We distribute eight numbers among 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 at the vertices of a rectangular block in such a way that the sum of the numbers on any face is equal to 18.

(a) Which numbers were discarded in the distribution?

(b) Show a possible distribution.

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(a) Since the block has six faces, the sum of the numbers on all faces is $18 \times 6=108$, but the number assigned to each vertex is counted three times in this sum. Therefore, the sum of the distributed numbers is 108/3 = 36. Since the sum of all numbers from 1 to 10 is 55, the sum of the two discarded numbers is 19. We conclude that the discarded numbers are 9 and 10.

(b) A possible distribution is shown in Figure 72.1.