Question

这个立方体各面上的数字是连续的整数。三对相对面上的两个数字之和都相等。这个立方体六个面上数字的总和是

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

$75$

$76$

$78$

$80$

$81$

答案：E

知识点：正方体分类讨论

难度评级：1090

解答：

六个连续整数包含 $11, 14, 15$ ，所以其中五个数是 $11, 12, 13, 14, 15$ ，第六个数是 $10$ 或 $16$ 。

如果第六个数是 $10$ ，为了让相对面和相等，配对应为 $(10,15), (11,14), (12,13)$ ，这会使 $11$ 和 $14$ 相对。但图中 $11, 14, 15$ 在同一个顶点相遇，所以没有两个是相对面。因此第六个数是 $16$ ，相对面可配成 $(11,16), (12,15), (13,14)$ ，每对和为 $27$ 。

六个面的总和是 $3 \cdot 27 = 81$ 。

所以正确答案是 E。

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 第 11 题

答案：E

难度评级：1090

解答：

其他年份的第 11 题