Let's say you are cornered by your worst enemy. They will kill you, but being an oddball enemy, they'll give you one request first. (And the request can't be "Don't kill me." They're on to that one.) I have a solution for you. Ask them to show you how DeLand's Paradox works. That will keep them busy for years while you live a happy and productive life. It will, however, make them justified when they finally get around to killing you.

A graphically obvious version of DeLand's Paradox is Curry's Triangle. It's a triangle sketched out on a backing of graph paper that, when rearranged on the same paper into the seemingly same shape, will suddenly be missing a square in its center. Careful observation will show you that the hypotenuses of the first and second triangles are slightly curved. One is curved up and one curved down, giving them slightly different areas, and accounting for the missing square. It's a good trick.

So of course the puzzle-makers changed it into a torture device. Theodore DeLand was a designer and amateur magician, and he did his own version of the triangle. He sketched out fifteen playing cards (maybe); the image he sketched was meant to be cut out and rearranged into an image with the overall same dimensions, but with slightly juggled pieces. In this new image, there were sixteen playing cards.

There are versions of this illusion everywhere. The more simple ones are lines, bars, or eggs. Put the pieces of the image one way and there is one amount of objects, put it another way and there is one extra. They all work on the same principle. The objects are put at strange intervals and strange elevations across the image. When the image is rearranged, one piece of each object is given to the next, but each object receives a smaller piece than it gives. All those small slices add up to one extra object. You're just taking a little mass out of each. Sounds simple enough.

But people kept going. Instead of simple geometrical objects, they decided to make the illusion work with people, making the images cartoony enough so that the slight loss of mass was not noticeable — except for the extra person who popped up out of nowhere. The most famous illusion is of leprechauns, created by well-known puzzle-maker Pat Lyons, but there are versions with stick figures, with runners racing over a globe — there's even a pin-up calendar version of the illusion. And as you can see if you click the youtube clip, any attempt to count or see the extra person will drive you totally mad. Even if the two images of this illusion are shown side by side, revealing the trick, it's still hard to see how it works. The brain balks at the idea that the extra mass can make a person.