Question

All the possible squares are made using combination of smaller squares in the grid. What is the probability of picking a square that is formed by at least two different-colored squares?

• A. 54/91
• B. 37/91
• C. 5/13
• D. 1/13

Answer 1

The correct option is C 5/13

Given is a $6\times 6$ grid.
The total number of squares in an $n\times n$ grid is calculated using:
$Totalsquares=\frac{n(n+1)(2n+1)}{6}$

So,
$Totalsquares=\frac{6(6+1)(2\times 6+1)}{6}$
$=7\times 13=91$

Similarly, for each differently colored grid,
Squares in yellow $3\times 3$ grid
$=\frac{3(3+1)(2\times 3+1)}{6}$ = 14

Total 3 x 3 squares = 4 x 14 = 56

Hence,
$\text{Squares with at least two colors = 91 - 56}$
= 35

Therefore,
$Probability=\frac{35}{91}=\frac{5}{13}$