
Your Answers 
(21/06) Joshua Crammer : If there are x ninjas, than each person, fighting a different number of people, will follow a number in this pattern: 1,2,3…x2,x1,x. However, since the x th person would have to fight x people, that person would have to fight himself. That of course is impossible, and so that person must pick another number in that series. That number has already been picked by someone else, and thus we come to the conclusion that two people are fighting the same number of people.
Shika : You're right ! Maybe I should make this more challenging.

(22/06) Cecil Artavion : The idea is this:
Let's say you have 10 boxes and 11 oranges. If you place oranges in boxes, at least 1 box must have 2 oranges.
If you have more objects than boxes, one of the boxes contains at least 2 objects. It seems obvious (and I guess it is), but the consequences are astounding.
For example, in this problem: The boxes are the number of kids a particular kid is fighting. The objects are the kids themselves. The boxes are labeled 1 to N1 (where N is the total number of kids). If kid A is fighting 3 kids, he gets placed in box 3. If kid B is fighting 5 kids, he is placed in box 5.
There are only N1 boxes because there are N kids. It's possible for one of them to fight all the others, which is N1 people the kid is fighting. But that's the maximum number of people the kid could be fighting.
The idea is that there are more kids than boxes, so two kids must be placed in the same box which means they are fighting the same number of kids (even if they aren't fighting the same kids).
Shika : Congratulations! You're right again!

(24/06) Warbringer : Ok, sorry for making you read all that crap, it wasn't very brilliant. Guess I was bored... :)
Here goes :
Let's suppose that there aren't at least 2 ninjas that are fighting the same number of ninjas, which means that none is fighting the same number.
There are N ninjas. Each of them can fight against a maximum of N1 ninjas and a minumum of 1. That makes N1 choices for N ninjas, which will obviously lead to a duplicate.
Example : 5 ninjas A, B, C, D, E
A fights 4 ninjas
B fights 1 ninjas
C fights 2 ninjas
D fights 3 ninjas
E fights ? ninjas
E has to fight either 1, 2, 3, or 4 ninjas.
Therefore it is impossible for all of the ninjas to fight a different number of ninjas, which means that at least 2 of them are fighting the same number of ninjas.
Shika : That's right! I should seriously make this harder...

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