how many number of rectangles which we can form on a chess board?

Question

its a question of permutation and combination(maths). do help me to get the answer of this question. thank you.

Answer 1

We have 9 horizontal and 9 lines available,to form a rectangle we have to select 2 horizontal and 2 vertical lines
Number of ways in which this can be done is
C(9,2)*C(9,2)
= 36 * 36 = 1296

Answer 2

First, when you say "rectangle" I'll assume you're counting squares in that category.

Rather than thinking of a chess board as 8 squares by 8 squares, we'll define it as 9 points by 9 points (height ; width). Point (1 ; 1) could be the bottom left; (9 ; 9) the top right A rectangle is formed by any two pairs of points where the height values are not the same as each other and the width values are not the same as each other. To avoid redundant counting (e.g. 1,1 ; 2,2 would be the same square as 2,1 ; 1,2), each new rectangle should only be formed upward and outward from the previous counted starting points.

From point (1,1_ rectangles would be formed with points ( 2 thru 9 ; 2 thru 9) or 8 times 8 = 64
From 2,1 (3-9 ; 2-9) or 7 times 8 = 56
From point (3;1): ( 4-9; 2-9) 6 *8 = 48
ditto 5*8 = 40
ditto 4*8 = 32
ditto 3*8 = 24
ditto 2*8 = 16
From (8 ; 1): (9 ; 2-9) 1 * 8 = 8
So the sum of rectangles (including squares) from the first column of points is 288
The second column sum would be the same, minus 36, or 252
or calculated 8*7 +7*7 + 6*7 + 5*7 +4*7 +3*7 + 2*7 +1*7
The third column of points would be 252 -36 = 216
(or 8*6 +7*6 + ... 1*6)
ditto = 180
ditto = 144
ditto = 108
ditto = 72
The 8th column of points would yield 36 final rectangles.

So, I think THE TOTAL = 1.296

Answer 3

The number of squares is 204. I know nothing of permutation and combination, but I can tell you that the no. of rectangles is more that 400.

No. of squares = 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1
= 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1
= 204

Answer 4

The number of squares is got by the formula (n² + p).
n - number of squares on the board.
p - the previous value.

For one square:
(n² + p) = (1² + 0) = 1

For two squares:
(n² + p) = (2² + 1) = 5

For three squares:
(n² + p) = (3² + 5) = 14

For four squares:
(n² + p) = (4² + 14) = 30

↓ ↓ ↓ ↓ ↓

For eight squares:
(n² + p) = (8² + 150) = 214


For the number of rectangles ?
I need to see if I can come up with a formula and get back to you.

Answer 5

Here's a guess :

Σ ( 1 → 7 ) 16 n +
Σ ( 1 → 6 ) 14 n +
Σ ( 1 → 5 ) 12 n +
Σ ( 1 → 4 ) 10 n +
Σ ( 1 → 3 ) 8 n +
Σ ( 1 → 2 ) 6 n +
Σ ( 1 → 1 ) 4 n

Answer 6

65 rectangles can be formed.

Answer 7

in calculation i ting is abt more than 500

Answer 8

That would be a(8). 2044.
http://www.research.att.com/~njas/sequences/A085582

Answer 9

32 exactly.