Josefina the Quail dances by the marsh, using steps of double length — short steps measure $45 \mathrm{~cm}$, long steps $60 \mathrm{~cm}$. Over time, she has worn an oval path, which she dances around repeatedly during long nights. If she repeats three long steps forward and one short step back, then the ninetieth step will bring her exactly back to where she started. If she repeats three short steps forward and one long step back, she also ends up exactly where she started.

Which step will bring Josefina back to the original spot in the second case?

(M. Petrová)
With one large four-step (three long steps forward and one short step back), Josefa moves $135 \, \text{cm} (3 \cdot 60 - 45 = 135)$. Ninety steps consist of 22 large four-steps and two long steps $(90 = 22 \cdot 4 + 2)$. Therefore, Josefa's circuit measures $3090 \, \text{cm} (22 \cdot 135 + 2 \cdot 60 = 3090)$.

With one small four-step (three short steps forward and one long step back), Josefa moves $75 \, \text{cm} (3 \cdot 45 - 60 = 75)$. Forty such four-steps—i.e., 160 steps—move her $3000 \, \text{cm} (40 \cdot 75 = 3000)$, ensuring she does not exceed her original position (before the last backward step, she is at a distance of $3060 \, \text{cm}$). The remaining $90 \, \text{cm}$ she covers in the next two steps $(2 \cdot 45 = 90)$.

Josefa will dance back to her original position on the 162nd step.

Note. To illustrate the situation, we provide some distances (in $\text{cm}$) corresponding to the dance steps used in the second case:

| step | 1 | 2 | 3 | 4 | 5 | 6 | $\ldots$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| distance | 45 | 90 | 135 | 75 | 120 | 165 | $\ldots$ |

| step | $\ldots$ | 158 | 159 | 160 | 161 | 162 | 163 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| distance | $\ldots$ | 3015 | 3060 | 3000 | 3045 | 3090 | 3135 |

From this, it is also clear that different steps correspond to different distances.

Evaluation. 2 points for the length of Josefa's circuit; 2 points for the number of steps in the second case; 2 points for clarity and quality of the commentary.