Question

1600 coconuts are distributed among 100 monkeys (it is possible that there
are monkeys receiving no coconut at all, so the distribution can be made anyway, there is no
fairness implied). Prove that at least 4 monkeys must have received the same number of coconuts.

Answer 1

pigeonhole principle

Answer 2

Well, I know that if you tried to give each monkey a different number of coconuts, say 1 to the first, 2 to the second, etc. It would take over 2500 coconuts.

Answer 3

(sorry, that should be over 5000 coconuts. I think I divided by 2 too many times..)

Answer 4

Suppose you tried to rig it so that, at most, 3 monkeys had the same number:
We can split up the 100 monkeys into 3 groups of 33 and then a single monkey.
Each of the three groups gets coconuts distributed thus: 0 to the first, 1 to the second, 2 to the third, and so on so that the 33rd monkey gets 32 coconuts. The single hundredth monkey gets 33. This is the only way to ensure that no more than three monkeys receive the same number of coconuts.
However, if you add up all the coconuts needed to do this, it would take 1,617 coconuts.
Therefore, with only 1600 coconuts, it cannot be done.