Competition problem — Combinatorics

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Combinatorics Difficulty 4.7 AIME Find the answer Taiwan

Can we find ten sets $A_1, A_2, \dots, A_{10}$ such that
(i) Each set is in the form of $\{a, b, c\}$ , where $a \in \{1, 2, 3\}$ , $b \in \{4, 5, 6\}$ , $c \in \{7, 8, 9\}$ .
(ii) Each set is different to any other.
(iii) If we place the sets into a circle ( $A_1, A_2, \dots, A_{10}$ ), then any pair of neighbouring sets has no common element, but any pair of non-neighbouring sets does? (Remark. $A_{10}$ is a neighbour of $A_1$ .)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Yes. Consider
$(1,4,9)$ , $(2,5,7)$ , $(3,4,8)$ , $(1,5,9)$ , $(2,4,8)$ , $(3,5,9)$ , $(2,4,7)$ , $(1,5,8)$ , $(3,4,7)$ , $(2,5,8)$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from zh; metadata (topic, difficulty) added by this project.