How many rectangles on the average chess board?

Answer 1

Rectangles made up of squares on the Chess board come in the 28 "flavors" listed here: 1x2, 2x3, 3x4, 4x5, 5x6, 6x7, 7x8 1x3, 2x4, 3x5, 4x6, 5x7, 6x8 1x4, 2x5, 3x6, 4x7, 5x8 1x5, 2x6, 3x7, 4x8 1x6, 2x7, 3x8 1x7, 2x8 1x8 Need another way to see it? Okay, try this: 1x2, 1x3, 1x4, 1x5, 1x6, 1x7, 1x8 2x3, 2x4, 2x5, 2x6, 2x7, 2x8 3x4, 3x5, 3x6, 3x7, 3x8 4x5, 4x6, 4x7, 4x8 5x6, 5x7, 5x8 6x7, 6x8 7x8

Answer 2

... on a regular chessboard? Well, as the chessboard contains 8x8 small squares, we see that the smallest possible rectangle has size 1x2 and largest ones have size 7x8. (We do not count squares as rectangles.)

* Rectangles of size 1x2: If we align this rectangle horizontally then on one 8-square row there are 7 such rectangles, on 8 rows there are 56 rectangles. Vertically, also 56 rectangles. In total 112 rectangles.

* Similarly for rectangles 1x3, only for each row there are 6 rectangles. So in total (6x8)x2 = 96 rectangles.

* It is better to generalize a formula for rectangles size 1xn where n is from 2 to 8. Total number of 1xn rectangles is: (9-n)x8x2 rectangles.

* Hence the total number of rectangles having either side of length 1:

(7 x 8 x 2) + (6 x 8 x 2) + ... + (1 x 8 x 2) = (7 + 6 + ... + 1) x 8 x 2 = ½(7 x 8) x 8 x 2 = (7 x 8) x 8 = 56 x 8.

In a similar way, you can calculate total number of rectangles having one size of length 2 and the other side longer than 2:

(6 x 7 x 2) + (5 x 7 x 2) + ... + (1 x 7 x 2) = (6 + 5 + ... + 1) x 7 x 2 = ½(6 x 7) x 7 x 2 = (6 x 7) x 7 = 42 x 7.

and total number of rectangles having one size of length 3 and the other side longer than 3:

(5 x 6) x 6 = 30 x 6.

... rectangles having one size of length 4 and the other side longer than 4:

(4 x 5) x 5 = 20 x 5.

... rectangles having one size of length 5 and the other side longer than 5:

(3 x 4) x 4 = 12 x 4.

... rectangles having one size of length 6 and the other side longer than 6:

(2 x 3) x 3 = 6 x 3.

... rectangles having one size of length 7 and the other side longer than 7: -- that is the rectangle 7 x 8:

(1 x 2) x 2 = 2 x 2.

The final answer is the sum of all results in boldabove.

(Well this problem can be solved faster using combinatorics or some programming but I would think you need some details otherwise you must have calculated yourself.)

Answer 3

It depends how big you want the rectangles to be ! A chess board consists of 64 squares, arranged in eight rows of eight. The smallest rectangle would be a 1 x 2 - which would yield 32 in total. However - if you made the rectangles 2 x 3, you would get 10 rectangles (with 4 squares left over).

Answer 4

There are no rectangles within the chessboard itself but there are 64 squares in a chessboard .

Answer 5

204 squares and 1,296 rectangles. That's a total of 1,296 rectangles and squares because squares are rectangles.

Answer 6

Zero, but there are 64 squares.

Answer 7

How many squares there are?

64

Answer 8

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