The spiral mole is digging a new tunnel: first it heads $10 \mathrm{~cm}$ north, then $11 \mathrm{~cm}$ east, then $12 \mathrm{~cm}$ south, $13 \mathrm{~cm}$ west, etc. (each segment is $1 \mathrm{~cm}$ longer than the previous one, directions repeat according to the given pattern). The mole's colleague maps the work: the start of the tunnel is marked with coordinates $[0,0]$, the first turn with coordinates $[0,10]$, the second turn with $[11,10]$, etc.

Determine the coordinates of the end of the segment that is $100 \mathrm{~cm}$ long.

(I. Jančigová)

Hint. In which direction did the mole dig the segment that is $100 \mathrm{~cm}$ long?
The lengths of segments (in $\mathrm{cm}$) for each direction are:

| direction | 1st round | 2nd round | 3rd round | $\ldots$ | $k$-th round |
| :---: | :---: | :---: | :---: | :---: | :---: |
| north | 10 | 14 | 18 | $\ldots$ | $6+4 k$ |
| east | 11 | 15 | 19 | $\ldots$ | $7+4 k$ |
| south | 12 | 16 | 20 | $\ldots$ | $8+4 k$ |
| west | 13 | 17 | 21 | $\ldots$ | $9+4 k$ |

A segment of $100 \mathrm{~cm}$ was struck by the mole in the 23rd round in the southern direction $(8+4 \cdot 23=100)$.

The coordinates of the ends of these segments are $[11,-2],[13,-4],[15,-6], \ldots$ The first coordinate increases by 2 progressively, while the second decreases by 2.

![](https://cdn.mathpix.com/cropped/2024_04_17_d278c4c03a37be47319bg-1.jpg?height=771&width=791&top_left_y=1579&top_left_x=632)

Generally, in the $k$-th round, the coordinates of the end of the segment in the southern direction are $[9+2 k,-2 k]$. For $k=23$, we get $[55,-46]$, which are the coordinates of the end of the meter-long segment.