In which base number systems is it true that the rule for divisibility by four is the same as the rule for divisibility by nine in the decimal system, and the rule for divisibility by nine is the same as the rule for divisibility by four in the decimal system?
In the decimal system, divisibility by 9 depends only on the sum of the digits because (10 - 1) is divisible by 9. Therefore, the base $A$ of the sought number system must be one greater than a number divisible by 4:

$$
A=4 k+1
$$

Divisibility by 4 in the decimal system depends on the number formed by the last two digits, because 10 is not, but 100 is divisible by 4. Therefore, $A$ must not be divisible by 9, but its square must be, so $A$ is divisible by 3 but not by 9.

Among the numbers of the form (1), 9 is the first divisible by 3, and from this point on, increasing the value of $k$ by threes yields numbers divisible by 3, that is,

$$
A=12 n+9
$$

Among these, only those are divisible by 9 that correspond to $n$ values divisible by 3. The sought numbers $A$ are therefore obtained from (2) by substituting for $n$ positive integers not divisible by 3.