There's a problem in my textbook that I'm having difficulties understanding. The solution given skips steps and is hard to decipher. The problem is as follows:
"During a month with 30 days, a baseball team plays at least one game a day, but no more than 45 games. Show that there must be a period of some number of consecutive days during which the team must play exactly 14 games."
I don't know how to arrive at the conclusion of 14 games. Anyone who can help me understand this application of the pigeon-hole principle would be greatly appreciated, thank you.
Best Answer
Let $\{a_i\}_{i = 0}^{30}$ denote the number of games up until and including the $i$-th day of the month (put $a_0 = 0$). Given the conditions of the task, $\{a_i\}$ is an increasing sequence with all members distinct and $0 \leq a_i \leq 45$.
Now, consider the sequence $\{a_i + 14\}_{i = 1}^{30}$ (add $14$ to every member of the original sequence). It is also increasing with all members distinct, but with $14 \leq a_i + 14 \leq 59$. Together these sequences consist of $62$ integers between $0$ and $59$, so by the pigeonhole principle, two of them must be equal.
Since both the sequences $\{a_i\}$ and $\{a_i + 14\}$ have distinct members, there must be indices $j$ and $k$ with $j \ne k$ such that $a_j = a_k + 14$. This means that 14 games were played in the period between the $(k + 1)$-st and the $j$-th day.