Chess Anyone

How many rectangles of any size are on an 8\times 8 checker board?
Source: mathcontest.olemiss.edu 2/8/2010

SOLUTION
A smaller chessboard 
4\times 4

How many 1\times 2 rectangles in a 4\times 4 chessboard?
First, we draw a 1\times 2 rectangle called the base rectangle at the upper left corner of the chessboard. We notice that there are 2 empty units under it. We move the base rectangle down one unit at a time to form two additional 1\times 2 rectangles for a total of three. The pattern is 1+\left (4-2\right )=3.
Likewise there are 3 empty units to the right of the base rectangle. We move it to the right one unit at a time to form 3 additional 1\times 2 rectangles for a total of four. The pattern is 1+\left (4-1\right )=4.
In general, let x\times y be a rectangle in a n\times n chessboard, where x denotes the horizontal dimension and y denotes the vertical dimension. The number of possible rectangles is
\left [1+\left (n-x\right )\right ]\left [1+\left (n-y\right )\right ]
Examples
Number of possible 1\times 2 rectangles in a 4\times 4 chessboard; x=1;y=2;n=4
\left [1+\left (4-1\right )\right ]\left [1+\left (4-2\right )\right ]=4\times 3=12
Number of possible 2\times 1 rectangles in a 4\times 4 chessboard; x=2;y=1;n=4
\left [1+\left (4-2\right )\right ]\left [1+\left (4-1\right )\right ]=3\times 4=12
Notice that there are as many 1\times 2 rectangles as 2\times 1 rectangles.
A bigger chessboard 8\times 8
Let’s use the general formula to calculate all possible rectangles of any size in an 8\times 8 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 x and the columns represent the vertical dimension y.
Examples
Number of 1\times 1 rectangles
\left [1+\left (8-1\right )\right ]\left [1+\left (8-1\right )\right ]=8\times 8=64
Number of 1\times 2 rectangles
\left [1+\left (8-1\right )\right ]\left [1+\left (8-2\right )\right ]=8\times 7=56
Total number of rectangles of any size in an 8\times 8 chessboard
288+252+216+180+144+108+72+36=1296

Answer: 1296

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , . Bookmark the permalink.

One Response to Chess Anyone

  1. Nick says:

    You could also argue that a rectangle of width n has 9-n 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!

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