There are 50 rooms in a line. If there are 26 rooms with girls, prove there are two girls exactly 5 rooms apart.

Question

My idea was place 25 girls in into pairs of rooms, and there is no scenario which there are 5 girls apart. Then, if you place one more, by pigeonhole principle, there's a girl exactly 5 away - but I want a more formal way of explaining this. Does anyone have a way? Cheers.

The description of the rooms and girls includes no geometric or topological features, so it seems to be undefined what you mean by "$5$ rooms apart", "$5$ girls apart" and "exactly $5$ away". — joriki, May 23 '13 at 10:39
This is same as http://math.stackexchange.com/questions/40960/prove-that-2-students-live-exactly-five-houses-apart-if — sam_rox, Nov 16 '14 at 13:22

score 5 · Accepted Answer · answered May 23 '13 at 11:00

5

A hint: Color the rooms with five colors according to the remainder mod 5 of the room number.

answered May 23 '13 at 11:00

Christian Blatter

226,825

There are 50 rooms in a line. If there are 26 rooms with girls, prove there are two girls exactly 5 rooms apart.

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