Show that some 5 consecutive chairs must be occupied.

Question

A group of 25 people are seated in a row of 30 chairs. Show that some 5 consecutive chairs must be occupied.

Related post: http://math.stackexchange.com/questions/121561/pigeonhole-principle-question It may help. — Ellie, Aug 22 '14 at 00:01

score 7 · Accepted Answer · answered Aug 22 '14 at 00:04

Since there are 25 people and 30 chairs, there are 5 empty seats. Divide the row of 30 chairs into 6 groups of 5 consecutive seats. Since there are 6 groups and 5 empty chairs, at least one of the 6 groups has no empty seats. So in that group with no empty seats, all five consecutive seats are full.

score 0 · Answer 2 · answered Aug 22 '14 at 00:03

0

Place single empty chairs so there's 4 full chairs between them. Obviously they can't be pushed further apart, as a group of 5 full chairs would then exist. How much do you have stuck on the end after you've placed them all?

answered Aug 22 '14 at 00:03

Dan Uznanski

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Show that some 5 consecutive chairs must be occupied.

2 Answers2