China Western Mathematical Olympiad

Question

Put numbers 1, 2, 3, 4, 5, 6, 7 and 8 at the vertices of a cube, such that the sum of any three numbers on any face is not less than 10. Find the minimum sum of the four numbers on a face. (posed by Qiu Zonghu) FIGURE_0

Accepted Answer

Suppose that the four numbers on a face of the cube are

a_{1}, a_{2}, a_{3}, a_{4}

such that their sum reaches the minimum and

a_{1} < a_{2} < a_{3} < a_{4}

. Since the maximum sum of any three numbers less than

5

is

9

, we have

a_{4} \geq 6

, and then

a_{1} + a_{2} + a_{3} + a_{4} \geq 16

.

As seen in the figure, we have

2 + 3 + 5 + 6 = 16

, and that means the minimum sum of the four numbers on a face is

16

.