The shortest path across a cube, found by flattening it

0:00There is an ant on a cardboard box, and a crumb somewhere else on it. The ant walks on the surface. It cannot fly, and it cannot tunnel through. What is the shortest way from the ant to the crumb? On a flat table that is not a question. Straight line. Done. On a box it is a real one, because the surface turns corners, and so must she. And here is the trap. You can draw a route that looks good.

0:30Looking is not the same as knowing, and the whole of this is about the difference. Say you have drawn a route and it looks short. To call it the shortest, you have to beat every other route there is. How many are there? Infinitely many. Nudge any route sideways and you have another one. So you cannot check them one at a time. Not in a lifetime. That is what makes shortest a hard word. It is a claim about everything you did not draw.

1:02There are two ways out of that. Find one reason that covers all of them at once. Or move the problem somewhere the answer is already known. Here is the move, and it is the only idea in this video. Cut the box along some of its edges and open it out flat. Nothing was stretched. Nothing was torn. Lay a piece of string along a route on the box, then open the box out. The string is the same length it was.

1:34Every distance measured along the surface survives the flattening. Watch what happens at a single fold. Roll the box over one edge, and that edge does not move at all. The new face comes down on the far side of it, and the two ends of the edge are exactly where they were. That is why the flattening is safe. It is a move, not a distortion. So the route is now a drawing on a flat sheet of paper.

2:05And on flat paper we already know the answer. Straight line. Take a box. The ant is halfway up one side. The crumb is at the middle of the top. Open out the side and the top, and the two points line up, one directly above the other. Two up the side face, two across the top. The straight route measures four. Now a second route. It crosses the same edge, but further along.

2:33Flattened, that one is not straight. It has a corner in it. That corner is the whole verdict. A route with a corner in it is longer than the straight one between the same two points. Always. And that is not a measurement. It is the fact that two sides of a triangle beat the third. So nothing has to be measured at all. Show that the corner is off the straight line, and you are finished.

3:02For the arithmetic anyway: the straight route is four, and this bent one squares to thirty-two, against sixteen. But four is not merely better than that one route. Four is the best there is, and that was settled by opening the box out every way it opens and looking at all of them. Move the crumb onto an edge of the box, and nothing about the method changes. A point on an edge belongs to two faces at once.

3:35Unfold towards one of them, or towards the other. Same answer either way. Six this time. Two up the side, four across the top. Longer than before, which is what you would expect, because the crumb has moved further away. The picture changes, the reasoning does not, and that is usually the sign that you have the right idea. So far, so easy. Now watch the method break. Same box. The ant is at the middle of one small square end.

4:10The crumb is down on the bottom edge, two along from that end. Open the box out, draw the straight line between them, and look at where it goes. It leaves the paper. Part of it runs across empty space, where there is no cardboard at all. That line is not a bad route. It is not a route. There is nothing on the box it corresponds to, so its length is not the length of anything.

4:37And that is not a freak. Of the twenty-six ways of opening this box between those two faces, twenty do exactly that. The trap is sharper than it looks, too. One of those useless lines measures shorter than a perfectly good route does. So you cannot even line the numbers up and take the smallest one. Checking is not optional. Six of the twenty-six do work. Cut a different set of edges, and the same two points land somewhere else on the paper.

5:12Now the straight line between them stays on the cardboard the whole way across. That one is a route. Fold it back up and you can watch her walk it. So the method has a second half, and it is the half people forget. Draw the line. Then check that it stayed on the paper. A number read off a line that wandered into empty space is a number about nothing.

5:37That is the moral, and it is worth more than either answer in this video: the way you open the box matters. Now a box worth the trouble. Thirty long, and square at both ends, twelve by twelve. The ant is on one end, on its middle line, one below the top edge. The crumb is stuck to the far end, also on the middle line, one above the bottom.

6:05About as far apart as two points on this box can be. The obvious route goes over the top. Up, along, and down the other end. Up one, to get out of her face. Thirty along the top. Then down the far end. How far down? The crumb sits one above the bottom of a face twelve high, so eleven. Open the box out that way, and those three pieces lie in a single straight line.

6:35One, plus thirty, plus eleven. Forty-two. And it really is straight on the flat, so nothing that goes over the box that way beats it. Forty-two. Write it down. Now open the same box out differently. Not over the top. Around. Roll the box along its length, so that all four long faces lie in a strip, one under the next. The ant's end face hangs off one side of the strip. The crumb's hangs off the other side, further down.

7:10The straight line between them is now the long side of a right triangle. Across: one to get out of her face, thirty along the box, one into his. Thirty-two. Down: twelve, and then twelve again. Two face widths, because going round means crossing two of them. Twenty-four. And now the Baudhayana theorem, the rule about the squares on the sides of a right triangle. Thirty-two squared is a thousand and twenty-four. Twenty-four squared is five hundred and seventy-six.

7:46Together, sixteen hundred. And sixteen hundred is forty times forty. The route is forty. Forty against forty-two. Going round the box beats going over the top, by two. Which is already worth stopping on. The obvious route is not the best one. But there is a bigger question sitting there, and it is the honest one. Two ways of opening the box gave two different numbers. What about the third way? The tenth?

8:18There is only one way to answer that, and it is to list them. There are twenty-eight ways of walking from her end to his without stepping on a face twice. Sixteen of them give a real route. The shortest of those sixteen is forty. So forty is the answer, and now we know it rather than hope it. Not forty because two pictures agreed. Forty because twenty-eight were tried and none of them beat it.

8:48And the winning route crosses five of the six faces. A search that only looked four faces deep would have stopped, returned something longer, and given no sign at all that anything was missing.