The image shows a sequence of triangles, each with numbers at their vertices and a number inside. The task is to determine the missing number in the last triangle based on the pattern observed in the first two triangles.

Analyzing the Pattern:

1. First Triangle:

   • Vertices: 5, 10, 7
   • Inside Number: 20
   • Calculation: $5 \times 10 - 7 = 50 - 7 = 43$ (This doesn't match the inside number directly, so this isn't the pattern.)

2. Second Triangle:

   • Vertices: 6, 12, 4
   • Inside Number: 38
   • Calculation: $6 \times 12 - 4 = 72 - 4 = 68$ (Again, this doesn't match directly.)

After reconsidering the operations, it appears the operation might involve addition rather than subtraction:

• First Triangle:

  • Vertices: 5, 10, 7
  • $5 \times 7 + 10 = 35 + 10 = 45$ (Still doesn't match the 20 inside the triangle)

  However, trying another pattern:

  • $5 \times 10 + 7 = 50 + 7 = 57$ doesn't work either.

Perhaps, the actual operation is more direct:

• Sum of Vertices:
  • $5 + 10 + 7 = 22$
  • $6 + 12 + 4 = 22$

The triangles might be using the difference or another operation. Based on more typical puzzle setups:

Hypothesis:

• The pattern might involve multiplying all vertices and then subtracting a constant value.
  • First Triangle: $5 \times 10 + 7 \times 2 = 50 + 14 = 64$ then subtract by a certain constant to get 20.
  • Second Triangle: $6 \times 12 + 4 \times 2 = 72 + 8 = 80$, then possibly subtracting a larger constant for 38.

But based on the options, trying the possibilities:

Final Hypothesis:

The pattern used seems more complex. Therefore, guesswork or further steps would typically be needed.

Final Choice:

Given standard puzzle patterns, option B: 64 seems correct assuming a straightforward decrease by a constant from the products or sums.

Would you like to delve further into the operations, try a different approach, or explore related mathematical concepts? Here are some follow-up questions:

1. How would you approach finding the pattern in different mathematical puzzles?
2. Can you think of an alternative pattern that might work?
3. What is the importance of testing multiple hypotheses in problem-solving?
4. How does changing the operations (e.g., from addition to multiplication) affect outcomes in sequences?
5. Why might guessing based on patterns sometimes yield the correct answer?

Tip: When facing pattern recognition problems, always test the simplest operations first, like addition, subtraction, and multiplication, before moving on to more complex combinations.