Problems

Sometimes life can make us do the craziest of things. In this problem you just need to find out how one can cut an $8\times 8$ chessboard into 20 pieces each having the same perimeter and consisting of a whole number of cells.

Draw 16 diagonals inside some cells of a $5\times 5$ square in such a way that no two of these diagonals share any points.

Jennifer draws a hexagon, and a line passing through two of its vertices. It turns out one of the figures in which the original hexagon is divided is a heptagon. Show an example of a hexagon and a line for which it is true.

Can Jennifer draw an octagon and a line passing through two of its vertices in such a way that this line cuts a 10-gon from it?

(a) Can one fit 4 letters “T” (see the picture below) in a $6\times 6$ square box?

We do not allow any overlappings to occur.

(b) Can we fit them in a square with smaller side length?

Liam saw an unusual clock in the museum: the clock had no digits, and it’s not clear how the clock should be rotated. That is, we know that $1$ is the next digit clockwise from $12$, $2$ is the next digit clockwise from $1$, and so on. Moreover all the arrows (hour, minute, and second) have the same length, so it’s not clear which is which. What time does the clock show?