44 monkeys have a total of 1407 bananas. No two monkeys have the same number of bananas. Show that there is a monkey that has exactly twice as many bananas as another one.

Well, I'm not great with doing math proofs or anything, but the smallest sequence of 44 unique integers in which no term is twice another adds up to 1408 (I assumed that there could be one monkey with no bananas):

0
1
3
4
5
7
9
11
12
13
15
16
17
19
20
21
23
25
27
28
29
31
33
35
36
37
39
41
43
44
45
47
48
49
51
52
53
55
57
59
60
61
63
64
----
1408

So in order to make the sum lower, you would need to remove one of the numbers and put one in that was lower, but all of the numbers that have been excluded were left out because they were twice the amount of one of the other numbers.  This would be impossible, so by contradiction you can see that if the sum is only 1407, then there must be a monkey that has twice as much as another monkey.

Posted by tomarken on 2006-11-13 12:13:30