Four bronze weights together weigh 40 kilograms; with their help, any whole number weight from 1 to 40 kilograms can be measured; what are the weights?
Let the values of the four bronze weights be $x, y, z$, and $u$. Then, every number from 1 to 40 can be expressed in the following form according to the problem.

$$
a x+b y+c z+d u
$$

where the quantities $a, b, c$, and $d$ can only take the values -1, 0, and 1, because each weight can appear at most once, not at all, or once (in the latter case, when the weight is placed on the same pan as the object being measured) in any single measurement. The values of $x, y, z$, and $u$ can thus be considered as consecutive powers of a base number in a number system, but one in which only three digits appear. The base number of this number system can therefore be nothing other than 3, and thus,

$$
x=3^{0}=1, \quad y=3^{1}=3, \quad z=3^{2}=9, \quad u=3^{3}=27
$$

For example: $6=1 \cdot 9+(-1) \cdot 3 ; \quad 32=1 \cdot 27+1 \cdot 9+(-1) \cdot 3+(-1) \cdot 1$. etc.