how many rectangles are there in chess board (including squares) ?show working?

Answer 1

A rectangle can be described by its left, right, down, up borders.

A choice of left and right is a choice of i and j, both between 1 and 8 with i smaller or equal to j.
There's 1+2+...+8=8*9/2=36 possibilities because
if i is 1, j can be 1,2,..,8
if i is 2, j can be 2,3,..,8
and so on.

There's just as many choices for up and down

So that's 36*36=1296 rectangles.

Answer 2

A total of 1296 rectangles:

There are 64 one-by-one squares,
49 two-by-two squares, ...
(8-n)^2 n-by-n squares, ...
1 eight-by-eight square;

2 x (7x8) one-by-two rectangles,
2 x (6x8) one-by-three rectangles,
...
2 x (1x8) one-by-eight rectangles;

2 x (6x7) two-by-three rectangles,
...
2 x (1x7) two-by-eight rectangles;
...;
2 x (1x2) seven-by-eight rectangles.

This can all be simplified to the sum of the first 8 cubes (1^3 + 2^3 + 3^3 + ... + 7^3 + 8^3).

You can add this up or use the short cut method for the sum of the first 8 cubes:
(8 x 9 / 2)²

This simplifies to:
36² = 1296

It might help to think of the general case of an n x n chessboard:

Count the rectangles by rows.

In row 1 there are n 1 by 1s , n-1 1 by 2s, ... two 1 by n-1s and
one 1 by n.

Row Total = n + (n-1) + (n-2) + ... + 3 + 2 + 1 = n(n+1)/2 rectangles.

But of course there are n rows, giving n (row sum).

Now count all rectangles of height 2. Start in the bottom row. There are n 2 by 1s and n-1 2 by 2s and ... and one 2 by n. Row Total = n + (n-1) + (n-2) + ... + 3 + 2 + 1 = n (n+1)/2

The row total is the same, as it will be for all the rectangles of height 3, 4, ... n because they all share the same bases at the bottom of the board. However, there are only n-1 rows of height 2 and n-2 rows of height 3 etc.

Thus,
number 1 by any totals: n [n(n+1)/2]
number 2 by any totals: (n-1) [n(n+1)/2]
number 3 by any totals: (n-2) [n(n+1)/2]
...
number n by any totals: 1 [n(n+1)/2]

Each row total is equivalent to 1 + 2 + 3 + 4 + ... + n-1 + n, which can be replaced by the short cut of n(n + 1)/2.

The total number of rectangles is that sum [n(n+1)/2] times
1 + 2 + 3 + 4 + ... + n-1 + n. This is the same series.

So the answer is:
Total = [n(n+1)/2]²

In this case n = 8 and the total is [9x8/2]² = 36² = 1296.

Answer 3

64 squares... 8 by 8.

Answer 4

89 i counted

Answer 5

oh God....

crazy question... i knew the equation for that....

but now, i already forgot it...

i'm sorry...

Answer 6

ALOT