The Vanishing Leprechaun - Answer

Ok . . . here we go. There is no magic nor slight of the hand. It is merely mathematics.

"Yeah right!" you say. Well, did you notice that one leprechaun, in going from 14 to 15 leprechauns, lost a knee-cap?
Another guy got the knee-cap, but gave up a foot.

So what is happening, is in going from 14 to 15, the 14 guys each give up a small piece of themselves.
If 14 guys give up 1/14th of themselves that equals one whole new guy!

Try rearranging the leprechauns in the order of give-and-take. Start with the knee-cap-loser.
Then comes the knee-cap-gainer/foot-loser. Then the chap who gained that foot, and so on.
This continues to a little man with crossed arms who acquires hair.

The 14 complete leprechauns are in order here.

14 in a row

Instead of three pieces, the above image has two pieces - top and bottom. Slide the bottom over,
giving the second man the first man's knee. What do we find?

15 in a row

15 leprechauns! (they are all a tad smaller though)
Each of the original 14 men gives some of himself to the next man, but each man receives less than he gives.
The extra leprechaun pieces add up to form a new man!

That's it! Pretty cool huh?