geometry – The crumpled paper problem

I seen a theorem recently (and have seen similarly phrased problems before), which I found quite surprising. Unfortunately, I’m not sure what its name is (or if it has one). I’ll refer to it as the “crumpled paper problem”. It goes something like this:

Suppose you were to take two pieces of paper, identical in size and shape, and place one on top another such that all 4 corners and all 4 edges line up. Now, every point along the top sheet can be “mapped” to its corresponding point on the bottom sheet. If you were to then crumple up the top sheet of paper, and toss it on top of the bottom sheet, there necessarily exists a point on the top sheet that lies precisely above its “mapping point” on the bottom sheet, no matter how it was crumpled or where it was placed.

This result, to me, at least, is very surprising. Intuitively, it seems to me that it must be that there is some way to crumple the paper, or location to toss it, such that no points line up. I can sort’ve imagine progressively folding the paper (analogous to crumpling), and kind of see how this might be, however its still quite challenging.

My question is, how is this proven, does the problem have a name, and to what field of mathematics does this belong? I’m very curious how this can be shown.


I’ve added those tags which I thought may be relevant. Unfortunately, I have no clue to what precise area of mathematics this belongs. Please do correct tags as necessary.