Tangrams: rearranging pieces without changing the area

0:00This is a square, and it is about to stop being one. Cut it like this, and you get seven pieces. Not seven pieces chosen at random — seven pieces that came out of that one square and still add up to it. Two big triangles. One middle-sized triangle. Two little ones. A square that sits on its corner, and a slanted four-sided piece. Seven, and that is the whole set for the rest of this.

0:30Nothing gets added, nothing gets thrown away, and nothing gets cut again. Look at how the sizes compare, because getting this wrong is the commonest way a puzzle stalls. Each big triangle is a quarter of the whole square. The middle triangle is an eighth, and so is the little square, and so is the slanted piece. Each small triangle is a sixteenth. A quarter and a quarter, three eighths, and two sixteenths.

1:02Add those up and you get exactly one whole square, which is the only answer they could add to. And notice the three pieces that are each an eighth are three completely different shapes. Now the rules, and there are only two of them. When you put the pieces down, nothing is left bare. And nothing is doubled up — no piece sits on top of another, not even a sliver of overlap along an edge.

1:31Edge meets edge, and the whole of the shape you are aiming for gets covered. That is it. Those two sentences are the entire game. They sound obvious, and they are about to do a surprising amount of work. So take the seven pieces and move them. Swing the two big triangles out, one below and one to the side. The outline you started with was a square. The outline you are looking at now is a slanted parallelogram.

2:04Same seven pieces. Every one of them still exactly the piece it was — nothing cut, nothing stretched. And the shape on the outside is completely different. So the outline is free. You can push it around as much as the pieces allow, and there are a great many outlines the pieces allow. But now ask a different question about those two pictures. How much surface is covered? In the square it is the four quarters and the eighths and the sixteenths, all seven of them, adding to one whole.

2:41In the parallelogram it is the same seven pieces, so it is the same total. It has to be. Nothing was added and nothing was taken away, and no piece was allowed to overlap another. So the amount of surface underneath did not move at all, and it could not have. That is the strange thing here: the outline is free and the surface is fixed, and they are both true at the same time.

3:11Watch it happen again, more dramatically. Take the same seven and lay them out as a triangle. One big triangle, one flat side, three corners — about as far from a square as an outline gets. Count the pieces. Still seven. Measure the surface. Still exactly the same as the square you started from. The picture changed completely and the quantity underneath did not flicker. You could keep going all day and it would keep not flickering.

3:46And it is not only smooth outlines. Move one big triangle over to the other side and the outline develops a notch — a corner that turns back on itself instead of going round. Something like a dart, or an arrowhead. That is a genuinely awkward outline. It is not convex, it has a dent in it, and it has six corners rather than four. Same seven pieces. Same surface underneath, to the last sixteenth.

4:18The rule does not care what the outline looks like, and that is exactly what makes it useful. What are you actually allowed to do to a piece? You can slide it anywhere. You can turn it to any angle. And you can pick it up and turn it over, which is a normal move and not cheating. For six of the seven pieces, turning one over changes nothing you could not have got by just turning it round.

4:47The slanted four-sided piece is the exception. Turn that one over and it genuinely leans the other way, and no amount of spinning it will get it back. So flipping is a real move, on exactly one piece. Here is what you are not allowed to do, and it is the thing everybody reaches for. When a solution nearly works, the temptation is to make one piece a little bigger.

5:16And there is a very good reason that temptation exists. Take a small triangle and double the length of every side. What you get is exactly a big triangle — the same shape, right down to the corners. But doubling the length did not double the surface. It made it four times as much. So that swap turns a sixteenth into a quarter, and now the seven pieces cover more than the square they came from.

5:45The shape was legal. The amount of surface was not, and the amount of surface is the thing that has to stay put. The same goes for quietly leaving a piece out. Drop the smallest one, the sixteenth, and you are covering fifteen sixteenths of what you were. It is a small difference and it is a real one. Drop a big triangle instead and you are down to three quarters, which is not subtle at all.

6:16Either way you are no longer answering the question you were asked. Keep the set whole. That is what makes the whole thing decidable. Now put those two facts side by side and see what they buy you. The outline can be anything. The surface is pinned. So when somebody hands you an outline and asks whether these seven pieces fill it, that is not a matter of taste. It is a question with an answer, and the answer is yes or no.

6:48If the outline covers a different amount of surface, the answer is no before you have moved a single piece. And if it covers exactly the right amount, then it might be possible — you still have to find the arrangement, but at least you are no longer guessing whether one exists. Close enough does not count, because the pieces have exact edges and exact corners, and either they close up or they do not.

7:19If you are actually trying to solve one, here is the practical advice. Put the two big triangles down first. Between them they are half of everything, so once they are placed, half the problem is already decided. Then the middle triangle, then the square and the slanted piece, and leave the two small triangles for last. The small ones are the most forgiving, because they are the ones that fit into whatever corner is left.

7:50And if it will not close, do not shave a piece. Move a big one. The big pieces are where the real decisions live. One last thing, because this is not just a puzzle. Read those two rules again: nothing left bare, nothing doubled up. That is not a rule about seven pieces cut from a square. It is a rule about covering anything with anything. Give it a name — call it covering exactly — and you can ask it about floor tiles, about hexagons, about shapes that have nothing to do with this square.

8:28The seven pieces were the training ground, because they let you feel the thing in your hands. The outline is free. The surface is not. Everything that comes next is built on that one sentence.