Question

The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is

$75$

$76$

$78$

$80$

$81$

Solution:

The six consecutive numbers include $11, 14, 15$ , so five of the faces are $11, 12, 13, 14, 15$ and the sixth is $10$ or $16$ .

If the sixth were $10$ , equal opposite sums would force the pairs $(10,15), (11,14), (12,13)$ , making $11$ and $14$ opposite. But the figure shows $11, 14, 15$ meeting at one corner, so no two of them are opposite. Hence the sixth number is $16$ , with pairs $(11,16), (12,15), (13,14)$ , each summing to $27$ .

The total is $3 \cdot 27 = 81$ .

Thus, the correct answer is E .

1990 AMC 8 Problem 11

Solution:

Problem 11 in Other Years