Question number 26. Mary wrote a number on each face of a cube. Then for each vertex, she added the numbers on each face of a cube. Then for each vertex, she added the numbers on the three faces on which the vertex lies. For example, for vertex B, here on my larger diagram, here's vertex B, she adds the numbers on the top face, B, C, D, A, on the front face, B, A, E, F, and finally on the right face, which is B, F, G, C. She obtained eight numbers. The numbers obtained by Mary for vertices C are 14, for D is 16, and for E is 24. And I have labeled that here on my larger diagram. The question is, what number did she obtain for vertex F? So let us mark that vertex F here with the variable x, and that is the value that we're looking for. Now, how do we actually obtain one of these numbers? Following Mary's example, we would have to add the numbers on faces, sharing these three edges over here, to obtain 16. And we would have to then add the numbers sharing these edges to obtain 14. And we see that there is an overlap. The back face, which is face D, C, G, H, is a term in both of those sums, and also the top face, A, B, C, D, is a term in both of those sums. So subtracting the two pairs of numbers, two pairs of numbers, we have 16 minus 14 is equal to 2, and that is the value on the left face minus the value on the right face. So let me roughly label these. The right face over here is that one, that is face B, C, G, F. And I'll say with value, let's call it, let's not call it A, let's call it B. And then the left face would be over here, that is face A, B, H, E, let's say with value Q. So we have a little equation here. We have 2 is equal to Q minus P. Now, if we follow the same logic and perform the same calculation with the other two vertex numbers, we would have 24 minus X, and that is also difference of the left face with the right face, Q minus P. And we can see that by studying the same sort of a diagram. The faces that make up the sum we would call F are the right face with value P, the bottom face, and the front face with value P, the bottom face, and the front face. Now, both the front face and the bottom face here are terms in the sum E, and the last term is the left face with value Q. So subtracting 24 minus X gives us the same difference, Q minus P. And with that setup, we are ready to finish the problem. So what we have is Q minus P is equal to 2 from the top equation and also equal to 24 minus X, and that finishes the problem. We have X is equal to 22, which has to be the number Mary obtained for the vertex F. And the answer is C, 22. C, 22.

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