$\underline{\text { Akulich I.F. }}$.

Avant-garde artist Zmey Kletochkin painted several cells on a $7 \times 7$ board, following the rule: each subsequent painted cell must share a side with the previously painted cell, but must not share a side with any other previously painted cell. He managed to paint 31 cells.

Beat his record - paint a) 32 cells; b) 33 cells.

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-09.jpg?height=340&width=346&top_left_y=1614&top_left_x=862)
Do not try to completely color the sides, leave the middle of the sides uncolored.

## Solution

a, b) If we can color 33 cells, then 32 cells can be colored by stopping in time. Three examples where 33 cells are colored are shown in the figure (in fact, there are many more such examples). It is impossible to color more than 33 cells - this has been verified by a computer.
![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-09.jpg?height=344&width=1056&top_left_y=2391&top_left_x=502)

## Answer

a, b) See the figure. Other examples are provided in the solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-10.jpg?height=341&width=348&top_left_y=1&top_left_x=858)

For problem 2002-6-4 b)

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