The teacher gave the children a task to multiply the number written on the board by three and add the number seven to the result. Kamča solved the task correctly. Her friend Růža, however, calculated a different problem: she multiplied the given number by seven and added the number three to the result. Růža's result was 84 greater than Kamča's.

Which number was written on the board?

(M. Petrová)
Let's consider what would happen if the girls added the same numbers to their different intermediate results. That is, let's consider the situation where Růža, instead of adding 3 to the sevenfold of the given number, added 7 like Kamča (or the situation where Kamča, instead of adding 7 to the triple of the given number, added 3 like Růža):

In such a case, the difference in the results of the girls would increase by 4, i.e., to $84 + 4 = 88$. At the same time, this difference would correspond to four times the number on the board (Růža multiplied by seven and Kamča by three). The number on the board was 22, because $88 \div 4 = 22$.

Evaluation. 2 points for auxiliary considerations or transformations; 2 points for the result; 2 points for the quality of the commentary.

Note. If we denote the unknown number on the board by $n$, then Kamča's result was $3n + 7$, while Růža's was $7n + 3$. The previous thoughts can then be written as follows:

$$
\begin{aligned}
(7n + 3) - (3n + 7) & = 84, \\
(7n + 7) - (3n + 7) & = 88, \\
4n & = 88, \\
n & = 22.
\end{aligned}
$$

The problem can also be solved by systematically testing possibilities and comparing Kamča's and Růža's results.