How many rectangles of any size are on an checker board?
Source: mathcontest.olemiss.edu 2/8/2010
SOLUTION
A smaller chessboard
How many rectangles in a chessboard?
First, we draw a rectangle called the base rectangle at the upper left corner of the chessboard. We notice that there are empty units under it. We move the base rectangle down one unit at a time to form two additional rectangles for a total of three. The pattern is .
Likewise there are empty units to the right of the base rectangle. We move it to the right one unit at a time to form 3 additional rectangles for a total of four. The pattern is .
In general, let be a rectangle in a chessboard, where denotes the horizontal dimension and denotes the vertical dimension. The number of possible rectangles is
Examples
Number of possible rectangles in a chessboard;
Number of possible rectangles in a chessboard;
Notice that there are as many rectangles as rectangles.
A bigger chessboard
Let’s use the general formula to calculate all possible rectangles of any size in an chessboard as follows
1 
2 
3 
4 
5 
6 
7 
8 
Number of rectangles 

1 
64 
56 
48 
40 
32 
24 
16 
8 
288 
2 
56 
49 
42 
35 
28 
21 
14 
7 
252 
3 
48 
42 
36 
30 
24 
18 
12 
6 
216 
4 
40 
35 
30 
25 
20 
15 
10 
5 
180 
5 
32 
28 
24 
20 
16 
12 
8 
4 
144 
6 
24 
21 
18 
15 
12 
9 
6 
3 
108 
7 
16 
14 
12 
10 
8 
6 
4 
2 
72 
8 
8 
7 
6 
5 
4 
3 
2 
1 
36 
The rows represent the horizontal dimension and the columns represent the vertical dimension .
Examples
Number of rectangles
Number of rectangles
Total number of rectangles of any size in an chessboard
Answer:
You could also argue that a rectangle of width n has 9n possible horizontal placements, so the total number of horizontal placements for all rectangle widths is 8+7+6+5+4+3+2+1 = 36. Similarly for vertical placements, giving 36*36 = 1296 rectangles!