Saudi Arabia Mathematical Competitions 2012

Partition the set $\{1,2,3,\dots ,30\}$ into 20 subsets as follows: $$\begin{array}{rl}& \{1,3,9,27\},\\ & \{2,6,18\},\\ & \{4,12\},\{5,15\},\{7,21\},\{8,24\},\{10,30\},\\ & \{11\},\{13\},\{14\},\{16\},\{17\},\{19\},\{20\},\\ & \{22\},\{23\},\{25\},\{26\},\{28\},\{29\}.\end{array}$$ A subset of $\{1,2,3,\dots ,30\}$ is delicious if and only if it does not contain two consecutive elements from any one of the 20 sets listed above. Thus, a delicious set contains at most 2 elements from each of the sets in the first two rows, and at most 1 element from each of the sets in the last two rows. It is possible for a delicious set to have exactly 2 elements from each of the sets in the first two rows, and exactly 1 element from each of the sets in the last two rows; therefore, a super delicious set must have this property. So there are $$3\cdot 1\cdot 2^5\cdot 1^{13}=96$$ super delicious sets, because there are 3 ways to choose 2 non-consecutive elements from $\{1,3,9,27\}$, and 1 way to choose 2 non-consecutive elements from $\{2,6,18\}$, and so on.