Pigeon hole principle?

Question

A guy reads a book. He read for 81 hours last 10 days. Prove that there has been two consecutive days when he read for at least 17 hours.

81 hours / 10 days equals 8,1 hour a day. 2 * 8,1 = 16,2. It isn't 17 hours or I'm missing something. Any suggestions please?

It appears the problem statement assumes that you cannot read a boo kfor fractional hours ... (which is of course nonsens, but for the sake of the problem statement ...) — Hagen von Eitzen, Feb 19 '14 at 18:56
Are fractional hours allowed? Perhaps the problem is looking at this from the Natural numbers only, so he can't read 8.1 hours but would either read 8 hours or 9 hours. — JB King, Feb 19 '14 at 18:56

Asinomás · Answer 1 · 2014-02-19T21:49:35.657

1

Suppose the hours are undivisible (like marbles)

First we divided the ten days into 5 pairs of 2 consecutive days(we call these 5 consecutive pairs: bins). since $\frac{81}{5}>16$ then by the pidgeonhole principle at least one of these 5 bins has 17 or more days.( since if all of these had 16 or less hours we would have at most 80 hours in the whole period of 10 days).

edited Feb 19 '14 at 21:49

answered Feb 19 '14 at 20:44

Asinomás

105,651

You didn't even use that definition... – chubakueno Feb 19 '14 at 20:49
Which definition? – Asinomás Feb 19 '14 at 20:51
You said you would call them pair bins, but that name wasn't used at all.(The proof is ok, though). You just say bins, that's all. It is just that I don't think that defining a thing just to use it half a time is the best thing to do. – chubakueno Feb 19 '14 at 20:54
Oh, ok, ok thanks, i'll fix it after lunch, thanks for input anyways, – Asinomás Feb 19 '14 at 20:57
fixed, I now use the term bin. – Asinomás Feb 19 '14 at 21:49

Pigeon hole principle?

1 Answers1