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Arrange 10 sets in a circle

Source: Taiwan 2014 TST1 Quiz 1, P2

July 18, 2014
combinatorics proposedcombinatorics

Problem Statement

Determine whether there exist ten sets A1A_1, A2A_2, …\dots, A10A_{10} such that (i) each set is of the form {a,b,c}\{a,b,c\}, where a∈{1,2,3}a \in \{1,2,3\}, b∈{4,5,6}b \in \{4,5,6\}, c∈{7,8,9}c \in \{7,8,9\}, (ii) no two sets are the same, (iii) if the ten sets are arranged in a circle (A1,A2,…,A10)(A_1, A_2, \dots, A_{10}), then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note: A10A_{10} is adjacent to A1A_1.)
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