Problem 4. (6 points) From a sheet of squares, a rectangle $10 \times 12$ has been cut out. It has been folded several times along the lines of the square grid until a $1 \times 1$ square is obtained. How many pieces can be obtained if this square is cut along the segment connecting

a) the midpoints of two opposite sides; (2 points)

b) the midpoints of two adjacent sides. (4 points)

(Find all possible answers and prove that there are no others.)
Solution: a) Let the cutting be vertical, and in all unit squares, let us draw vertical segments connecting the midpoints of opposite sides. When folding along the lines of the square grid, these segments overlap. Therefore, the cutting is done along these and only these segments. As a result, we get either $10+1=11$ or $12+1=13$ pieces.
![](https://cdn.mathpix.com/cropped/2024_06_03_549d507bf9e35c472301g-2.jpg?height=314&width=1172&top_left_y=1818&top_left_x=521)

b) Note that each unit square will be cut exactly once. This cut can be made along one of the four segments connecting the midpoints of adjacent sides. If we choose one of the four ways to cut any square, the cuts in the other squares are uniquely determined (see the figure). This results in $6 \cdot 7 + 1 = 43$, $5 \cdot 6 + 1 = 31$, $6 \cdot 6 + 1 = 37$, and $5 \cdot 7 + 1 = 36$ pieces, respectively.

The problem was solved by Nikolay Belukhov from Stara Zagora, Nikolay Chipov and Alisa Kovacheva from NPMG, Radostin Bakrdzhiev, Stoyan Stoyanov, Ivo Petkov, Todor Stoyanov, Georgi Simeonov from Burgas, Kremena Marinova from Ruse, and Slavina Kitova from AK Sofia.