100 monkeys have together 1600 bananas. Show that at least four monkeys have the same number of bananas.
We distribute the bananas such that monkeys #3n, #3n+1, #3n+2 get n bananas each for n=0,1,2,.......
Substituting n=32, we have, the total number of distributed bananas as:
3*(0+1+2+.......+32+33) = 3*(33*34)/2 = 1617
Therefore, monkeys #3n, #3n+1, #3n+2 get n bananas each for n=0,1,2,.....,32, which is distributed amongst 3*33=99 monkeys. Monkey #100 gets 33 bananas.
However, we have 1600 bananas. So, there must be a reduction of 17 bananas so that some monkeys must have less bananas.
Clearly, such an assignment will entail at least four monkeys having the same number of bananas.
Edited on July 23, 2022, 8:17 am
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