Problem 885

Let the number thought of by András and Béla be $a$ and $b$, respectively. At the beginning of the game, András knew $a$, knew that $b$ was a positive integer and either $b=a-1992$ or $b=a+1992$; similarly, Béla knew that $a$ was a positive integer and either $a=b-1992$ or $a=b+1992$. Both of them had to choose the correct one from two possibilities.

I. Let's examine why András couldn't guess $b$ at first. If $a \leq 1992$, it would have been easy for András: $b=a-1992$ is not possible because $a-1992 \leq 0$, and $b$ is a positive integer. If, however, $a>1992$, then both $a-1992$ and $a+1992$ are positive integers, so in this case, András could not decide. Therefore, the reason András couldn't guess $b$ at first is that $a>1992$.

II. Then Béla started to think. Similarly, he realized that $a>1992$, but this information was not enough for him. Why? Because his number was greater than $2 \cdot 1992=3984$. If $b \leq 3984$, then $a=b-1992$ would not have been possible, since $b-1992 \leq 3984-1992=1992$. In this case, however, András and Béla would have guessed each other's number in the first round.

If, however, $b>3984$, then both $b-1992$ and $b+1992$ are greater than 1992, so Béla could not guess $a$.

The only reason, therefore, for Béla to "pass" was that $b>3984$.

III. The third time, András could have guessed Béla's number by excluding one of the cases between $b=a-1992$ and $b=a+1992$. Clearly, he also realized that $b>3984$, and this could have been enough for him because his number was not greater than $3 \cdot 1992=5976$.

If $a>5976$, then András could not have guessed Béla's number, because both $a-1992$ and $a+1992$ would have been greater than 3984. Thus, $a \leq 5976$, so András could exclude the case $b=a-1992$, because then $b=a-1992 \leq 3984$.

Therefore, András's number was not greater than 5976, and from this, he concluded that Béla's number was the larger one: $b=a+1992$.

IV. If both had thought of a number one greater, that is, András thought of $(a+1)$ and Béla thought of $(b+1)$, then

- András would not have guessed $(b+1)$ at first, because $(a+1)>1992$;

- Béla would not have been able to decide in the second round, since $(b+1)>3984$;

- András, according to his conclusion, would not have been able to decide in the third round either, so $(a+1)>5976$.

We have concluded that $a \leq 5976$, $b=a+1992$, and $(a+1)>5976$. This is only possible if $a=5976$ and $b=7968$.