Pigeonhole Principle - Consecutive Days

Question

If an athlete trains for 20 days in a 31-day month, how can I prove that he/she will need to train on consecutive days at least once using the Pigeonhole Principle?

Answer 1

For training days to be nonconsecutive, you need at least a number of rest days equal to the number of training days minus one. If $t$ is the number of training days, then $t-1$ is the minimum number of rest days so that every training day is nonconsecutive.

$$t+(t-1) \le 31 \Longrightarrow t\le 16$$

So, since $t>16$, by the Pigeonhole Principle, it must be that at least two training days are consecutive.

Answer 2

Note that in order to not train on consecutive days, each training day must not be followed by another training day. So we are trying to allocate $20$ two-day periods into a month which has $15$ two-day periods. Since $20 > 15$, by the Pigeonhole Principle, we must allocate two of our training days to the same two-day period—meaning we are training on consecutive days.