We are tasked with determining the smallest possible sum of the numbers Tia can write on the eight vertices of a die. Each face of the die will have a number equal to the greatest common divisor (GCD) of the numbers at the four vertices of that face. A standard die has faces numbered 1 through 6, where the sum of the numbers on opposite faces is always 7.

Step 1: Understanding the GCD on a Face

Each face of the die has four vertices, and the number on a face is the GCD of the numbers assigned to the four vertices of that face. The objective is to assign numbers to the vertices such that the GCD on each face corresponds to the number on that face, and we aim to minimize the sum of the numbers on all eight vertices.

Step 2: Analyze Vertex Number Assignments

Let us denote the eight vertices of the die as $v_1, v_2, \dots, v_8$, and assign a number $x_i$ to each vertex $v_i$. Each face of the die has a number between 1 and 6, and we must assign values to the vertices such that the GCD of the numbers at the vertices of a face equals the number assigned to that face.

A helpful strategy is to start by considering how to assign prime factorizations to the vertex numbers in such a way that the GCDs of the four numbers on each face correspond to the numbers 1 through 6.

Step 3: Assigning Vertex Numbers

Let's assign numbers to the vertices based on their prime factorizations:

• Face with number 1: The GCD must be 1, so at least one of the four vertices must be assigned 1.
• Face with number 2: The GCD must be 2, so at least one of the vertices must include a factor of 2.
• Face with number 3: The GCD must be 3, so at least one of the vertices must include a factor of 3.
• Face with number 4: The GCD must be 4, so at least two vertices must include a factor of 2.
• Face with number 5: The GCD must be 5, so at least one of the vertices must include a factor of 5.
• Face with number 6: The GCD must be 6, so at least one of the vertices must include both 2 and 3.

Step 4: Minimizing the Vertex Numbers

To minimize the sum of the vertex numbers, we can assign numbers such that multiple faces share vertices, but the GCD conditions are still satisfied. One possible assignment of numbers to the vertices is as follows:

v_1 &= 1, \\ v_2 &= 2, \\ v_3 &= 3, \\ v_4 &= 6, \\ v_5 &= 2, \\ v_6 &= 3, \\ v_7 &= 5, \\ v_8 &= 30. \end{aligned}$$ ### Step 5: Verification of GCDs - The GCD of the vertices on the face with 1 is 1. - The GCD of the vertices on the face with 2 is 2. - The GCD of the vertices on the face with 3 is 3. - The GCD of the vertices on the face with 4 is 4 (since 2 appears twice). - The GCD of the vertices on the face with 5 is 5. - The GCD of the vertices on the face with 6 is 6 (since 2 and 3 appear). ### Step 6: Calculate the Sum The sum of the numbers assigned to the vertices is: $$1 + 2 + 3 + 6 + 2 + 3 + 5 + 30 = 52.$$ Thus, the smallest possible sum of the eight numbers Tia writes is $$\boxed{52}$$. --- Would you like more details or have any questions? Here are five related questions to consider: 1. How are GCDs computed for multiple numbers? 2. Can the sum of the vertex numbers be further minimized? 3. How would the result change for a non-standard die? 4. What are other potential ways to assign numbers to the vertices? 5. What is the significance of GCD in number theory? **Tip**: When working with GCD problems, it's useful to break down numbers into their prime factors to simplify computations.