The game played by 'M'
The game played throughout the film by 'M' is a so-called 'nim' game, which is not difficult to analyze mathematically. The first player always loses because the second player (regardless of the first player's moves) can always leave a 'safe' or winning position on the table and via a series of safe positions (known to him) he can leave the last card (or match or whatever object) for the first player, who then loses.
What puzzles me is that 'M' in the second of the four games shown in the movie does not play his winning strategy. He still wins, because his opponent makes a mistake, but he could have lost. The game goes as follows:
1-3-5-7 to 0-3-5-7 to 0-2-5-7 to 0-2-5-6 to 0-2-4-6 (all correct, still winning position for 'M') to 0-2-3-6 and then 'M' plays 0-2-3-5, whereas he should have played 0-2-3-1, which is a winning position. His opponent however fails to play 0-2-3-1 and 'M' still wins.
In the third and fourth game 'M' starts (so he could be beaten by a good player), but again his opponents do not know how to beat him.
Does anyone have a clue why 'M' leaves his winning strategy in the second game? Maybe pure arrogance in the face of his incompetent fellow players?