[ Processes and operations ] $]$ [equations in integers]

Shagreen leather fulfills wishes, but after each wish, its area decreases: either by 1 dm² in the ordinary case, or by half - if the wish was cherished. Ten wishes reduced the area of the leather threefold, the next few - sevenfold, and after a few more wishes, the leather disappeared altogether. What was the original area of the leather?
Let the original area of the skin be $S$ dm² ($S>0$). After the area was reduced threefold and then sevenfold, it became $S / 21$ dm². This number must be an integer; otherwise, it would be impossible to get 0 by dividing by 2 and subtracting 1.

Consider the first 10 wishes. Among them, there cannot be more than one ultimate wish, since after them, the area decreased threefold, and $S / 4 < S / 3$. Consider two cases.

1) There were no ultimate wishes. Then $S-10=S / 3$, so $S=15$. But this number is not divisible by 7, which contradicts the condition of the area being reduced sevenfold.
2) Suppose there were $k$ ordinary wishes first, then an ultimate wish, and then another 9 - $k$ ordinary wishes. Then $1 / 2(S-k)-(9$ $-k)=S / 3$, from which $S=3(18-k)$. Considering that $k$ is an integer from 0 to 9, we find the only value of $k$ for which $S$ is divisible by 7: $k=4$. Therefore, $S=42$.

## Answer

42 dm $^{2}$.