chess board

Question

number of rectangles in a chess board that are not sqaure

a)1296

b)1092

c)1096

d)1292

Comments

b) 1092

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how?

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https://math.stackexchange.com/questions/178693/analysis-of-how-many-squares-and-rectangles-are-are-there-on-a-chess-board

Answer 1

in the chessboard, There are 9 horizontal lines and 9 vertical lines. Choose two distinct horizontal lines, and two distinct vertical lines.  (any rectangle determines a pair of horizontal lines and a pair of vertical lines)

so # rectangle = ⁹C₂ * ⁹C₂  = 1296

u can see that there are 8² small 1×1  squares, 7²   2×2 squares, and so on down to 1²     1×1 squares,,, u need to add to find total # squres

so # squres = 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8²  =  204

number of rectangles in a chess board that are not sqaure = 1296 - 204 = 1092

Comments

Very well, great job.

Answer 2

Explanation:-

No of Squares :- 

Dimension	No of Squares
1X1	64
2X2	49
3X3	36
4X4	25
5X5	16
6X6	9
7X7	4
8X8	1

So no of squares = 204

No of Rectangles :- 

In chess board we have 9 vertical lines and 9 horizontal line, now to construct rectangle of any dimension we have to take 2 horizontal line and 2 vetical line, which can be done in

= C(9,2) * C(9,2)

= 36 * 36

= 1296

To get total no of Rectangles which are not square simply subtract total no of squares from total no of rectangle.

= Total no of rectangles - total no of squares

= 1296 - 204 

= 1092

So, answer is 1092