3. Some leprechauns have more of their body above the horizontal line Other leprecahuns have more of their body below the horizontal line.

4. In order, the 14 complete leprechauns are... 15 leprechauns! Using this ordering of the leprechauns, it is clear where the extra man comes from. Each of the original 14 men (but the last) 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!

5. Theorem: There are at least two leprechauns that have the same number of leprechaun friends. Three assumptions & indirect proof by contradiction: 1.Assume there are n leprechauns. 2.Assume friendship is mutual, i.e., if I’m your friend, then you’re my friend and vice-versa. 3.Now, assume that no two leprechauns have the same number of leprechaun friends. (This is the indirect method.)

6. leprechaun# of leprechaun friends 11 22 33 44...... n - 1? n? A list of all leprechauns and the number of their leprechaun friends

7. This conclusion contradicts our assumption that no two leprechauns have the same number of leprechaun friends. Thus, there must be at least two leprechauns with the same number of leprechaun friends. A leaf for... happiness health Love inspiration