TITLE: Your friend has given you his list of 115 best Doctor Who episodes in order of greatness
QUESTION [7 upvotes]: Your friend has given you his list of 115 best Doctor Who episodes (in
order of greatness). It turns out that you have seen 60 of them. Prove
that there are at least two episodes you have seen that are exactly four
episodes apart on your friend’s list

REPLY [2 votes]: I like the $\bmod 4$ answers,
but here’s another alternative. 
Partition the episodes into $14$ blocks of eight consecutive episodes each, plus three extra episodes. At least $57$ of your episodes are among the blocks of eight. 
By the Pigeonhole Principle, one of the blocks must include at least five of your episodes. Use either $\bmod 4$ arithmetic or brute force to show that five episodes in a block of eight must include two episodes exactly four places apart.