A group of children held a grape-eating contest.
When the contest was over, the winner had eaten N grapes, where N is a positive integer.
The child in Mth place had eaten N + 2 – 2M grapes.
The total number of grapes eaten in the contest was 2016.
Find the smallest possible value of N.
(In reply to I could win this contest (spoiler)
by Steve Herman)
If N=2 and M=1 for the first place child, then he must eat N+2-2*M = 2+2-2*1 = 2 grapes. Then for a second place child N=2 and M=2, meaning he must eat N+2-2*M = 2+2-2*2 = 0 grapes.
The same applies to your N=4 case, a third place child must eat 0 grapes. The 2*M term in the grape function means that each place eats exactly two less grapes than the immediately higher place.
The big tie for second place can be salvaged if we say that the first place child ate 4 grapes and there was 1006 way tie for second place with 2 grapes.
All of that aside, I vaguely recall seeing a version of this problem with the added condition "Each child eats a different number of grapes."