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.