- 28 Apr '13 16:28

I'm not much for Combinatorics, but here is what I got*Originally posted by talzamir***If I pick four different squares from a chessboard.. say, D1, F2, E4 and C3.. their center points can (and in this case do) form a square. If four different squares are chosen randomly, how likely is it that they do form a square?**

First the number of distinct ways of selecting 4, boxes out of 64

C(4,64) = 64!/(4!*60!) = 635376

Next, looking at each column

The number of ways of selecting 2, of 8 boxes

C(2,8) = 8!/(2*6!) = 28

In order to make a square we need to select 4 points, 2 points in each column.

Number of ways of doing this

"Possible" square configurations = 28*7 = 196

of those 196 combinations, 1/7 are square configurations

Square configurations = 28

Probability of selecting a square configuration = 28/635376 - 28 Apr '13 17:44

Nevermind, pretty sure my answer is wrong. Back to the drawing board!*Originally posted by joe shmo***I'm not much for Combinatorics, but here is what I got**

First the number of distinct ways of selecting 4, boxes out of 64

C(4,64) = 64!/(4!*60!) = 635376

Next, looking at each column

The number of ways of selecting 2, of 8 boxes

C(2,8) = 8!/(2*6!) = 28

In order to make a square we need to select 4 points, 2 points in each column.

Number of ...[text shortened]...

Square configurations = 28

Probability of selecting a square configuration = 28/635376 - 28 Apr '13 17:51 / 1 edit

Yes, it seems like you're missing all of the diagonal squares that you can make on the board.*Originally posted by joe shmo***Nevermind, pretty sure my answer is wrong. Back to the drawing board!**

e.g.:

B1 - A2 - B3 - C2

B1 - A3 - C4 - D2

B1 - A4 - D5 - E2

B1 - A5 - E6 - F2

etc... - 29 Apr '13 22:39 / 2 editsI'm trying to do this without writing a program, so I'm determining some limitations first.

Including reflections and rotations, there are a total of 10 unique starting locations on the chessboard.

(A1, A2, A3, A4, B2, B3, B4, C3, C4, D4)

Including reflections and rotations, there are a total of 19 edges that make it possible to form a square on the chessboard.

(0, 1-7)

(1, 1-6)

(2, 2-5)

(3, 3-4)

Now, to pare down which of the 19 edges can be used with each of the 10 starting squares, and I should have the number.

Feel free to correct me if I'm wrong on either of these points. - 05 May '13 13:34I wonder if this is correct, or if it could be made more elegant.

Types of squares:

a1-b1-b2-a2 (1,0 square. from first square to the next, one step to the right, none towards the 8th row)

..

a1-h1-h8-a8. (7,0 square)

Then there are some in a 45 degree angle.

b1-c2-b3-a2. (1,1 square).

..

d1-g4-d7-a4 (3,3 square).

And the rest.

b1-d2-c4-a3 (2,1 square)

A square with dimensions (x,y) as defined above are possible as long as x + y < 8, and fit on the board in (8 - (x+y))^2 positions. Also, due to symmetry squares (0,1) and (1,0) are the same square, but (1,2) and (2,1) are not.

Thus,

(0,1): 49

(0,2) and (1,1): 36

(0,3), (1,2), and (2,1): 25

(0,4), (1,3), (2,2) and (3,1): 16

(0,5), (1,4), (2,3), (3,2), (4,1): 9

(0,6), (1,5), (2,4), (3,3), (4,2), (5,1): 4

(0,7): 1

1x49 + 2x36 + 3x25 + 4x16 + 4x9 + 6x4 + 1x1 = 50 + 72 + 124 = 266 different quadruplets that form a square out of a total of 64! / (60!4!) = 635,376 so the probability of randomly picking a foursome that forms a square on the chessboard is about 0.04%.