Take the first five letters in the alphabet. Then create every possible two-letter combination (order disregarded; AB is the same as BA for the purpose of this problem) using only those letters. This creates ten pairs:
AB AC AD AE BC BD BE CD CE DE
Then divide them into three groups, so that Group 1 has four pairs in it, while Group 2 and 3 have three pairs each. Example:
Group 1: AB AC AD AE
Group 2: BC BD BE
Group 3: CD CE DE
First problem: Is it possible to divide them into groups so that all possible three-letter combinations can be created by picking one pair from group 1, one pair from group 2 and one pair from group 3? For example, if AB is in group 1, BC is in group 2 and AC is in group 3, the three-letter combination ABC can be created in this way.
Second problem: Is it possible to divide them into groups so that, no matter which pair you pick from group 1, you can still create a three-letter combination by picking the right pairs from group 2 and 3?
Third problem: Is it possible to divide them into groups so that both of the above are true?
I suspect the answer to all three is “no”. If so, can this be proved without actually trying all possible combinations?