Rectangles

To expand slightly on the author's solution, a rectangle in this figure would have to be contained within one of the two 3 x 3 square grids, or it would have to be contained within the bottom row AND include the red square (as a rectangle in the bottom row not containing the red square would have to be contained by one of the 3 x 3 yellow square grids.)

To count the rectangles in the 3 x 3 squares, note that a rectangle is uniquely determined by the two horizontal lines and the two vertical lines that make up its sides; thus we can determine the number of rectangles by counting how many ways we can choose two horizontal and two vertical sides. Since a 3 x 3 square is made up of four horizontal and four vertical lines, each of those two choices can be made ${4 \choose 2}$ ways. Thus there are ${4 \choose 2} \times {4 \choose 2} = 36$ rectangles in each 3 x 3 square, for a total of 72 rectangles.

In the bottom row, the two horizontal sides of our rectangles are already determined; in choosing the vertical sides, to make sure we include the red square, the left side of our rectangle has to be either the left side of the red square or one of the three lines to the left of it, and a similar choice has to be made on the right. Thus we have four choices for a left side and four for a right side, for a total of 4 x 4 = 16 rectangles.

Therefore we have a total of 88 rectangles.

A $3\times 3$ yellow square has $\binom{4}{2} \times \binom{4}{2} = 36$ rectangles. We have two of those, so $36\times 2 = 72$ . Next, we need to calculate how many rectangles crosses the red bridge square. We have 4 choice for the left end as well as the right end, hence there are $4\times4=16$ rectangles over there.

Add'em up we get $72+16=88$ squares.

I counted 19 1X1, 26 1X2 (don't forget the 2 with the red square), 15 1X3, 4 1X4, 3 1X5, 2 1X6 and 1 1X7. all these last 10 are across the bottom using the red square. then 8 2X2 and 8 2 X 3 and the 2 3X3. 19 + 26 + 15 + 4 + 3 + 2 + 1 + 8 + 8 + 2 = 88

Another way how to approach this problem is to count all rectangles in the 3 x 7 grid and subtract the ones that would have the missing tile inside. There is in total ${8}\choose{2}$ * ${4}\choose{2}$ rectangles. To get the number of rectangles with the missing tiles we need to choose one edge line on the left, another on the right, making it 4 x 4 options times the number of bottom and top lines, which is 5 (calculated as ${4}\choose{2}$ - 1). Together it is 2 x 3 x 4 x 7 - 4 x 4 x 5 = 168 - 80 = 88.