148 Reasoning

I'm waiting for the MU vs. Barca match to start, and I got a little bored, so I've decided to present a Maths problem that should be rather interesting and puzzling at the same time. Here's a small idea of the problem you may encounter.. Do you know Zeno's Paradox?
.

And as we all know, the guy will never catch up to the tortoise, because the distance between them will reduce to half, and half again, ad infinitum and ad nauseam...

.

But Zeno's Paradox was so last millenium, duh...

.

Sorry, I just had to do that. I was wondering how Valley Girl talk would sound like in the middle of a Math conversation, but really, I digress...

.

So, back to the Math Problem I was talking about earlier. This is known as the Level-k Thinking Problem, but I shall rename it, The Average Game's Problem. Go, Google or Wiki it if you're free, but let me present the problem in the most entertaining way possible...

.

Remember The Kindergarten Teacher's Addition Problem. Well, since you did a great job solving that puzzle, the Principal decided to transfer you to the local Primary School. As your luck would have it, you wound up teaching Maths again. Well, the Maths still isn't too hard and all in all, you're pretty contented. So, one day, you were teaching averages (Average, as in, the average of 5, 7 and 9 is (5 + 7 + 9)/3 = 7) , and to reinforce the concept, you came up with a game...

.

The rules were simple, everyone in the class would pick a number from 0 to 100, and the person whose chosen number is closest to the HALF the average of all the numbers picked will win a nice gift... But in the middle of the game, you realised that you had just created a very big problem... What was it?

.
PS: Assume that the kids in your class are actually quite intelligent....