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# counting question

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How many different rectangles can be drawn on an 8*8 chess board?

Aug 16, 2020

#1
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First of all, let's see how many one-row rectangles there are.

1 x 1 rectangles = (9–1)*(9–1) = 64

1 x 2 rectangles = (9–1)*(9–2) = 56

1 x 3 rectangles = (9–1)*(9–3) = 48

1 x 4 rectangles = 5 * 8 = 40

1 x 5 rectangles = 4 * 8 = 32

1 x 6 rectangles = 3 * 8 = 24

1 x 7 rectangles = 2 * 8 = 16

1 x 8 rectangles = 1 * 8 = 08

Total = 64+56+48+40+32+24+16+8 = 288 1-row rectangles!

There's a pattern here :D

Total one row rectangles = 8*1+8*2+…+8*8

= 8 * (1+2+…+8)

= 8 * (8*9/2) Since,n(n+1)/2 = sum of all numbers below n

=8 * 36

Similarly, Two row rectangles = 7 * (1+2+…+8)

=7 * 36

So all rows rectangles = 8*36+7*36+…+1*36

= 36 * (1+2+…+8)

= 36 * 36

It's \(\fbox{1296}\) :)

BTW, this also counts all the squares!

Aug 16, 2020
#2
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Question:   How many different rectangles can be drawn on an 8*8 chessboard?

36 different rectangles ( squares included ) can be drawn on a chessboard. 