61 hours of sleep over 10 nights, and the pigeonhole principle

Question

A student in MATH 2P71 class slept for 61 hours over 10 nights. Show that on some pair of consecutive nights, he slept at least 13 hours. Assume that the student only sleeps an integer number of hours.

Could someone take me step by step on how I would apply the pigeon principle to this question and achieve the right answer? I am very lost… I understand the principle itself, I just have trouble applying it to the question.

if he sleeps 1 hour per day then he doesn't reach 61 hours in ten days... — C.c, Oct 04 '16 at 04:01

score 0 · Accepted Answer · answered Oct 04 '16 at 04:01

0

Divide the ten nights as follows:

1  2 | 3  4 | 5  6 | 7  8 | 9  0

Since we have five pairs of nights but 61 hours of sleep, by the pigeonhole principle there must be at least one pair of nights with $\left\lceil\frac{61}5\right\rceil=13$ or more hours of sleep. This proves the statement.

answered Oct 04 '16 at 04:01

Parcly Taxel

103,344

Interesting. That solves an even stronger result that there must be an odd to even pair. – fleablood Oct 04 '16 at 06:17
its not true there are $9$ pairs of consecutive nights – Blaise Thunderstorm Dec 22 '16 at 15:09

61 hours of sleep over 10 nights, and the pigeonhole principle

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