Which rectangles break into identical squares, and which do not

0:00Draw a rectangle that breaks into three identical squares in a row. That is the whole task. Now notice what it does not give you. No length for the long side. No length for the short one. Not a single number anywhere. Which feels like far too little to start with. You cannot pick up a ruler if nobody has told you what to measure. But read it once more. It asks for a rectangle with that property. Not one particular rectangle. Any one at all.

0:33So the size is yours to choose. Only the shape is fixed. And that turns out to be the easy half of this problem, not the hard half. Before doing three, do two. When a problem will not give way, make it smaller and see whether the smaller one does. So: a rectangle that breaks into two identical squares, sitting side by side. If you can build that, the three-square version is unlikely to need a new idea from you.

1:05And if you cannot build it, you have saved yourself a great deal of wasted effort on the harder one. Two it is. So where do you start drawing? You cannot start with the long side, because you do not know how long it is. You cannot start with the short side either, for exactly the same reason. So do not start by drawing at all. Draw a plan instead. A rough diagram: not to scale, not measured, deliberately sloppy.

1:38Its only job is to hold the labels still while you think about them. Everything difficult about this problem gets settled on that rough diagram, and the real construction afterwards takes about a minute. Here is the plan. A rectangle. Along the top, three points: A, B and C. Along the bottom, three more: F, E and D. One line straight down the middle, from B to E, cutting the whole thing into two pieces.

2:10The left piece is A B E F. The right piece is B C D E. And now say out loud the two things you were promised about those pieces. They are identical to each other. And each one of them is a square. Nothing here is drawn accurately. B is nowhere near halfway along. Neither piece is remotely square. It does not matter in the slightest. The labels are right, and the labels are what you are about to reason with.

2:43So follow it round. The two pieces are identical. So the top edge of the left one equals the top edge of the right one. A B equals B C. Now use the other promise. Each piece is a square, and a square has all four of its sides equal. In the left piece, that gives you A F equals A B, and B E equals A B, and F E equals A B. Four sides, one length.

3:13In the right piece, the same again. B C and C D and D E and B E are all equal to one another. And look at B E. It belongs to both pieces at once. Which hooks the two chains together into one. Every short line in the entire figure — and there are seven of them — is the same length. That is worth writing onto the diagram itself, and there is an agreed way of doing it.

3:43A short tick drawn across a line means this: any two lines carrying a tick are the same length. So put one on A F. One on B E. One on C D. And along the top and bottom: A B, B C, F E, E D. Seven ticks, and the diagram stops being a picture of the question. It is now carrying the answer. Someone who picks it up can read the reasoning off it without doing any of the reasoning again.

4:16Now choose. Pick a length for A F. Anything you like — that is the thing the problem deliberately left open. Say four centimetres. Every ticked line in the figure is now four centimetres. So A B is four. And B C is four. And the long side, A C, is A B followed by B C. Eight. Eight by four. That is your rectangle, and you can go and draw it.

4:45Notice how many decisions you actually made. One. After that, nothing was left open. And here is where the ruler becomes optional. You never have to measure A F at all. Draw it whatever length you like. Raise the two sides. Then open a compass to span A F exactly, lift it, and set that span down along the top to place B. Once more from B, and you have C.

5:15Why go to the trouble, when the ruler is lying right there? Because a ruler has to be read, and a reading can be off — and the second reading can be off differently from the first. Then your two pieces are not identical, and everything you just proved stops applying. A compass reads nothing. It carries the length it is already holding. Set it slightly wrong at the start and you get a smaller figure than you intended. You still get two identical squares.

5:49Now three. Same rough diagram, one more cut. Four points along the top, four along the bottom, three pieces between them. Run the identical chain. All three pieces the same as each other, each of them a square, so every short line in the figure is equal to every other. The long side is now three of those instead of two. Four centimetres across gives you twelve along. And with the compass: open it to the short side, and step it along the top three times instead of two.

6:27One extra copy of a length you were already holding. That is the entire difference between the two problems. So which rectangles refuse to split? Take one that is four centimetres by two and a half. For two squares, the long side has to be exactly twice the short side. Twice two and a half is five. You have four. It fails. And it fails by half a centimetre, which sounds small and is not.

6:58Nearly is not a category here. Either the two pieces are identical squares or they are two different rectangles. Try seven by two against the three-square test. Three twos are six. You have seven. Also out. And no amount of cleverness about where to cut will rescue either of them. Which raises a question about why that argument went so smoothly. It worked because of something about the numbers two and three that is easy to walk past.

7:32Two identical squares can be joined into a rectangle in exactly one way. Side by side. There is no second arrangement to be clever with. Three is the same. One row of three, and nothing else is available. But four is a different animal. Four identical squares make a long rectangle, four times as long as it is wide — or they make a square, two by two. With four, the number stops deciding the shape.

8:05So what you proved was not a general law about splitting rectangles into squares. It was a fact about two, and a fact about three. And step back to see what the answer actually was. Not a size. A comparison. Two identical squares means the long side is twice the short one. Three means three times. Which is why no measurement was ever required, and why the problem could hand you a compass and quietly take the ruler away.

8:39Eight by four works. Two by one works. Twenty by ten works. So does a rectangle a mile long and half a mile wide. As far as this question is concerned they are all the same rectangle. The size was never part of the answer. It was only ever the shape.