Friends Jarda, Premek, and Robin were playing marbles. Jarda wasn't doing well, so after the game, he had the fewest marbles of all. The boys felt sorry for him, so Robin gave Jarda half of all his marbles, and Premek gave him a third of his. Then Jarda had the most marbles, so he returned seven marbles to each of his friends. After these exchanges, they all had the same number, which was 25 marbles.

How many marbles did Jarda have after the game (before the exchanges)?

(M. Petrová)
We can solve the task advantageously from the end:

- Before the second (last) exchange round, Premek and Robin had seven marbles less, while Jarda had 14 more. So Premek and Robin had $25-7=18$ marbles, while Jarda had $25+14=39$.
- Before the first exchange round, Robin had double the number of marbles (he gave half to Jarda, leaving him with half), and Premek had three halves of the marbles (he gave a third to Jarda, leaving him with two-thirds). So Robin had $2 \cdot 18=36$ marbles and Premek had $\frac{3}{2} \cdot 18=27$ marbles.
- The total number of marbles remained constant throughout the redistribution. The sum after the second, respectively, first exchange was $25+25+25=18+18+39=75$ marbles. Before the first exchange (after the game), Robin had 36 and Premek had 27 marbles. So after the game, Jarda had

$$
75-36-27=12 \text { marbles. }
$$

Note. The visualization of the previous considerations, or the verification of the results, might look like this:

| states | Robin | Premek | Jarda | exchanges |
| ---: | :---: | :---: | :---: | :--- |
| after 2nd exchange | 25 | 25 | 25 |  |
|  |  | $\uparrow 7$ | $\downarrow 7$ | Jarda to Premek |
|  | $\uparrow 7$ |  | $\downarrow 7$ | Jarda to Robin |
| after 1st exchange | 18 | 18 | 39 |  |
|  | $\downarrow 18$ |  | $\uparrow 18$ | Robin to Jarda |
|  |  | $\downarrow 9$ | $\uparrow 9$ | Premek to Jarda |
| after the game | 36 | 27 | 12 |  |