Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:

In a standard 52 deck of cards, there are 13 cards of each of four suits. Kevin guesses the suit of the top card, and the top card is revealed and discarded. This process continues till there are no cards remaining.
If Kevin always guesses the suit of which there are the most remaining (breaking ties arbitrarily), prove that he will get at least 13 guesses right.

Solution

Solution:

Imagine that the cards have been given ranks $1, 2, \ldots, 13$ and moreover that within each rank the cards have been sorted in ascending order (i.e. Kevin will encounter $1,2, \ldots, 13$ of hearts in that order).
Then, observe that Kevin will always guess the last card of rank $r$ correctly, for any $r=1, \ldots, 13$ . This completes the proof.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.