Problems

a) In the construction in the figure, move two matches so that there are five identical squares created. b) From the new figure, remove 3 matches so that only 3 squares remain.

How many squares are shown in the picture?

Cut the shape (see the figure) into two identical pieces (coinciding when placed on top of one another).

On a \(100 \times 100\) board 100 rooks are placed that cannot capturing one another.

Prove that an equal number of rooks is placed in the upper right and lower left cells of \(50 \times 50\) squares.

On the grid paper, Theresa drew a rectangle \(199 \times 991\) with all sides on the grid lines and vertices on intersection of grid lines. How many cells of the grid paper are crossed by a diagonal of this rectangle?

In the \(6\times7\) large rectangle shown below, how many rectangles are there in total formed by grid lines?

Imagine a \(5\times6\) rectangular chocolate bar, and you want to split it between you and your \(29\) closest friends, so that each person gets one square. You repeatedly snap the chocolate bar along the grid lines until the rectangle is in \(30\) individual squares. You can’t snap more than one rectangle at a time.

The diagram shows a couple of choices for your first two snaps. For example, in the first picture, you snap along a vertical line, and then snap the left rectangle along a horizontal line.
How many snaps do you need to get the \(30\) squares?