 # 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. Avery Stewart 2022-07-18 Answered
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.