10-4-1. Anya and Borya are playing rock-paper-scissors. In this game, each player chooses one of the figures: rock, scissors, or paper. Rock beats scissors, scissors beat paper, and paper beats rock. If the players choose the same figure, the game ends in a tie.

Anya and Borya played 25 rounds. Anya chose rock 12 times, scissors 6 times, and paper 7 times. Borya chose rock 13 times, scissors 9 times, and paper 3 times. There were no ties in any of the rounds. How many times could Anya have won? List all possible answers.
Answer: 16.

Solution version 1. Note that, since there were no draws, when Anya chose rock, Borya must have chosen scissors or paper. Anya chose rock 12 times. Borya chose scissors or paper a total of $9+3=12$ times. Therefore, all these cases must have occurred in the rounds where Anya chose rock.

This means that in all the other 13 rounds (when Anya chose scissors or paper), Borya chose rock.

From the rounds where Anya chose rock, she won 9 times: when Borya chose scissors. From the rounds where Borya chose rock, Anya won 7 times: when she chose paper. Therefore, Anya won 16 times.