26 Aug '06 21:25>
How many squares do you think are on a chess board?
Originally posted by Fat LadyI didn't think that the first person to reply would get it right.
There are:
8*8 squares of side 1.
7*7 squares of side 2.
6*6 squares of side 3.
5*5 squares of side 4.
4*4 squares of side 5.
3*3 squares of side 6.
2*2 squares of side 7.
1 square of side 8.
Making a total of 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204.
Originally posted by Fat LadySorry, this was posted just the other day (in the correct forum). 🙄
Here's a slightly trickier one, one of Martin Gardner's.
Originally posted by ThudanBlunder1x2=2
So before this thread dies, how many rectangles are there on a chessboard?
Originally posted by lauseyI have no idea what you are doing here. This problem can be bruteforced (there are 2*(9-m)*(9-n) mxn rectangles on a chessboard where m!=n [if so then remove the 2*]).
1x2=2
3x2=6
4x2=8
5x2=10
6x2=12
7x2=14
8x2=16
Total: 68
1x3=3
2x3=6
4x3=12
5x3=15
6x3=18
7x3=21
8x3=24
Total: 99
1x4=4
2x4=8
3x4=12
5x4=20
6x4=24
7x4=28
8x4=32
Total: 128
1x5=5
2x5=10
3x5=15
4x5=20
6x5=30
7x5=35
8x5=40
Total: 155
1x6=6
2x6=12
3x6=18
4x6=24
5x6=30
7x6=42
8x6=48
Total: 180
1x7=7
2x7=1 ...[text shortened]... 24
4x8=32
5x8=40
6x8=48
7x8=56
Total: 224
224 + 203 + 180 + 155 + 128 + 99 + 68 = 1057
Originally posted by XanthosNZI realised I made a mistake in my bruteforce approach. Wrote an algorithm that did indeed get 1296. If you are talking about oblongs, it is 1092 (previously assumed that rectangles aren't squares).
I have no idea what you are doing here. This problem can be bruteforced (there are 2*(9-m)*(9-n) mxn rectangles on a chessboard where m!=n [if so then remove the 2*]).
However a more elegant solution is that a chessboard has 9 vertical boundaries and 9 horizontal boundaries and to form a rectangle we choose any two of each.
There are 36 ways of choosing 2 from 9 (9C2) and therefore there are 36^2 rectangles.
36^2 = 1296