Using row and column sums to prove a grid cannot be filled

0:00Here is a three by three grid, filled with the numbers one to nine, each of them used exactly once. Ringed around the outside are six more numbers. Down the right-hand side: sixteen, nine, twenty. Along the bottom: thirteen, seventeen, fifteen. Nobody has said what they are. That is the puzzle. Take a moment, because the answer is going to look almost too simple to be worth the trouble. It is worth it. In a few minutes those six numbers are going to prove something that no amount of trying ever could.

0:38They are going to show that a certain grid cannot be filled in at all. Not that it is hard. That there is no filling. The four, the seven and the five along the top come to sixteen. And there is a sixteen sitting at the end of that row. Six, one, two makes nine. And a nine at the end of that one. Three, nine, eight makes twenty. Twenty.

1:04So each ring on the right is the total of the row it sits beside. Now read downwards instead. Four, six, three is thirteen, and thirteen is what is written under that column. Seven, one, nine is seventeen. Five, two, eight is fifteen. Both of them are there. Six little sums. Three rows read across, three columns read down, and their totals written round the edge. Now the same idea, run in reverse.

1:37Here is a grid with almost nothing in it. A nine in the top-left corner, a five in the bottom-right, and that is all. But the totals are all there. Thirteen, fourteen, eighteen down the side. Twenty-four, nine, twelve along the bottom. And you are still filling it with one to nine, each of them used once. So where do you start? Not just anywhere. Start where the numbers have the least room to move.

2:07The top row has to come to thirteen, and it already holds a nine. So the other two cells have to make four between them. Two different numbers, out of what is left, adding to four. There is exactly one way. A one and a three. That is the whole method. Find the place where the total leaves almost no choice, and take it. Here is another one, and this time it bites even harder.

2:36The row totals are twenty-four, fifteen and six. Six. From three different numbers, none of them repeated. The smallest three you have are one, two and three, and they make exactly six. So that bottom row is not a puzzle. It is one, two and three in some order, and nothing else will do. Twenty-four does the same thing at the other end. The largest three you have are seven, eight and nine, and they make exactly twenty-four.

3:07Two rows settled, without a single guess. Though be careful. Settling which numbers go in a row is not the same as settling where each one goes, and this grid has two finished answers, not one. Now this one. One cell filled in. A six, in the middle row, on the right. The rings down the side read five, twenty-one, nineteen. Along the bottom, nine, eleven, twenty-six. Have a go at it. Genuinely have a go, because failing is the point.

3:43You will not manage it. Neither will anybody else. But here is the trap, and it catches almost everyone. If you try for ten minutes and give up, what you have learned is something about your ten minutes. You have learned nothing at all about the grid. To say a thing is impossible you have to rule out the attempts nobody has made yet, including all the clever ones. So let us stop attempting, and start bounding.

4:13Forget the grid for a moment. Just ask how small one of those totals could possibly be. It is three different numbers from one to nine. The three smallest you have are one, two and three. One and two and three make six. You cannot go lower. There is nothing smaller left to reach for. Now the other end. The three largest are nine, eight and seven. Nine and eight and seven make twenty-four.

4:43So every one of those six ringed numbers, in every grid of this kind that has ever been drawn or ever will be, sits somewhere between six and twenty-four. Back to the grid that would not be filled. Look at the ring on the top row. Five. Five is below six. There are no three different numbers from one to nine that add up to five. The very best you can do is one, two and three, and that is already six.

5:14Now look along the bottom. Twenty-six. Twenty-six is above twenty-four. It is two more than the very best you have got. That grid was never going to work, and we know it without touching a single cell. And that is the difference. Not, I could not do it. There is nothing to do. Something odd turns up if you go back over the grids that did work. Add all six ringed numbers together. Every time, ninety.

5:45Sixteen, nine, twenty, thirteen, seventeen, fifteen. Ninety. Thirteen, fourteen, eighteen, twenty-four, nine, twelve. Ninety again. And it splits neatly down the middle. The three row totals on their own come to forty-five. The three column totals on their own come to forty-five. Every time. In every grid of this kind. Which is a strange thing for a puzzle-setter to have arranged, six times over, entirely by accident. They did not arrange it. They could not have stopped it.

6:25Take the three rows and lift them straight out of the grid. The top row is three of your numbers. The middle row is three more. The bottom row is the last three. Between them, that is all nine of them, once each. So adding the three row totals is just adding one to nine in a slightly roundabout order. One, two, three, four, five, six, seven, eight, nine. Forty-five.

6:53And notice what that argument never mentioned. It never said how the grid was filled. Shuffle it however you like and the three rows still hold all nine. Now do it again, cutting the other way. The left column is three of the numbers. The middle column is three more. The right column is the last three. The same nine numbers. Just gathered into different piles. Forty-five. So the ninety was never a pattern at all. It is forty-five twice, because the nine numbers have been counted twice. Once across, and once down.

7:34And that has a consequence the puzzle-setter would rather you did not spot. They do not get to choose six numbers. Once five of them are down, the sixth is already decided for them. I checked that against every grid of this kind there is, and it never once came out otherwise. Which gives us a second look at the grid that refused. Its three row totals were five, twenty-one and nineteen.

8:04Add them up. Forty-five. So on that count it is perfectly respectable. Its three column totals were nine, eleven and twenty-six. Add those. Forty-six. Forty-six, and it is not allowed to be forty-six, because those three numbers are the same nine numbers cut downwards. So that grid fails twice over, for two completely separate reasons, and either one on its own would have finished it. Here is what we have, and it is worth being exact about it.

8:40One quantity that cannot move. Forty-five. And one range that cannot be escaped. Six to twenty-four. Neither of them came from trying anything. They came from what the numbers one to nine simply are. Which is why they cover every attempt at once, including all the attempts that nobody has made. But be careful what you claim in the other direction. A set of totals passing both tests does not mean the grid can be filled. It only means these two arguments have run out of things to say about it.

9:17And even when it can be filled, there may be more than one way. That second grid had two finished answers. An argument like this tells you where the answers are not. Finding one is still your job.