A problem related to projective plane

Question

Design 25 cubes so that each cube's six faces display integer numbers in the range 0,1,...,31 such that:
On each cube, all the numbers are different.
Any given two cubes share exactly one common number.
Please give your answer as 25 lines of six numbers. Bonus question: Use as small a range of numbers as possible.

After seeing the solution, I read that this can be solved using "Projective plane". So my questions are:

1. How to solve such a question? (What is the strategy..)
2. How can I decide that a given problem (like this) has a solution? For example, given 25 cubes and a set from 1 to 10.. Logically it's impossible to be solved.
3. What field of maths does "Projective plane" belong to? I mean, what is (more general) topic in maths that I have to study to understand "Projective plane"?
   
   
   Note: I don't know what tags do I have to choose for this question.

Answer 1

Here is a simplified version of your problem with the geometrical diagram.

It might be phrased like, how many committees of 3 can be formed by a group of 7 people such that no pair of people serve on two committees.

The figure is called the "Fano Plane" and a finite protective plane.

Your figure will be several orders of magnitude more complicated, but a similar idea.

You might want to look up "the Sunday golfer problem" as it is a well known and very similar problem.

Answer 2

1. As you should already be suggested, :-) (as far as I know) there is no direct plan or program of solution, it is a creative task. Math contests winners are good in that. They proceed by trials and errors, based on their experience, knowledge, intuition, conjectures, guesses, using symmetries, brute force search, trying to adapt known constructions for the problem solution. Sometimes you may be just lucky.

2. It depends on the problem, what is (much) harder to find its solution or to prove that it does not exist

I’ll illustrate the above ideas by a story about the magic hexagon told by M.A.B. Deakin.

3. Projective planes which you are referenced are studied by linear algebra over finite fields. As instances of relevant topics your may read about Steiner systems and Kirkman’s schoolgirl problem. Topological projective planes are very different topic.

Note: I’ll retag your question.