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how many rectangles are in the chess board?

infinisa
 guissmo wrote: Now let's screw everyone and ask how many rectangles are in the chess board. )

Hi guissmo

I think this problem deserves a thread of its own!

Any rectangle on a chess board will be w by h units, where 1≤w≤8 and 1≤h≤8.

Obviously there are 8 x 8 = 64 different possible shapes, but they can be positioned in many different ways. To describe the position of a rectangle, let's use an x-y coordinate system on the chess board, taking ats bottom left hand corner as the origin, so that (x,y) represents the point x units to the right and y units up, where 0≤x≤8 and 0≤y≤8.

If we place a w by h rectangle so that its bottom left hand corner is at (x,y), then it's top right corner will be at (x+w, y+h), so we must have x+w≤8 and y+h≤8. That is, x≤8-w and y≤8-h. So we have 0≤x≤8-w and 0≤y≤8-h. This gives 9-w possibilities for x and 9-h possibilities for y, or (9-w)x(9-h) ways of placing a w by h rectangle altogether.

Taking all different possible sizes of rectangle w by h, this gives:
Sum (w from 1 to 8 ) Sum (h from 1 to 8 ) of (9-w)x(9-h)
= Sum (w from 1 to 8 ) of (9-w) x Sum (h from 1 to 8 ) of (9-h)
= Sum (w from 1 to 8 ) of (w) x Sum (h from 1 to 8 ) of (h)
= (½ x 8 x (8 -1))^2
= 28^2
= 784

In more readable mathematical notation, this is:

aningbo
never thought of that. rectangles in a chess board.

but why bother anyways?
jabce85
nice, and interesting.
wrong. i knew there are more than 1000 rectangles years ago before the Internet (i mean before i start use the Internet, in the 1980s). i read it somewhere. that 1 thing i stilll memorize, so when i saw your answer, i knew immediately it was wrong.

and today i also found out there are several ways to calculate it, and all end up with 1296 rectangles.
chiragpatnaik
that was easy...
chatrack
infinisa wrote:
 guissmo wrote: Now let's screw everyone and ask how many rectangles are in the chess board. )

Hi guissmo

I think this problem deserves a thread of its own!

Any rectangle on a chess board will be w by h units, where 1?w?8 and 1?h?8.

Obviously there are 8 x 8 = 64 different possible shapes, but they can be positioned in many different ways. To describe the position of a rectangle, let's use an x-y coordinate system on the chess board, taking ats bottom left hand corner as the origin, so that (x,y) represents the point x units to the right and y units up, where 0?x?8 and 0?y?8.

If we place a w by h rectangle so that its bottom left hand corner is at (x,y), then it's top right corner will be at (x+w, y+h), so we must have x+w?8 and y+h?8. That is, x?8-w and y?8-h. So we have 0?x?8-w and 0?y?8-h. This gives 9-w possibilities for x and 9-h possibilities for y, or (9-w)x(9-h) ways of placing a w by h rectangle altogether.

Taking all different possible sizes of rectangle w by h, this gives:
Sum (w from 1 to 8 ) Sum (h from 1 to 8 ) of (9-w)x(9-h)
= Sum (w from 1 to 8 ) of (9-w) x Sum (h from 1 to 8 ) of (9-h)
= Sum (w from 1 to 8 ) of (w) x Sum (h from 1 to 8 ) of (h)
= (½ x 8 x (8 -1))^2
= 28^2
= 784

In more readable mathematical notation, this is:

n= no. Of small squre box
then, N is
N = 1+ n + n ^(n-1)
inchirieri
1000?
pscompanies
LOL who'd have thought of calculating rectangles on a chess board. So what's the answer in the end?
Afaceinthematrix
 badai wrote: wrong. i knew there are more than 1000 rectangles years ago before the Internet (i mean before i start use the Internet, in the 1980s). i read it somewhere. that 1 thing i stilll memorize, so when i saw your answer, i knew immediately it was wrong. so i google up and got 1296 as the answer. and today i also found out there are several ways to calculate it, and all end up with 1296 rectangles.

Well that would make sense... It's an eight by eight board, so you can add (1+2+3+4+5+6+7+8) and square it to get 1296... Is that one of the ways to calculate it?

So then that would mean that there are 784 rectangles on a 7x7 chessboard which means that infinisa just made a slight error in his/her calculations...