Colourful Squares

We look at three cases, depending on whether the three points are in the same square, two different squares, or 3 different squares.

The probability the points are in the same square is $\frac{1}{81}$ , since there is a $\frac{1}{9}$ chance that each of the second and third points are in the same square as the first. The probability that all these points have the same colour is 1.

The probability that the points are all in different squares is $\frac{56}{81}$ , since there is a $\frac{8}{9}$ chance that the second point is in a different square than the first, and a $\frac{7}{9}$ chance that the third point is in a different square than both of them. The probability that all these points have the same colour is $\frac{1}{5^2}$ , since this is the probability that all three squares received the same colour.

The probability that two points are in one square and one is in another is $1 - \frac{1}{81} - \frac{56}{81} = \frac{24}{81}$ . The probability that the points all have the same colour is $\frac{1}{5}$ .

Thus, the probability that all the points are the same colour is $\frac{1}{81} + \frac{56}{81 \times 5^2} + \frac{24}{81 \times 5} = \frac{25 + 56 + 120}{2025} = \frac{201}{2025} = \frac{67}{675}$ . So $a + b = 67 + 675 = 742$ .