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

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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.
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**.

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

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