Can you give examples of combinatorial construction riddles, approachable by gifted high school students?
Examples:
Find a finite set $A$ and $B \subset 2^A$ such that any element of $A$ is covered by at least 100 sets in $B$, but $B$ cannot be partitioned into 2 disjoint sets, each covering $A$.
Find the shortest sequence of numbers between 1 and K (K=99) such that each pair of numbers are adjacent somewhere in the sequence (1231 is an answer if K=3).
Find a systematic way to arrange a round robin tournament between N teams (N even or odd).
Find (different methods) to cover the complete graph on 50 vertices by disjoint spanning trees.
Motivation: I'm working with gifted high school students. This is part of an excellence program in computer science. We have found these kind of problems adequate for stimulating and identifying the better students, before we teach them algorithmic methods. We also let them find winning strategies for mathematical games at this level of their knowledge.