Wednesday, 27 November 2013

Vanishing area and infinite chocolate

There is a paradox or puzzle in mathematics sometimes known as the vanishing area paradox. There are several variations, but they all share the same principle, so I'll just talk about the one in the example video below:

You take a rectangle (in this case, made of chocolate), 4x6 units, and cut it diagonally between 2 and 3 units from the bottom. Make a couple more cuts, swap two of the pieces and suddenly you have a piece left over. You've created extra chocolate out of nowhere!

But you haven't, actually. It's a rounding error and sloppy measurement. The main rectangle is now actually only 5.75 units high, and I can prove it. Think about the right-hand edge. When you make that second cut, 1 unit in from the left, the cut goes partway through that first piece on its right-hand edge. The piece that moves from the top is only 0.75 units tall at that point, but the cut on the original right-hand edge takes the full piece away. You're replacing a full piece with a 3/4 piece, making the whole block of chocolate shorter. Because it's a fairly small difference, you don't notice.

If you're still not convinced, do it for yourself, but do it four times and count the resulting rectangle. You'll probably notice it getting shorter long before you get through all four times. After the second time, one of the rows will be suspiciously half-sized. After the third time, it will be a ridiculously stubby little quarter-size row.

Mokalus of Borg

PS - It's a neat trick, though.
PPS - The shorter row gets less noticeable the wider the block.

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