Recently, during one of my less exciting meetings, I started idly doodling on a notepad. For some reason, I drew a small 3x3 square, divided it into 9 subsquares, and started filling in numbers so all the sums in each row, column, and diagonal would match, no doubt drawing on vague memories from doing such puzzles in my childhood. This is the classic “magic square” puzzle, where you try to fill in an NxN square with numbers from 1 to N^2 and produce such matching sums.
The 3x3 magic square is relatively trivial to solve, but is interesting in that it’s the only size square (other than the unsolvable 2x2 case and the silly 1x1 case) where the “magic” solution is unique. If you create a 3x3 square with 4 9 2 in row 1, 3 5 7 in row 2, and 8 1 6 in row 3, that is literally the only possible solution: any other solution you come up with will be some combination of reflections and rotations of that one. But surprisingly, the number of solutions climbs very rapidly as the square size grows: there are a whopping 880 4x4 squares, over 275 million 5x5s, and over 10^19 6x6s. Apparently the full set of magic squares of arbitrary size has not been characterized in detail by mathematicians, though there are procedures known for generating arbitrarily large examples.
There are a number of ways you can transform a magic square and preserve its “magic” property, of all the row, column, and diagonal sums matching. You can probably figure out some of these off the top of your head. The most obvious one is to add, subtract, or multiply all numbers by a constant, though that does result in violating the rule of using the numbers 1 thru n^2. A less obvious one is to choose two cells at opposite points on a diagonal, and interchange their rows and columns— after playing with a few examples, it’s not hard to see why this works. There are various other related row/column interchange methods; the Wikipedia page describes numerous variations, as well as a generalization.
One of the most amusing things I discovered about these squares is that the “magic” part isn’t just a cute name: for thousands of years, people have attributed mystical properties to these squares, especially the unique 3x3 one. For example, in ancient China, this was known as the “Lo Shu” square, said to have first been discovered on the back of a magical turtle that emerged from the Luo river during a large flood. The 8 outer cells of the square are associated with the 8 trigrams used in the I Ching, and Feng Shui practitioners associate each of the squares with one of the 5 classical Chinese elements: Earth, Wood, Water, Metal, and Fire. Feng Shui, as you might recall, is the ancient Chinese art of properly organizing and arranging your house so the elements are in harmony. Since there are more squares than elements, a few are repeated: 4, 3, and 8 are connected to Wood, 2 and 5 to Earth, 7 and 6 to Metal, 9 to Fire, and 1 to Water.
Here’s how you use the magic square in Feng Shui: lay it out over a floorpan of your house, such that the number 1 corresponds to your front door. It’s a bit confusing how they decide to scale the square, but I guess you’re suppose to try to fit it to a best approximation of just overlapping your house. Unless your house is perfectly square, parts of some of the sub squares will be outside your walls, or in little-used areas like storage closets. Then for each square, decide if it’s well “energized” in your home— if not, you may need to compensate by putting more of that element in your house. As one Feng Shui site states, if square 6, which corresponds to the Metal element, needs enhancing, “Wearing gold is recommended, as well as hanging a gold-toned metal windchime.” I’m not quite sure what makes gold a better representative of “metal” than, for example, tin, but such ignorance probably explains why my karma is so poor these days.
Actually, what always amuses me about these New Age uses of “elements” is that they ignore the last 200 years of chemistry, where we have discovered the true elements, as you know them from the periodic table. But this is fixable: in fact, scientists currently know of 118 elements, which falls very close to the square number of 121, equal to 11x11. So for a modern, accurate Feng Shui, we should use an 11x11 magic square for this divination. The fact that enormous numbers of such squares exist might add some confusion, but since we apparently believe in stuff like the I Ching anyway if we’re using this method, just cast some yarrow sticks to let the universe determine which construction method it wants you to use to build your square. Then you can make each square correspond to the atomic number of an actual element in the Periodic Table, starting over at 119 for a handful of duplicates. (Actually a much better ratio of duplicates than the 3x3 method in any case.)
Now you can lay this 11x11 square over your house’s floor plan, and identify which elements to adjust in your house to truly enable it to vibrate in sync with all the energies of the world. Perhaps your kitchen needs more of element 11, Sodium— just add more salt to your food. Or maybe your child’s room is overlapping the square of element 82, Lead— better check it for old paint. If square 86 falls in your house, better call a Radon inspector ASAP. I’m not sure quite what to do if you detect a deficiency of square 117, Tennessine, though— given that the longest-lived samples have lasted a few hundred milliseconds, you may need to get some extra help from the spirits on that topic, or build a nuclear reactor. But you can’t argue with the math.
And this has been your math mutation for today.