Question

The faces of a standard die are numbered 1, 2, 3, 4, 5, and 6 such that the sum of the numbers on any two opposite faces is 7. Tia writes one number on each vertex of the die such that the number on each face of the die is the greatest common divisor of the numbers at the four vertices of that face. What is the smallest possible sum of the eight numbers Tia writes?

Answer 1

To find the smallest possible sum of numbers on a standard six-sided die's vertices such that each face's number is the GCD of its vertices' numbers, one must consider opposite faces and choose the smallest possible values for vertex numbers, ensuring proper GCDs for faces.

Step-by-step explanation:

The student is asking about a mathematical problem related to the properties of numbers and probability using a standard six-sided die. As each face of a die adds up to seven with its opposite, Tia can assign numbers to each of the eight vertices in such a way that the number on each face represents the greatest common divisor (GCD) of the numbers at the four vertices of that face. To find the smallest possible sum of the eight numbers, consider the following:

1. 
   
3. Identify pairs of opposite faces and their corresponding assigned vertex numbers ensuring the face value is their GCD.
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6. Choose the smallest possible numbers that can be assigned to the vertices.
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9. Ensure the GCDs for relevant faces are correct. Considering opposite faces sum up to seven, a good strategy would involve using the numbers 1, 2, and 3 so that the GCDs are correctly represented.
10. 
    
12. Calculate the sum of the eight vertex numbers ensuring it's the minimum sum possible while satisfying the GCD condition.

With numbers carefully chosen, the sum can be minimized, although the actual calculation is left to the student as an exercise. This problem combines elements of number theory and combinatorics.