Question

A painter draws 64 equal squares of 1 square inch on a square canvas measuring 64 square inches. She chooses two squares (1 square inch each) randomly and then paints them. What is the probability that two painted squares have a common side?

• A. 112/2016
• B. 1/3
• C. 512/10034
• D. 3/97
• E. 7/108

Answer 1

The Correct Option is A

To solve this problem, we must calculate the probability that two randomly chosen squares, each of 1 square inch, on a 64-square-inch canvas share a common side.

Let's break down the problem step-by-step:

1. First, determine the total number of ways to choose any two squares out of the 64 squares. This is given by the combination formula: \(C(64, 2) = \frac{64 \times 63}{2} = 2016\).
2. Next, calculate the number of ways to choose two adjacent squares. On an 8x8 grid:
   • In each row of 8 squares, there are 7 pairs of adjacent squares. Since there are 8 rows, this gives us \(8 \times 7 = 56\) horizontal adjacent pairs.
   • In each column of 8 squares, there are 7 pairs of adjacent squares. Since there are 8 columns, this gives us \(8 \times 7 = 56\) vertical adjacent pairs.
   • The total number of adjacent pairs is then \(56 + 56 = 112\).
3. Therefore, the probability that the chosen squares share a common side is given by the ratio of the number of favorable outcomes (adjacent squares) to the total number of outcomes (any two squares): \(\frac{112}{2016}\).

The fraction \(\frac{112}{2016}\) can be simplified, but it is in its correct form as per the question options. Thus, the correct answer is the first option:

112/2016

.

 

Hence, the probability that two painted squares have a common side is indeed \(\frac{112}{2016}\).

Answer 2

To solve this problem, we must calculate the probability that two randomly chosen squares on a 64-square-inch canvas have a common side.

The canvas consists of 64 squares arranged in an 8x8 grid.

The total number of ways to select any 2 squares out of 64 is given by the combination formula \( \binom{64}{2} \):

\(\binom{64}{2} = \frac{64 \times 63}{2} = 2016\)

Next, we determine how many pairs of neighboring squares share a common side:

• Each row in the grid has 7 horizontal pairs of adjoining squares. Therefore, with 8 rows, there are \(8 \times 7 = 56\) horizontal pairs.
• Each column in the grid also has 7 vertical pairs. Thus, with 8 columns, there are \(8 \times 7 = 56\) vertical pairs.

This results in a total of:

\(56 + 56 = 112\) pairs of squares sharing a common side.

Finally, the probability that two randomly chosen squares have a common side is calculated as:

\(\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{112}{2016} = \frac{112}{2016}\)