Problem with the application of the pigeonhole principle.

Question

A football team plays at least one match per day in a month of $30$ days , but no more than $45$ matches in that month. Is it true that in some consecutive days in the month, the team will play exactly $14$ matches ?

Answer 1

Let $a_k$ denote the number of football matches the team has played after $k$ days. Since the team plays at least one match per day, the sequence $\{a_k\}$ is strictly increasing. Moreover, since the team plays at most $45$ matches in the month of $30$ days, $a_{30} \leq 45$. Consider the sequence defined by $b_k = a_k + 14$. Then the sequence $\{b_k\}$ is also strictly increasing. Moreover, $b_{30} \leq 45 + 14 = 59$. Consider the union of the sequences $\{a_k\}$ and $\{b_k\}$. It has $60$ terms, each of which is a positive integer which does not exceed $59$. Hence, by the Pigeonhole Principle, there exist $i, j$, with $i > j$, such that $a_i = b_j = a_j + 14$, which means that there is a stretch of $i - j$ consecutive days in which the team plays exactly $14$ matches.