The vertices of a cube are labeled with the numbers 0,1,2,3,4,5,6 and 7, such that no two vertices are labeled with the same number. The sum of any 2 numbers on an edge is prime. What is the maximum sum of the 4 numbers on a face?
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To find the maximum sum of the four numbers on a face, we need to look for a pattern among the possible pairs of numbers on an edge that add up to a prime number.
First, let's list out the pairs of numbers on the cube that add up to prime numbers:
(0, 1), (0, 3), (0, 5), (0, 7), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 7), (5, 6), (6, 7)
Now, we can see that the maximum prime number we can get by adding two numbers is 7 (e.g., 3 + 4 = 7).
To find the maximum sum of four numbers on a face, we need to choose four numbers from the given list of pairs without repetition.
The key observation here is that no matter which four pairs we choose, we can't avoid using at least one number twice. Hence, the maximum sum of the four numbers on a face would be 7 + 7 + x + y, where x and y are two distinct numbers chosen from the list of pairs.
Since x and y are chosen from the same list, the maximum sum of the four numbers on a face is 7 + 7 + 3 + 4 = 21 + 7 = 28.
Therefore, the maximum sum of the four numbers on a face is 28.