In a school playground, n children are standing so that the distances between ea

Question

In a school playground, n children are standing so that the distances between each pair of children are all distinct. Each child is holding a ball. When their teacher blows a whistle, each child simultaneously throws his or her ball to the closest child. If n is odd and n is greater than 3, prove that there is now at least one child without a ball.

I understand that since the distances are distinct there must be one child who is standing the furthest distance away and therefore must be left without a ball. But mathematically how can this be represented?

Explanation / Answer

Let there be n students numbered from 1,2,3,....n standing in a line. It can be clearly understood that each student will either be in possession of either[0 or 1 or 2] balls. It is clear that if atleast one of the student has 2 balls, then atleast one student will have 0 balls or is left out.

Step:1 On the blow of the whistle, we know that 1 has no option but to throw it to 2. Similarly n has to throw to n-1.

If 1-2 distance is lower than that of 2-3 then 1 gets the ball or else is left out. Similar is the case with (n-1)-(n-2) and (n-1)-n.

If either of '1' or 'n' doesn't get the ball then end of the problem.

If both '1' and 'n' are in possession of balls i.e.(1 gave to 2; 2 gave to 1 & n-1 gave to n and n gave to n-1), then '1' has 1 ball and 'n' has 1 ball.

Now '3' and 'n-2' can either have 0 or 1 balls. If it is 0 then end of the problem, else '3' and 'n-2' will have to get the balls from '4' and 'n-3' respectively(since '2' gave to '1' and 'n-1' gave to 'n')

You can see a pattern emerging. Now '4' and 'n-3' can either have 1 or 2 balls(but not 0 ( since '3' has given to '4' and 'n-3' has given to 'n-2'.)).

You can see pairs exchanging. '1' to'2' and '2' to '1' ; '2' to '3' and '3' to '4'. similarly 'n' to 'n-1' and 'n-1' to 'n' ; 'n-2' to 'n-1' and 'n-1' to 'n-2'.

Now the middle is left out. He can't have the ball.

The problem can end at any stage. Ultimate step is the middle man not having the ball.

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