Exercise 10. 22 cards bearing the numbers $1,2, \ldots, 22$ are on the table. Alcindor and Benoît take turns removing a card of their choice from the table, until there are none left. Alcindor goes first. Then each calculates the units digit of the sum of their cards. The winner is the one with the highest result.

Does one of the two players have a sure way to win? If so, determine which one.
Solution to Exercise 10: Let's show that Alcindor can win for sure.

First, we note that the outcome of the game is not affected if we replace the values of the cards with their units digit. All cards are then in pairs, except for the cards 1 and 2, which are in triples.

Alcindor chooses the following strategy: he starts by drawing the card 2. Then, each time Benoît takes a card, he chooses a card of the same value if possible; if none are left, he chooses any card.

The game proceeds as follows, from the second move: Benoît and Alcindor take turns picking cards of the same value, until Benoît draws the last 1. Then Alcindor draws a card of any value $v$. Benoît and Alcindor take turns picking cards of the same value, until Benoît draws the last $v$, and so on.

At the end, Alcindor has two "2" cards and one card of each of the other values, while Benoît has two "1" cards and one card of each of the other values. They obtain respective scores of 7 and 6, so Alcindor wins.