## 20. Nicolas plays

## with tin soldiers

Nicolas has the same number of tin Indians, Arabs, cowboys, and Eskimos. After his cousin Sebastian's visit, he indignantly discovered the disappearance of a third of his soldiers. Suppose the number of Eskimos remaining is the same as the number of cowboys that disappeared, and that only two Indians out of every three remain. How many Arabs did cousin Sebastian take?
20. Let $x$ be the number of soldiers of each type that Nicolas had initially, and $y$ be the number of cowboys taken by Sebastian (or the number of remaining Eskimos); in this case, the number of Eskimos taken is $x-y$.

On the other hand, it is known that the number of Indians taken is $x / 3$. Let $z$ be the number of Arabs taken. The total number of soldiers taken is $4 x / 3$, since it constitutes one third of the total number of soldiers. But it is easy to see that it also equals

$$
y+(x-y)+x / 3+z
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-212.jpg?height=497&width=1054&top_left_y=1259&top_left_x=498)

From this, we get the equation

$$
4 x / 3=4 x / 3+z
$$

from which it follows that $z=0$.

Thus, cousin Sebastian did not take any Arabs.