# Al has 75 days to master discrete mathematics. He decides to study at least one hour every day, but no more than a total of 125 hours. Assume Al always studies in one hour units. Show there must be a sequence of consecutive days during which he studies exactly 24 hours.

Al has 75 days to master discrete mathematics. He decides to study at least one hour every day, but no more than a total of 125 hours. Assume Al always studies in one hour units. Show there must be a sequence of consecutive days during which he studies exactly 24 hours.
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Step 1
Let ${a}_{k}$ be the number of hours of work Al has done after k days. Then $\left\{{a}_{1},{a}_{2},{a}_{3},\dots ,{a}_{75}\right\}$ is an increasing sequence of distinct positive integers since Al does at least one hour of work each day. Observe that $1\le {a}_{k}\le 125$ since Al does at most 125 hours of work over the 75 days.
Let ${b}_{k}={a}_{k}+24$. Then the sequence $\left\{{b}_{1},{b}_{2},{b}_{3},\dots ,{b}_{75}\right\}$ is also an increasing sequence of distinct positive integers. Observe that $1+24=25\le {b}_{k}\le 149=125+24$.
Step 2
Now consider the union of the two sequences. It consists of 150 numbers that are at least 1 and at most 149. Thus, two of them must be the same. Hence, ${b}_{k}={a}_{k}+24={a}_{j}$ for some j,k. Thus, ${a}_{j}-{a}_{k}=24$, so Al does exactly 24 hours of work from day ${a}_{k+1}$ to day ${a}_{j}$.
This is a clever application of the Pigeonhole Principle that forced me to consult my combinatorics notes.
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