Q. Four players - Anand, Karpov, Ivanchuk, and Polgar - participated in a chess competition in which each player played exactly one game with each of the other three players. At the end of the competition the results were as follows :-

1) Karpov won against Ivanchuk, who did not win against Anand.
2) The number of games won by Anand is equal to the number of games won by Ivanchuk.
3) No game ended in a draw.

Then which of the following definitely is true :-

a) Ivanchuk lost to Polgar.
b) Anand lost to Karpov.
c) Polgar lost to Anand.
d) Karpov lost to Polgar.

Ans. The given observations about the individual Round Robin games are as follows:
1) Karpov won against Ivanchuk, who did not win against Anand.
2) The number of games won by Anand is equal to the number of games won by Ivanchuk.
3) No game ended in a draw.

Let us represent Anand, Polgar, Karpov and Ivanchuk by A, P, K and I respectively.

The from the given data we have

Game Winner
A-K
A-P
A-I A
K-P
K-I I
I-P
since Ivanchuk lost both the games against Karpov and Anand from observation 1.

Now from observation 1, Anand has definitely won one game since he beat Ivanchuk. From observation 2, the number of games won by Anand is equal to the number of games won by Ivanchuk. Since Ivanchuk has lost both games, he should at least win the last game so that the number of games won by him is equal to those won by Anand. Also since Anand has won one game, he cannot win games against Karpov and Polgar since Ivanchuk cannot equal him then, after having lost two out of three games. Hence Anand must lose to Karpov and Polgar and Ivanchuk must win against Polgar.

Hence among the given statement b which suggests that Anand lost to Karpov must definitely be true.