jrk23 wrote:
Bunuel wrote:
How many squares of any size are there on an 8x8 chessboard?
A. 4096
B. 1296
C. 204
D. 65
E. 64
Anyone please explain.
.1..2...3...4..5...6..7...8
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 2
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 3
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 4
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 5
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 6
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 7
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 8
The question is asking how many different squares of differing sizes you can make out of a chessboard.
For example, you can make 64 different 1x1 size squares, as there are 64 different squares.
Alternatively, you can make 1 single 8x8 size squares, as that is as large as the chessboard is.
e.g. For a 5x5 size square, how many different ways can you position it?
.1..2...3...4..5...6..7...8
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 2
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 3
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 4
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 5
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 6
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 7
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 8
In this example for a 5x5 square, you can see that there are 4 positions it can take vertically and horizontally.
The current position, and 3 others vertically and horizontally.
So, there are a total of 4x4 ways to position a 5x5 square.
Now to find how many 2x2, 3x3, 4x4, 5x5, 6x6, and 7x7 squares with different positioning on the chessboard.
Extrapolating this...
Size
..... Possible Squares
1
............64 (8x8)
2
............49 (7x7)
3
............36 (6x6)
4
............25 (5x5)
5
............16 (4x4)
6
............9 (4x4)
7
............4 (4x4)
8
............1 (1x1)
Sum of which is 204.