# Pigeonhole Principle

1. Feb 28, 2012

### Dschumanji

1. The problem statement, all variables and given/known data
A modem runs for 300 hours over 15 days. Prove that there exists 3 consecutive days where the modem ran for at least 60 hours.

2. Relevant equations
Pigeonhole principle

3. The attempt at a solution
By the pigeonhole principle, there must exist at least one day where the modem runs for 20 hours. If the modem runs for 20 hours every day for the 15 days, then it must run for 60 hours over any three consecutive days. The least amount of time the modem can run over three consecutive days is 12 hours, which implies that for the other 12 days the modem must run 24 hours a day. Therefore there exists three consecutive days where the modem runs for more than 60 hours in this case.

I feel that those two cases imply that the rest of the cases must also have three consecutive days where the modem runs for at least 60 hours. But I have no way to prove that. I'm thinking this approach is not that good. Is there a better way to tackle this problem using the pigeonhole principle?

2. Feb 28, 2012

### awkward

Consider days 1-3, 4-6, 7-9, 10-12, and 13-15. Can all of these intervals contain less than 60 modem-hours?