Squares of chess

Sample space will be total ways of selecting 2 squares (order doesn't matter) out of available 64 squares. So sample space = $^{64} C_2$

Now there are 7 unique adjacent square sets in each row and each column. i.e. favorable cases will be 7×(8 rows + 8 columns) = 112. Hence Required Probability= $\huge \frac{Favourable~ cases}{Sample ~Space} = \frac{112}{^{64} C_2} = \frac{1}{18}$

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Also there could be other solution too if we consider ALL possible squares and not only the smallest squares. (i.e. if we consider that total squares are more than 64).

In that case let us first calculate the sample space i.e. total number of squares on a chess board.

1, 8x8 square

4, 7x7 squares

9, 6x6 squares

16, 5x5 squares

25, 4x4 squares

36, 3x3 squares

49, 2x2 squares

64, 1x1 squares

Therefore, there are actually: $64+49+36+25+16+9+4+1= 8^2+7^2+6^2+...1^2=204$ squares on a chessboard!

So sample space = $^{204} C_2$

Now if we assume that 2 squares can have a side common only if they are of the same dimensions:

Let the length of the smallest square be 1 unit.

So the possibility of selecting 2 squares with a common side (As calculated in the first method above):

having the side length greater than 4 unit length = 0

having the side length of 4 unit = 1×(2 rows + 2 columns) = 4

having the side length of 3 unit = 3×(6 rows + 6 columns) = 36

having the side length of 2 unit = 5×(7 rows + 7 columns) = 70

having the side length of 1 unit = 7×(8 rows + 8 columns) = 112

So total number of favorable cases: $= 4+36+70+112=222$

Hence Required Probability= $\huge \frac{Favourable~ cases}{Sample~ Space} = \frac{222}{^{204} C_2}$

Consider all possible cases There are 64C2 ways to select any 2 squares... That is 2016 ways Now looking for favourable cases... Instead of looking for the blocks we count the edges... Every edge signifies the two adjacent blocks connected to it. There are 7 vertical and 7 horizontal lines... Each decided into 8 parts(edges). So we count and get (7x2)x8

So the probability is 112/2016 = 0.0555555.....

Any one with the solution