Fifteen monkeys share 100 bananas. Prove that two monkeys have the same number of bananas. Use PHP to solve. Label 15 pigeonholes 0 through 14. Each monkey has a name tag. One name tag in each pigeonhole would mean 105 bananas, so one pigeonhole is empty and one pigeonhole contains two name tags so two monkeys have the same number of bananas.
Fifty-one different numbers are each less than 100. Prove that one pair of numbers sum to 100. Use PHP to solve. Label 50 pigeonholes (1,49), (2,48), ...(50,50). If the 51 numbers fit into 50 pigeonholes, two numbers must fit into the same pigeonhole. Those two numbers sum to 100.
In a dark room, how many socks do you need to pick from a drawer of black and blue socks to make a pair? The answer is three. If one pigeonhole is black and the other blue, three socks means one pigeonhole has two socks.