The snail Leona fell into the middle of a square grid. She decided to crawl "in a spiral" as indicated in the picture; she will not visit any square twice and will not skip any square.

![](https://cdn.mathpix.com/cropped/2024_04_17_df9261f4862336f3ce8cg-1.jpg?height=489&width=494&top_left_y=835&top_left_x=815)

From the first square to the second, she crawled east, from the second to the third, north, from the third to the fourth, west, from the fourth to the fifth, also west, from the fifth to the sixth, south... In which direction did she crawl from the 81st to the 82nd square?

(M. Petrová)
We can imagine the entire situation as caterpillar Leona crawling over squares. We will track which square $\mathrm{v}$ she ends up on and how many squares she has already crawled over. It is important to determine which of these squares contains the 81st square.

1. round: The square numbered 1 is surrounded by eight adjacent squares. By doing so, she has already crawled over 9 squares, thus covering all the squares in a $3 \times 3$ square (the "starting" square is exactly in the middle). She is now on the square at the southeast vertex of this square. From there, she crawls in the eastern direction.
2. round: Now she has crawled around the previous square. By doing so, she has already crawled over all the squares in a $5 \times 5$ square, and she is again on the square at the southeast vertex of this square. This square, therefore, has the number 25, and she again crawls into the next square in the eastern direction.
3. round: Again, she has crawled around the previous square, so she has already crawled over all the squares in a $7 \times 7$ square. Once again, she ends up on the square at the southeast vertex of this square. This square has the number 49, and she crawls into the next square in the eastern direction.
4. round: Now she has crawled over all the squares in a $9 \times 9$ square and ended up on the square at the southeast vertex of the square. This square has the number 81, and we are interested in which direction she will crawl to the next square. Since she is moving to the next square from here, she must crawl in the eastern direction again.

![](https://cdn.mathpix.com/cropped/2024_04_17_df9261f4862336f3ce8cg-2.jpg?height=794&width=808&top_left_y=240&top_left_x=658)

Another solution. There are enough squares, but Leon's path can still be drawn:

| 65 | -64 | 63 | 62 | 61 | $-60=$ | -59 | 58 | 57 |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 66 | 37 | -36 | 35 | 34 | 33 | 32 | 31 | 56 |  |
| 67 | 38 | 17 | $-16^{*}$ | -15 | -14 | -13 | 30 | 55 |  |
| 68 | 39 | 18 | 5 | -4 | -3 | 12 | 29 | 54 |  |
| 69 | 40 | 19 | $\frac{v}{6}$ | $1-$ | $\rightarrow 2$ | 11 | 28 | 53 |  |
| 70 | 41 | 20 | 7 | $\rightarrow 8$ | $\rightarrow 9$ | $\rightarrow 10$ | 27 | 52 |  |
| 71 | 42 | $21-$ | -22 | -23 | -24 | 25 | $\underset{-26}{\uparrow}$ | 51 |  |
| 72 | 43 | -44 | -45 | -46 | $\rightarrow 47$ | -48 | $\rightarrow 49$ | $\rightarrow 50$ |  |
| 73 | $\rightarrow 74$ | $\rightarrow 75$ | $\rightarrow 76$ | $\rightarrow 77$ | $\rightarrow 78$ | $\rightarrow 79$ | $\Rightarrow 80$ | $\rightarrow 81$ | $\rightarrow 82$ |

Leona crawled from the 81st to the 82nd square in the eastern direction.