$A, B, C$ and $D$ are mathematicians. $D$ asks the others - one of whom is quite amused - about their ages. They decide to share the data determining their ages in a number system different from the decimal system.

$A$ says: "Take four numbers, each differing from the next by 2. Write the smallest followed by the next. The resulting two-digit number in a certain number system gives my age; and this is the same as the product of the two largest numbers."

- Now $B$ states that his age can be found by writing two consecutive numbers side by side (starting with the smaller one), and considering this number as a two-digit number in a number system whose value is exactly the product of the next two consecutive numbers.
- Finally, $C$, the youngest, declares that he can say the same about his age as $A$ did about his own.

$D$ can now determine the ages of $A, B$, and $C$, and also which one of them has an amusing nature.

Determine what it is!
I. Solution: The communications of $A$, $B$, and $C$ are independent of each other - this is not affected by the fact that $C$'s communication matches $A$'s -; among them, only the storyteller's communication provides a connection, that $C$ is the youngest; thus, we can initially deal with the communications separately. Since we are talking about base numbers and digits of number systems, these are implicitly natural numbers.

Let the smallest number used by $A$ be denoted by $a$, and the base by $b$:

$$
a \cdot b + (a + 2) = (a + 4)(a + 6)
$$

where $a + 2$ is also a digit, so $a + 2 < b$. From this,

$$
b = \frac{a^2 + 9a + 22}{a} = a + 9 + \frac{22}{a}
$$

can only be an integer if $a$ is a divisor of 22. Thus,

$$
\begin{aligned}
& a = 1 \text{ gives the } b = 32 \text{ system, } \quad \overline{13} \text{ number, } \quad 35 = 5 \cdot 7, \\
& a = 2 \text{ gives the } b = 22 \text{ system, } \quad \overline{24}, \quad 48 = 6 \cdot 8, \\
& a = 11 \text{ gives the } b = 22 \text{ system, } \quad \overline{(11)(13)}, \quad 255 = 15 \cdot 17 \\
& a = 22 \text{ gives the } b = 32 \text{ system, } \quad \overline{(22)(24)}, \quad 728 = 26 \cdot 28
\end{aligned}
$$

yields the possible ages of $A$ and $C$; the last two are clearly not possible, and since $C$ is the younger of the two, $A$ is 48 years old and $C$ is 35 years old.

Similarly, let the smallest number used by $B$ be denoted by $c$, and the base by $d$:

$$
\begin{gathered}
c \cdot d + (c + 1) = (c + 2)(c + 3), \quad (c + 1 < d) \\
d = \frac{c^2 + 4c + 5}{c} = c + 4 + \frac{5}{c}
\end{gathered}
$$

From this, $c = 1$ gives the 10-based system number $\overline{12}$, $12 = 3 \cdot 4$,

$c = 5$ gives the 10-based system number $\overline{56}$, $56 = 7 \cdot 8$

yields the age of $B$. Both ages are acceptable, but since $C$ is the youngest, $B$ is 56 years old. $B$'s amusement does not lie in providing incorrect data, but in using the decimal system despite the agreement.

Pál Gáti (Pécs, Nagy Lajos g. I. o. t.)

II. Solution: We can also arrive at the above results by trial and error. For $A$ and $C$, taking the smallest number as 1, the abbreviation $\overline{13}$ corresponds to a base number where this represents $5 \cdot 7$, specifically $(5 \cdot 7 - 3) : 1 = 32$ (we subtract the 3 that does not fit into the higher unit from the number $5 \cdot 7$, leaving the base number once). Similarly, from $\overline{24} = 6 \cdot 8$, the base number $(6 \cdot 8 - 4) : 2 = 22$ is obtained, which and the age of 48 are acceptable. On the other hand, the assumptions $\overline{35} = 7 \cdot 9$, $\overline{46} = 8 \cdot 10$, $\overline{57} = 9 \cdot 11$, $\overline{68} = 10 \cdot 12$ would yield a fractional base number, which is not usable; trying larger numbers is unnecessary due to the known maximum human lifespan.

For $B$, we need to perform several similar trials, but only the observed 10 and 56 are acceptable.