The image shows a special dice with positive integers on all sides. If you multiply the numbers on opposite sides, you always get the same answer. What is the smallest possible sum of the six numbers on the dice?

To solve this problem, we need to analyze the properties of the dice and the relationships between the numbers on opposite sides.

### Step 1: Understanding the dice structure
A standard dice has six sides with opposite sides summing to 7 (e.g., 1 opposite 6, 2 opposite 5, 3 opposite 4). However, in this case, we are given the condition that **the product of numbers on opposite sides is always the same**. Let's call this product \( P \).

Let the numbers on the dice be denoted as \( a_1, a_2, a_3, a_4, a_5, a_6 \), and assume that opposite pairs of faces are:
- \( a_1 \) opposite \( a_6 \),
- \( a_2 \) opposite \( a_5 \),
- \( a_3 \) opposite \( a_4 \).

According to the problem, the products of these opposite pairs are equal, meaning:
\[
a_1 \times a_6 = a_2 \times a_5 = a_3 \times a_4 = P
\]
where \( P \) is a constant.

### Step 2: Minimizing the sum
We want to minimize the sum of the six numbers on the dice:
\[
S = a_1 + a_2 + a_3 + a_4 + a_5 + a_6
\]
Given the condition \( a_1 \times a_6 = a_2 \times a_5 = a_3 \times a_4 = P \), one possible solution is to make each product equal to the smallest square number, since equal products often lead to small sums.

### Step 3: Exploring the smallest possible values
Let's try \( P = 6 \) as a test case because the factors of 6 provide integers that keep the numbers small.

- For \( a_1 \times a_6 = 6 \), one pair could be \( a_1 = 1 \) and \( a_6 = 6 \),
- For \( a_2 \times a_5 = 6 \), one pair could be \( a_2 = 2 \) and \( a_5 = 3 \),
- For \( a_3 \times a_4 = 6 \), one pair could be \( a_3 = 2 \) and \( a_4 = 3 \).

The sum of these numbers is:
\[
S = 1 + 6 + 2 + 3 + 2 + 3 = 17
\]

Thus, the smallest possible sum of the numbers on the dice is \( \boxed{17} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. Can the product on opposite sides of the dice be different from 6 and still minimize the sum?
2. How would the solution change if the dice had more than 6 sides?
3. What is the significance of choosing small factors for the product \( P \)?
4. Can there be more than one solution for the numbers on the dice that yield the same sum?
5. What are the general mathematical properties of opposite faces on a fair dice?

**Tip:** When dealing with sums and products, always try small factor pairs first to minimize overall values.

Explore how to find the smallest possible sum of numbers on a special dice where the product of numbers on opposite sides is always the same. Learn the key concepts of number theory and algebra, using factorization to optimize sums. Suitable for Grades 9-12.

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