| 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:
