There are $m+n+1$ cards numbered $1,2,\ldots m+n+1$. Participants A and B respectively get $m$ and $n$ cards. Meanwhile, they only know what they get. The remaining card is face down on the desk. Starting with A, participants A and B take turns doing one of the following:
- Say a number. If the other participant has the card, he should show the card.
- Guess the remaining 1 card. If it is correct, the participant wins, otherwise he loses. In both cases, end the game.
What is the optimal strategy for each participant? What is the probability of each participant to win the game?