Question

How to calculate the number of rectangles in a chess board using permutation and combination ?

Answer 1

why do you need permutations or combinations for that? :O :O !!

Answer 2

not the number of squares..... the number of rectangles in it

Answer 3

oh oh oh sorry.. my bad :)

Answer 4

do u know how to do it ?

Answer 5

Squares would be a slightly easier problem, but it's the same basic concept.

Answer 6

m thinking wait.!

Answer 7

so when you say rectangles would you include squares too!?

Answer 8

of course....

Answer 9

seems quite overwhelming .. i got no clue how to start :-/

Answer 10

oh.... :(

Answer 11

maybe if we started with something smaller.. and pick up some knack.. what about counting the same for a 3 x 3 mini chess board (9 squares) ?

Answer 12

Sounds like a good way to get your head around it. You'd have the 1X1, the 1X2 (make sure you double the number that fit to account for the same pattern rotated by 90 degrees), the 1X3 (same thing as 1X2), the 2X2, the one 3X3, and the 2X3 (same as all others which are not square, you have to double the number)

Answer 13

hmm... yes....
1 box = 1 sq and 1 rect
4 boxes = 5 sq and 9 rect.....
getting any idea ?

Answer 14

1 box = 1 rect (including square :P )

4 box = 9 rect (including 5 squares)

Answer 15

@Mashy yes.... getting any idea ?

Answer 16

trying doing the Numb3r1's method.. that coudl be the actual way of doing it.. using perms and combos!

Answer 17

@Numb3r1 i'm not getting you clearly.....

Answer 18

is the answer 28^2 ?

Answer 19

sorry 36^2 ??

Answer 20

Actually, I'm seeing a pattern here. Each mxn box, with n>m, can fit in 2(9-n)^2 times (unless m=n, then it's only (9-n)^2 times

Answer 21

Mashy, if you were thinking C(n) (count the unique boxes which can be found in an nxn chessboard) is given by T(n) (triangular number n)squared, let's look at 3x3. I count 9 1x1+12 2x1 + 6 3x1 + 4 2x2 + 4 2x3 + 1 3x3 box(es)=36=T(3) squared! Looks like you got it!

Answer 22

I'm checking 4x4, just to be sure.

Answer 23

16 1x1
24 2x1
16 3x1
8 4x1
9 2x2
12 2x3
6 2x4
4 3x3
4 3x4
1 4x4
Adds up to 100! That's it!

Answer 24

yes you are right.... its 6^2 for 9 boxes and 10^2 for 4 boxes

Answer 25

@Mashy yes i think the answer is 36^2..... i'm getting it

Answer 26

That's just 1296.

Answer 27

so... if the board is in the order mxm then the number of rectangle in it is [m(m+1)/2] ^ 2

Answer 28

Final solution achieved!

Answer 29

yes..... thank you so much guys.... :)

Answer 30

thank you @Numb3r1 and @Mashy :)

Answer 31

i thought.. in order to get a rectangle.. you need to chose any two points in x axes and any 2 points in y axis.. each ordered pair of 2 points on x and y axes constitute a rectangle.

so .. how many ways can you chose 2 points from 9 points ? ( 8 squares 9 points).. that is 9C2.. so total ways is 9C2 times 9C2

Answer 32

@Mashy thats a cool one.... i mean the actual answer in terms of permutation and combination..... thank you so much....

Answer 33

your welcome :) :)!