Posts Tagged ‘magic figures’

Finally, a Magic Magic 45-omino

January 14th, 2017

In the figure below, the numbers in each row, column, and main diagonal sum to 115:

Quite a long time ago, I came up with the idea of representing the lo shu (3×3 magic square) as a set of squares in a 9×9 grid, partitioned into nine 3×3 cells. The number of squares in each cell would correspond to a number in the lo shu. The most “magic” way to arrange the cells would seem to be to have 5 squares in the set in each row, column, and main diagonal. (This can be done because the lo shu’s magic sum of 15 can be divided among three rows or columns.) Although it doesn’t affect the “magicality” of a figure, I thought it aesthetically desirable for such a figure to be connected (i.e., a single polyomino) and hole-free. There are 12 hole-free magic 45-ominoes, if my code for discovering them is correct.

A figure with the same number of squares in each row, column, and main diagonal makes an ideal canvas for a sparse magic square. But with 45 numbers to place, and 20 constraints to meet, we start to push on the edge of what’s computationally feasible. The solver I wrote (which, I admit, might not have been very good) could not find a solution. Bryce Herdt manually tweaked the output of my solver to make a semimagic solution, that is, one where the rows and columns add to the magic sum, but the diagonals still didn’t work.

When I discovered that the Numberjack constraint engine could easily be used to code solvers for magic figures, I tried it on this problem, but got nowhere. The solver would run for an arbitrarily long period of time without spitting out any solutions. Recently I tried it again, and this time I got solutions. Paradoxically, what made the problem easier to solve was that I added more constraints. I manually placed the numbers 1 through 9 in the 3×3 cells that they correspond to. This seems to have made the search space small enough that the solver would not be able to spend an inordinate amount of time stuck in a barren zone.

Faux Shu Follies

June 18th, 2016

The Lo Shu, or 3×3 magic square, was discovered in China in antiquity. It is the only way, (up to symmetry) to place the numbers 1 through 9 in a 3×3 grid such that the numbers in each row, column, and main diagonal add up to the same number (or magic sum). This fact seems to be universally known among recreational mathematicians. So when I had the chance to meet a number of them this spring at the fabulous 12th Gathering for Gardner conference, I told them that I knew a different way to do it. When they pronounced me mad, or a liar, I showed them one of these:

Mathematics is full of counterexamples that result when the simple way of understanding a conjecture is not exactly what the conjecture literally says, so this kind of cheating is totes legit.

If the fact of the uniqueness of the Lo Shu is new to you, a quick proof might be in order. First, let’s enumerate all of the sets of three numbers between 1 and 9 that sum to 15: {1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8}, {2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}. There are eight sets, so we’ll need all of them to fill the eight lines in the magic square. The number 5 appears four times, the other odd numbers appear twice, and the even numbers appear three times. Therefore the center square, being part of four lines, must be 5, the corner squares, being part of three lines, must be the evens, and the side squares, being part of two lines, must be the other odds. Choose any corner, and put a 2 in it. That forces 8 into the opposite corner. Choose one the remaining corners, and put a 4 in it. After that, the rest of the numbers are forced. No matter what corners you choose, the result can be rotated or flipped to get the square formed by choosing any different pair of corners. Q. E. D.

Well, wait, you say, what if the magic sum isn’t 15? Quite right, 14 and 16 also both have eight sets of numbers between 1 and 9 that sum to them, so our proof is not done. I will leave it as an exercise to the reader to show that they cannot be used to form a 3×3 magic square.

And then, once the reader is satisfied, I’ll say: there is a way to place the numbers 1 through 9 in a 3×3 grid, with exactly one number in each cell, (you didn’t think I’d try the same shenanigans twice?) so that they occupy eight lines that each connect exactly three numbers that sum to 14. And having followed me this far, you are now enough of a recreational mathematician to be able to call me mad, or a liar. But you might want to have a look at this before you wager money on it:

This result was adapted from one discovered by Lee Sallows, which is described in his book, Geometric Magic Squares.

Well, clearly the problem here is that you’re allowing me to draw my own graphics. If you forced me to use physical number tiles as in the first image, I couldn’t get up to any fancy tricks. So if I told you that I could arrange those exact same nine number tiles in a block of three rows of three tiles each, and make it so that for every line that passes through the center of three tiles that form a connected group, the sum of those tiles is 14, I would have to be mad.