Two Squares are Chosen at Random on a Chessboard. What is the GMAT Problem Solving

Question: Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?

1. 1/18
2. 64/4032
3. 63/64
4. 1/9

Correct Answer: A

Solution and Explanation.
Approach Solution 1:
The given case states that there are two squares at random on a chessboard. Accordingly, we need to find the probability that they have one side in common.

To find the solution to this problem, we need to understand that in 64 squares there are these following:

(1) There are 4 at-corner squares, each has only 2 squares each having a side in common with
(2) 6*4 = 24 side squares, each only has 3 squares such that each has a side in common with
(3) 6*6 = 36 inner squares, each has 4 squares such that each has a side in common with

Accordingly, the calculation stands out as-

P = (4/64)*(2/63) + (24/64)*(3/63)+ (36/64)*(4/63)
P = 1/18

Hence, the probability of two squares having a common side is 1/18

Approach Solution 2:

64 no of squares and we can choose first square by 64 ways

and second by 63ways.

There are a total of 64 * 63 = 4032

Corners have only 2 adjacent squares. we have 4 corners.

total we can choose using corner square are=4*2

another 24 squares along the boundaries they have 3 adjacent squares.

total no= 24*3

the remaing squares are 34 they have 4 adjacent sides.

total= 36*4

number of combinations = (4 * 2) + (24 * 3) + (36 * 4) = 224

 = .

Approach Solution 3:
Two squares can be chosen in 64C2 ways= 64*63/2= 2016

Now if we consider vertically say in the first column we can get 7 pairs of squares where each pair will have one common side. similarly for the second third….and eight column = 7*8=56 possibilities.

Again if we consider horizontally say in the first row then we get 7 pairs of squares where each pair will have one common side. Same goes for the second third ….eight rows = 7*8 =56 possibilities

favourable outcome= 56+56=112

So prob= 112/2016 = 1/18.

“Two squares are chosen at random on a chessboard. What is the”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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