# Problem on pigeon hole principle

This is a problem based on pigeon hole principle.

A tennis player has three weeks to prepare for a tennis tournament. She decides to play at least one set every day but not more than 36 in all. Show that there is a period of consecutive days during which she will play exactly 21 sets.

Let $r_i$ be the number of sets played up to and including day $i$. So

$0 = r_0 < r_1 < r_2 < ... < r_{21} \leq 36$

So there are 37 pigeonholes - possible values - for the 22 pigeons - the $r_i$.

However, we are looking for pairs $i, j$ with $r_j = r_i + 21$ - since such a pair would tell us that 21 sets were played on days $i+1$ through to $j$. So if we assume there are no such pairs, we can "combine" the pigeonholes $i$ and $i + 21$. So we have the following pigeonholes:

$\{0 \text{ or } 21\}, \{1 \text{ or } 22\}, ..., \{15 \text{ or } 36\}, \{16\}, ... \{20\}$.

And we have 22 pigeons, 21 pigeonholes, giving a contradiction.

This is very similar to the How are the pigeonholes calculated in this pigeon-hole problem?

To answer this specific question - using the same logic and process from the reference answer above:

$1 < r_1 < r_2 < r_3 \ldots < r_{21} \le 36$

Doing $r_i+21$:

$22 < r_1+21 < r_2+21 < r_3+21 \ldots < r_{21}+21 \le 57 \,(=36+21)$

$r_i$ is 21 values and $r_i+21$ is 21 values.

So we have 42 pigeons and 57 pigeonholes.

Edited to correct my calculation of pigeonholes. I then don't see how to solve the question and also note the difference between my number of pigeons/pigeonholes and the other answer.

I also note that if she runs once every day and twice on the last day she will have run 22 times - which is not exactly 21 times.

• It seems to me that there are $57$ pigeonholes, not $41$. Where do you get $41$ from? – Christopher Sep 16 '14 at 11:46
• You are correct - which I think invalidates the logic - back to the drawing board.... – AndyL Sep 16 '14 at 13:37