 # Problem of the Week Problem C and Solution Spin to Win

## Problem

Esa has created a game for his math fair using two spinners. One spinner is divided into four equal sections labeled $$1,~3,~4,$$ and $$5$$. The other spinner is divided into five equal sections labeled $$1,~3,~5,~9,$$ and $$12$$. To play the game, a player spins each spinner once and then multiplies the two numbers the spinners land on. If this product is a perfect square, the player wins. What is the probability of winning the game?

Note: A square of any integer is called a perfect square. For example, the number $$25$$ is a perfect square since it can be expressed as $$5^2$$ or $$5 \times 5$$.

## Solution

In order to determine the probability, we must determine the number of ways to obtain a perfect square and divide it by the total number of possible combinations of spins. To do so, we will create a table where the rows show the possible results for Spinner $$1$$, the spinner with four sections, the columns show the possible results for Spinner $$2$$, the spinner with five sections, and each cell in the body of the table gives the product of the numbers on the corresponding spinners.

 Spinner $$\mathbf{2}$$ Spinner 1 $$\mathbf{1}$$ $$\mathbf{3}$$ $$\mathbf{5}$$ $$\mathbf{9}$$ $$\mathbf{12}$$ $$\mathbf{1}$$ $$1$$ $$3$$ $$5$$ $$9$$ $$12$$ $$\mathbf{3}$$ $$3$$ $$9$$ $$15$$ $$27$$ $$36$$ $$\mathbf{4}$$ $$4$$ $$12$$ $$20$$ $$36$$ $$48$$ $$\mathbf{5}$$ $$5$$ $$15$$ $$25$$ $$45$$ $$60$$

From the table, we see that there are $$20$$ possible combinations of spins. Of these, the following result in products that are perfect squares:

• The number $$1$$ is a perfect square $$(1 = 1\times 1)$$, and it occurs one time.

• The number $$4$$ is a perfect square $$(4 = 2\times 2)$$, and it occurs one time.

• The number $$9$$ is a perfect square $$(9 = 3\times 3)$$, and it occurs two times.

• The number $$25$$ is a perfect square $$(25 = 5\times 5)$$, and it occurs one time.

• The number $$36$$ is a perfect square $$(36 = 6\times 6)$$, and it occurs two times.

Thus, $$7$$ of the $$20$$ products are perfect squares. Therefore, the probability of winning the game is $$\frac{7}{20}$$, or $$35\%$$.

Extension: A game is considered fair if the probability of winning is $$50\%$$. Can you modify this game so that it is fair?