Recently, the following game has become popular: $A$ says to $B$: "Write down your shoe size (as a whole number only!), add 9 to it. Multiply the resulting number by 2, and then add 5. Multiply the resulting sum by 50, and add 1708. Subtract your birth year from this sum. Tell me the final result!" $B$ (who has been calculating silently and unseen) replies: 4720. To which $A$ responds: "Your shoe size is 38, and you are 14 years old!" $A$ also did the same with a person $C$, except that $C$ was asked to add 7 instead of 9 to the shoe size, and to add 1711 instead of 1708 to the 50 times the sum. $C$'s final result: 5059. What shoe size does $C$ wear, and how old is $C$? What is the explanation for this game?
Let's denote the shoe size by $x$ and the age by $y$ years. The birth year is therefore $1952-y$.

Perform the operations commanded by $A$: $[(x+9) \cdot 2+5] \cdot 50+1708-(1952-y)=(x+9) \cdot 100+250+1708-1952+y=$ $100(x+9)+(y+6)$.

The result is a 4-digit number, the first half of which, as a 2-digit number, is the shoe size increased by 9, and the second half is the age increased by 6. Therefore, from 4720, $A$ can conclude that the shoe size is $47-9=38$ and the age is $20-6=14$ years.

In general, if we add $a$ to the shoe size and $(n-250+b)$ after multiplying by 50, where $n$ is the current year (in this example, 1952), then

$$
\begin{gathered}
{[(x+a) \cdot 2+5] \cdot 50+(n-250+b)-(n-y)=} \\
=100(x+a)+250+n-250+b-n+y=100(x+a)+(y+b)
\end{gathered}
$$

In the case of $C$, $a=7$ and $b=9$, and thus from 5059, it follows that the shoe size is $50-7=43$ and the age is $59-9=50$ years. $a$ and $b$ are integers, either positive or negative, but it must be ensured that $x+a \geq 0$. If we do not insist on a 4-digit number, then $a$ does not need to be limited from above, but with $b$, we must ensure that $y+b$ is not negative and not a 3-digit number, i.e., $0 \leq y+b \leq 99$.

Imre Szlanka (Aszód, Petőfi g. II o. t.)