How many rectangles are there on the chess board?

Question

Simple one.

Answer 1

If you have an a*b rectangle (and a is not equal to b) it can be found 2*(9-a)*(9-b) (you can turn each rectangle to form a b*a rectangle

For example, a 2*3 rectangle can be found 2*(9-2)*(9-3) =84
A 6*7 rectangle can be found 2*(9-6)*(9-7) = 12

If a =b (you have a square) then that square can be found
(9-a)(9-a) times
A 1*1 square can be found (9-1)*(9-1) =64
a 4*4 square can be found (9-4)*(9-4) =25

Now you need to sum up every possibility

First calculate rectangles with the shortest side =1
1*1, 1*2, 1*3 thru 1*8
64 +2*8*7 +2*8*6 +2 *8*5.... +2*8*1
64 +2*8(7 +6 +5 ...+1)

Now calculate rectangles with the shortest side =2
2*2, 2*3, 2*4 thru 2*8
49 +2*7*6 +2*7*5 + 2*7*4 +...2*7*1)
49 +2*7(6 +5 +4 ...+1)

The first number is a perfect square

And the series is of the form
2*(n+1)*(1 +2 +3 +...n )
n*(1+n)^2


So the total number of rectangles is the sum of the perfect squares (1 +4 +9 +...64)
with (1*2^2 +2+3^2 +3+4^2 +...7*8^2)

the sum of the perfecxt squares is 204

204 +4 +18 +48 +100 +180 +294 +448 =1296

Answer 2

Total number of squares=204
Total number of squares + rectangles= 1296
Therefore no: of rectangles = 1296- 204=1092

Answer 3

Well since all squares are rectangle, A chess board has 8 squares to a side (4 of each color).
So 8x8=64
But since the board itself is a square you have 65 total.

This is assuming you are not allowed to combine squares on the board to make more rectangles.

Answer 4

There are no rectangles on any chess board. They only have squares.

Answer 5

8 wide by 8 deep equals 64 squares.

Answer 6

None of the answers so far given are correct. There are many different rectangles from 1x1 to 2x1 to.... to 7x8 to 8x8.

I could tell you the correct answer, but you'd have to email me, and I'd have to charge you a lot of money. :-) :-)
.

Answer 7

(2080) ^2 = 4,326,400

where 2080 represents (64+63+62 +... + 1)^2

Answer 8

none, there are only squares.