Pigeonhole principle: 112 hrs over 12 days, then at least 19 hrs over some consecutive 2 days

Question

The problem I'm working on says:

A basketball player has been training for 112 hours during 12 days. He has trained an integer number of hours every day. Prove that there was two consecutive days where he has trained for at least 19 hours.

I'm following this rationale to try prove it:

As we are interested in two consecutive days we split the 12 days into 6 pairs: {1,2}, {3,4}, {5,6}, {7,8}, {9,10}, {11, 12}

Using a corollary of the Pigeonhole principle we know that the sum of one the pairs is greater than 18, but not 19.

Does this mean that I can't prove it this way?

Answer 1

This can be proved easily. Suppose the statement is NOT true. Then any given pair of consecutive days he trains for at most 18 hours. So he trains a total of at most 18 * 6 = 108 hours. This is a contradiction since we are told that he trains a total of 112 hours.

I hope this helps! :)

Answer 2

Your grouping works just fine. The average over the $6$ groups is $\dfrac{112}{6}$, which is greater than $18$. So at least one of the group sums is greater than $18$. Since all group sums are integers, at least one of these must be $\ge 19$.