On a fairy-tale island, dragons and cyclopes live. All dragons are red, three-headed, and two-legged. All cyclopes are brown, one-headed, and two-legged. Cyclopes have one eye in the middle of their forehead, and dragons have two eyes on each head. Together, the cyclopes and dragons have 42 eyes and 34 legs.

How many dragons and how many cyclopes live on the island?

(M. Petrová)
Since both cyclopes and dragons are bipedal, the total number of these creatures is $17(34: 2=17)$.

If all the creatures were cyclopes, they would have a total of 17 eyes. This is 25 fewer than the actual number $(42-17=25)$.

Each dragon has 5 more eyes than any cyclops, so there are 5 dragons among the creatures $(25: 5=5)$. The remaining 12 creatures are cyclopes $(17-5=12)$.

Another solution. The problem can also be solved by trying out possibilities: there are a total of 17 creatures on the island $(34: 2=17)$, among which there are at most 7 dragons $(42: 6=7)$.

In the following table, we list the number of cyclopes $(k=17-d)$ and the total number of their eyes $(6d+k=5d+17)$, which should be 42, depending on the number of dragons $(d)$:

| dragons | 1 | 2 | 3 | 4 | $\mathbf{5}$ | 6 | 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| cyclopes | 16 | 15 | 14 | 13 | $\mathbf{12}$ | 11 | 10 |
| total eyes | 22 | 27 | 32 | 37 | 42 | 47 | 52 |

The only solution that fits is marked in bold.

Note. With the notation used in the description of the above table, the number of dragons can be determined as the solution to the equation $5d+17=42$ (which also corresponds to the reasoning in the first solution to the problem).

Testing can also be based on a different principle, without needing to know the number of creatures on the island (but considering the number of legs).

Evaluation. 2 points for partial insights (e.g., the total number of creatures or the maximum number of dragons); 3 points for solving the problem; 1 point for completeness and quality of the commentary.