Magic grids of integers

0:00Here is a grid of sixteen numbers, and a game you play on it. Circle any number you like. Then strike out its whole row, and its whole column. Now circle any number that has not been struck out, and strike out that one's row and column too. Keep going until there is nothing left to circle. Then multiply together everything you circled. That is the entire game. No scoring, no opponent, nothing to remember.

0:31The interesting part is what happens when two people play it separately, without watching each other. They get the same answer. Every single time. So let us play it once, slowly, and watch what each circle costs. I will start at minus six, up in the top row. That is a completely arbitrary choice, and it is meant to be. Its row goes. Its column goes. Seven other cells died along with that one circle.

1:02Nine cells are still standing, and from those I take fourteen. Another row struck, another column struck, and now only four cells survive. I circle twenty, which kills the row and the column it sits in. And that leaves exactly one cell alive, so the last pick is not really a pick at all. Eighteen. Four circles, and the board is finished. Four numbers, then. Minus six, fourteen, twenty and eighteen.

1:35Minus six times fourteen is minus eighty four. Times twenty is minus one thousand six hundred and eighty. And times eighteen gives minus thirty thousand two hundred and forty. Write that number down somewhere and keep hold of it, because everything that follows is about it. It is big, it is awkward, and it is extremely specific. There is nothing round about it, nothing memorable, and nothing that looks as though it was chosen to come out neatly.

2:08Which is exactly what makes the next part strange. Play again, and this time choose differently on purpose. Start at eight, over in the corner, about as far from minus six as this grid will let me get. Then twenty one. Then minus six from the third row. Then thirty. Not one of those four cells is a cell I circled the first time. The two plays have nothing in common whatsoever.

2:37Eight times twenty one is one hundred and sixty eight. Times minus six is minus one thousand and eight. And times thirty is minus thirty thousand two hundred and forty. The same awkward number, out of four completely different factors. You could stop here and file it away as a curiosity. Play a third time, a fourth, a tenth, and it keeps on happening. There are twenty four different ways to play this grid, and every one of them lands in the same place.

3:13But no amount of that is an explanation. Evidence tells you something is true. It does not tell you why, and it cannot tell you when it would stop being true. Suppose I changed a single cell. Would it still work? Playing more games will never answer that question. So let us go and find the reason instead. Start by noticing what the striking out actually does. Every time you circle something, you kill one row and one column.

3:46So you can never circle twice in the same row, and never twice in the same column. Four circles, four rows, four columns. One from each. Always, whatever you choose, and whatever order you happen to choose it in. That is the only condition there is, and it is a loose one. Which cell you take inside a row is entirely up to you. Sixteen cells drop to nine, then to four, then to one.

4:16The striking is just a way of forcing that condition on you without ever announcing it. Now look at the rows themselves, laid one above another. Take the third row. Twelve, minus six, eighteen, minus nine. Every single cell in it is three times something. Three fours, three minus twos, three sixes, three minus threes. And the second row is those same somethings, every one of them multiplied by minus seven.

4:48The top row uses two. The bottom row uses five. Four rows, four multipliers, and underneath all of them one shared list: four, minus two, six, minus three. Nothing on the board tells you those two lists are there. You have to go hunting for them. Put each list where it belongs, and the grid stops being mysterious. The four multipliers down the left hand side. The shared list across the top.

5:20Every cell is now simply the number on its left, times the number above it. It is a multiplication table with its headers rubbed out. And that changes what a circled cell is. It is not really a number any more. It is a pair. Minus six is two times minus three. Fourteen is minus seven times minus two. So when you circled four cells, what you were actually picking up was eight numbers.

5:51And now the whole thing falls out. One cell from each row means you used each of the four side numbers exactly once. One cell from each column means you used each of the four top numbers exactly once. So whatever you circled, you collected all eight of them. Every play does. The only difference between my two games is how those eight got paired up. And multiplication cannot see a pairing.

6:22Swap two factors around and the product does not move; that is the first law you need. Bracket them differently and it does not move either, and that second law is what lets the pairs be broken up and reassembled at all. The same picture settles the sign, before you multiply anything. Look at the eight hidden numbers and count the minus signs among them. One down the side, at minus seven. Two along the top, at minus two and at minus three.

6:56Three altogether. Three is odd, so the answer comes out negative. Now notice what that is not about. My first play circled exactly one negative cell, and so did my second. But there are plays that circle three negative cells, and they land on the very same negative answer. The minus signs you can see are not the ones doing the work. Which means you can build one of these yourself.

7:27Choose any four numbers for the side and any four for the top. Fill in each cell with the product of its two headers, then rub the headers out. That is a grid, and it will behave exactly the way this one does. The numbers are entirely your choice. The behaviour is not. There is one loose end worth knowing about before you go. Your two lists are not the only pair that fits the grid you built. Make every number in both lists negative, and every cell comes out identical.

8:02But the two lists get multiplied together in the end, so those minus signs cancel each other and the answer never notices. So what makes a grid like this one special? Is it the numbers, or is it the arrangement? It is worth being careful here, because the honest answer is both, and the two do different jobs. The numbers have to be chosen so that a hidden pair of lists exists at all.

8:32Change a single cell. Turn that eight into a nine. The grid instantly stops being a multiplication table in disguise, and different plays start giving different answers. But once the numbers do sit in that shape, the arrangement takes over completely, and your choices stop mattering. Sixteen cells. Twenty four ways to play. Twenty four calculations that look nothing whatsoever like each other. And one answer, settled before anybody drew a single circle.