with weak magic (rows add up but not the columns) but the numbers 1-36 are all there; or all rows and columns make the magic number of 111, yet not all of the numbers are present and there are several duplicated (and even triplicated) numbers.

1 9 17 24 28 32 111
26 36 4 7 15 23 111
17 19 27 32 6 10 111
34 2 12 17 19 27 111
21 29 31 4 8 18 111
12 16 20 27 35 1 111
132 111 111 111 111 111 111 90

The above table shows the totals for the rows and columns for one attempt I made for a semi-magic square. Rows and columns work out to the correct total, but not the diagonals, as shown by the yellowed numbers. There are also duplicate entries, as well as missing entries. 1, 4, 12, 19, and 32 have duplicates, while there are three of 17 and of 27. Numbers missing are 3, 5, 11, 14, 22, 25, 30, and 33. That being said, the rows and columns add perfectly to 111, but not the diagonals. However, the average of the diagonals is the magic number 111 (this does not always work out). The sum of the missing numbers is 156, while the sum of the “excess” numbers (the sum of the numbers that occur twice plus double the sum of the numbers occurring thrice) is also 156 (could be a coincidence).

The above semi-magic square results from the sum of two squares where a knight’s tour is performed with the second square where the numbers 1 to 6 go in random order going down from top to botton. If I am too close to the bottom edge of a column, the knight’s tour wraps back to the top of the square. Beginning on the third column, I shift the next entry one extra square downward. The result is 6 of each number, each of these unique to its own row and column.

The first square are the multiples of 6 from 0 to 30 in random order going from left to right, also in a knight’s tour, wrapping from right to left and continuing. The third row is shifted by 1 to the right.

0 6 12 18 24 30
24 30 0 6 12 18
12 18 24 30 0 6
30 0 6 12 18 24
18 24 30 0 6 12
6 12 18 24 30 0
+
1 3 5 6 4 2
2 6 4 1 3 5
5 1 3 2 6 4
4 2 6 5 1 3
3 5 1 4 2 6
6 4 2 3 5 1

Note that the first square wasn’t really randomized.  When I tried to randomize it, the result was still semi-magic, similar to what was described. In the case I attempted, the average of the diagonals was not 111. The two are added, this time using actual matrix addition built into Excel. There is a “name box” above and at the far left of the application below the ribbon but above the spreadsheet itself. This is where you can give a cell range a name. I highlighted the first square with my mouse, and in the name box I gave a unique name like “m1x” (no quotes). The second was similarly selected and called “m2x”. I prefer letter-number-letter names so that the spreadsheet does not confuse it with a cell address (which it will). Then I selected a 6×6 range of empty cells on the spreadsheet and in the formula bar (not in a cell) above the spreadsheet (next to the name box), I entered =m1x+m2x, then I pressed CTRL+ENTER. The range of empty cells I selected is now full with the sum of the squares m1x and m2x, which is the first semi-magic square shown in this article.

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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 sci.bio.food-science.