We know that a chess game ends when a state called checkmate is reached. Considering games which end in checkmate and not considering games that go on forever, is the number of different possible chess games countable or uncountable? Could someone give me possible hints to attack the problem?
Recall a set $S$ is countable if $|S|$ $\leq |\mathbb{N}|.$
Since there are finitely many possible ways to place figures on the board, one can number them [the ways] with $\mathbb N$. Then it is possible to describe any game with a unique finite (since we consider only the games that were ended) sequence of natural numbers.
Is the set $F$ of finite sequences of natural numbers countable? Yes. Try to build an injection $F\hookrightarrow\mathbb N$. It is usefully to remember about different numeral systems.