OPEN MIDDLE - Perfect Squares Slrections: Using the digits 1-9, at most one time each, to fill expression evaluate to a perfect square number. \[ 18 \times \square \times 2 \]

To solve the problem, we need to fill the blank in the expression \( 18 \times \square \times 2 \) with a digit from 1 to 9 (each digit can be used at most once) such that the resulting product is a perfect square. ### Step-by-Step Solution: 1. **Identify Used Digits:** - The digits already used in the expression are **1**, **8**, and **2**. - Remaining available digits are **3, 4, 5, 6, 7, 9**. 2. **Formulate the Expression:** \[ 18 \times d \times 2 = 36 \times d \] We need \( 36 \times d \) to be a perfect square. 3. **Check Each Available Digit:** - **\( d = 3 \):** \( 36 \times 3 = 108 \) (Not a perfect square) - **\( d = 4 \):** \( 36 \times 4 = 144 \) (Perfect square: \( 12^2 \)) - **\( d = 5 \):** \( 36 \times 5 = 180 \) (Not a perfect square) - **\( d = 6 \):** \( 36 \times 6 = 216 \) (Not a perfect square) - **\( d = 7 \):** \( 36 \times 7 = 252 \) (Not a perfect square) - **\( d = 9 \):** \( 36 \times 9 = 324 \) (Perfect square: \( 18^2 \)) 4. **Valid Solutions:** - **\( d = 4 \)** - **\( d = 9 \)** ### Conclusion: You can fill the blank with either **4** or **9** to make the expression evaluate to a perfect square. - **If \( d = 4 \):** \[ 18 \times 4 \times 2 = 144 = 12^2 \] - **If \( d = 9 \):** \[ 18 \times 9 \times 2 = 324 = 18^2 \] **Final Answer:** The blank can be filled with 4 or 9. For example: \[ 18 \times 4 \times 2 = 144 \quad\text{or}\quad 18 \times 9 \times 2 = 324 \] Both are perfect squares.