Integer grids, and why the total comes out the same every time

0:00Here is a square of nine cells, and the cell in the middle of it is empty. Not zero. Empty. Nothing is written there, and nothing is meant to be. Around the outside sit eight numbers. Four, minus one and minus three along the top. Minus three and one down the sides. Minus one, minus one and two along the bottom. Add the top row. Four, take away one, take away three, and you land on zero.

0:31Now add the bottom row. Minus one, minus one, two. Zero as well. Two rows, two zeros. That could be luck. It is not, and the reason is worth the next ten minutes. Try the left column, reading downwards. Four, minus three, minus one. Zero again. And the right column. Minus three, one, two. Zero. Four totals, four zeros, and that shared number has a name. It is the border sum.

1:04Now a second square, with completely different numbers in it. Five, minus three, minus five along the top. Zero and minus five down the sides. Minus eight, minus two, seven along the bottom. Its four totals are minus three, minus three, minus three and minus three. Different numbers, a different answer, and exactly the same strange agreement. Before going any further, the one mistake that ruins every puzzle in this family.

1:39The border sum is not the total of all eight numbers around the edge. On the first square those eight come to minus two. Its border sum was zero. The reason is the four corners. A corner sits in a row and in a column at the same time, so walking round the outside collects it twice over. A border sum is one row, or one column, taken on its own.

2:06And the empty middle belongs to no row and no column that counts, which is precisely why it is drawn empty. Now turn the puzzle around. Instead of being handed the square and asked for its border sum, you are handed the border sum and most of the square is missing. Six in the top left, eight beside it, minus five at the middle right, minus two at the bottom centre. Everything else is blank, and the border sum has to be minus two.

2:40Start where you have the most. The top row already holds six and eight, which make fourteen, so the last entry there must be minus sixteen. That fills a corner, and filling a corner is what unlocks a column. The right column now holds minus sixteen and minus five, which make minus twenty one, so the bottom right must be nineteen. Two more steps of the same kind and the square is finished. Every blank was forced by a corner that had just been filled.

3:16But that is not what happens with every square. Here is another. Minus ten in the top left, nine in the bottom left, minus five at the middle right, and the border sum has to be four. The left column already holds two of its three entries, so the middle left is forced to be five. So far, the same as before. And then it stops. Choose anything you like for the top right corner, and the rest of the square arranges itself around your choice. There are dozens of answers.

3:52The difference is not the numbers. It is how many entries you were handed at the start. Eight cells to fill and four conditions to meet means four entries is what it takes. The square that was forced had four. This one has three. Second puzzle, and this one is a game. Here is a square of sixteen numbers, four rows across and four columns down. Circle any number you like. Then rule out the whole row it sits in, and the whole column it sits in, and those are gone.

4:28Circle any number that survived. Rule out its row and its column as well. Keep going until there is nothing left to circle. You will have circled four numbers, and the game is to add them up. Circle minus one. Then nine. Then minus seven. Then minus two. They add to minus one. Now play it again, and this time try to do better. Start with nine, the largest number on the board. Out goes its row, and out goes its column.

5:02Take minus five next. Then two, then minus seven, because after two turns there is not much choice left. Nine, minus five, two, minus seven. Minus one. Try a third route, and a fourth. Chase the big numbers, or chase the small ones, or close your eyes. Minus one, every time. There are twenty four different ways to play this board, and all twenty four of them come to minus one.

5:33So the total was never yours to choose. It was settled before you sat down. Look at the top row of the square: three, four, zero, nine. Now look at the second row. Minus two, minus one, minus five, four. That is the top row with five taken off every single entry. The third row is the top row with two taken off. The last row is the top row with ten taken off.

6:03So every number in that square is a column number plus a row number. Three, four, zero and nine going across. Nothing, minus five, minus two and minus ten going down. It is an addition table, and it was one the whole time. Now watch what the rules of the game actually do to you. The moment you circle a number, its entire row leaves the board. So you can never circle twice from the same row.

6:35The same is true of columns. Four circles, four different rows, four different columns. Which means the four numbers you circled are a row number plus a column number, four times over, and between them they use up every row once and every column once. So the total is all four row numbers plus all four column numbers, whichever four cells you happened to take. Three and four and zero and nine is sixteen. Nothing and minus five and minus two and minus ten is minus seventeen. Sixteen with minus seventeen is minus one.

7:19Here is another board of the same kind, and there is something wrong with it. Seven, ten, thirteen, sixteen along the top, and each row below is the row above it with nine taken off. Except in one place. The second entry along the bottom row ought to be minus seventeen. What is sitting there is minus seven. One digit. And with that digit missing, the puzzle stops working. Eighteen of the twenty four ways to play still come to minus eight. The six routes that pass through that one cell come to plus two instead.

7:59So the game is not magic, and it is not sturdy either. It works because somebody built it to work, and a single wrong entry is enough to break it. One more thing that is decided in advance. Here is a line of tokens. Three green, then two red. Three green, then two red. On and on it goes. Each green token is worth one, and each red token takes one away.

8:28There are a hundred tokens in the line, and the question is what the whole line comes to. You could count them one by one. Better to find the piece that repeats. Three green and two red is five tokens, worth three take away two, which is one. A hundred tokens is twenty of those pieces, so the line is worth twenty. Sixty green and forty red, and you never had to count past five.

9:00Which is the idea underneath all of it. What looked like a property of your choices turned out to be a property of how the thing was built. So build one. Choose four numbers for the columns and four for the rows, add them into a square, and hand it to somebody. Whatever they circle, they will land on the total of your eight numbers, and you knew that before they picked anything.

9:26For the other puzzle, choose a border sum first and work inwards, and remember that four given entries is what pins a square down. That is the difference between a trick and a piece of mathematics. A trick you have to perform. This one you can hand over, and it still works.