BrightKidz Library
Subjects
Three squares of a grid, and how many lines run through each one Three copies of the same three by three grid, side by side. In the first the middle square is filled in and four lines run through it: its row, its column and both long diagonals. In the second a corner square is filled and only three lines run through it: its row, its column and one diagonal. In the third an edge square is filled and just two lines run through it, its row and its column. The same grid, three different squares, and each one sits on a different number of lines.

Why Five Has to Sit in the Middle

About 11 minutes

Two things were handed to you and neither was explained. The five goes in the middle. And there is no genuinely different square hiding somewhere — every one you find is this one turned round or held up to a mirror.

Both are true. Both sound like things somebody discovered by trying for a long time.

They are not. One piece of counting settles both, and you can do it on the back of an envelope in about ten minutes.

Every line is a set of three that makes 15

Forget the grid for a moment. Just ask which three of the numbers 1 to 9 can add up to 15.

Hunt for them in order and you will find them all:

1 5 9 · 1 6 8 · 2 4 9 · 2 5 8 · 2 6 7 · 3 4 8 · 3 5 7 · 4 5 6

That is the whole list. There are eight of them, and there is no ninth — try to invent one and you will find the number you need has already been used or does not exist.

Now count the lines on the board: three rows, three columns, two diagonals. Eight.

Eight sets and eight lines. That is not a coincidence and it is the first hint that this puzzle has far less room in it than it looks.

Some numbers turn up far more often than others

Go back down the list and count how many of the eight sets each number appears in. Do it yourself — it takes two minutes and the answer is better if you find it.

The number How many sets it is in
5 4
2, 4, 6 and 8 3 each
1, 3, 7 and 9 2 each

Five is in more of them than anything else, and by a clear margin. The even numbers come next. The odd ones apart from five are in the fewest.

Nobody arranged that. It falls out of what adds up to 15.

Some squares sit on far more lines than others

Now do the same counting on the grid, which is the other half of the answer.

Put a finger on the middle square. How many lines run through it? Its row, its column, and both diagonals. Four.

Move to a corner. Its row, its column, and one diagonal — the other diagonal misses it completely. Three.

Move to a square in the middle of an edge. Its row and its column, and that is all. No diagonal comes near it. Two.

The three kinds of square in a three by three grid A three by three grid. One middle square, four corner squares and four edge squares are each shaded a different amount, because the three kinds sit on different numbers of lines. The middle square lies on its row, its column and both diagonals. A corner lies on its row, its column and one diagonal. An edge square lies only on its row and its column. middle, corner, edge

Pick a kind of square and see what it demands of the number that lives in it.

  • The middle square — sits on four lines. Every line through it is a different set of three that adds to 15, and every one of those sets contains this number. So the number here has to appear in four different sets.
  • A corner square — sits on three lines. Its row, its column and one diagonal. The number here has to appear in three different sets. There are four corners.
  • An edge square — sits on only two lines. Its row and its column; no diagonal reaches it. The number here need only appear in two sets. There are four of these.

Now put the two counts side by side

Here is the move, and it is worth going slowly.

Every line running through a square is one of those eight sets, and every one of them contains whatever number you put in that square. Four different lines means four different sets. So a number sitting in the middle must appear in four sets, or one of its lines has nowhere to come from.

Only one number in the whole puzzle appears in four sets. Five has to go in the middle. Not because it is balanced or central or feels right — because nothing else can do the job.

Keep going and the rest falls just as hard.

The four corners each need a number that appears in three sets. The numbers appearing in three sets are 2, 4, 6 and 8 — and there are exactly four of them for exactly four corners. So the corners are the even numbers, every time.

That leaves 1, 3, 7 and 9 for the four edges, which need only two sets each. Which is exactly what those four have.

Three counts, and every number in the puzzle has been sent to its seat.

So is there a different square?

No. And now you can see why rather than being told.

Once five is stuck in the middle and the evens are stuck in the corners, the only freedom left is which corner the 2 picks, and which way round the rest run from there. Four corners to choose from, and then two directions — eight arrangements in all.

Eight is exactly the number of ways you can turn a square and flip it: four quarter-turns, and each of those seen in a mirror.

So every one of the eight is the same square wearing a different pose. There is no ninth arrangement waiting to be found by somebody more patient than you. The puzzle only ever had one answer, and the counting knew it before a single number was written down.

Why can 9 never sit in a corner of the square?

Go and fail to break it

The best way to believe this is to try to beat it.

Draw a blank grid and put something other than 5 in the middle — 6, say. Now try to finish it. You need four lines through that middle square, all adding to 15, and every one of them has to contain the 6. Hunt back through the list of eight sets and you will find only three that contain a 6 at all. The middle square needs four.

You will not get stuck at the end. You will get stuck almost straight away, and the wall you hit is the counting you just did.