Well defined problems

For a problem to be well-defined it should have a clear start and goal states. And a well-defined problem can be represented as a ‘state space’ – a map of every possible ‘path’ of decisions from the start to the goal.

The Tower of Hanoi Problem

This is a classic well-defined transformation problem. Here is a diagram of its state space – i.e. all possible sequences of moves.

The right-most path shows the 7 step solution from start point to goal.

Means-Ends problem solving with sub-goals

If the problem is simple enough, you may solve it by drawing out the state space and finding the best line. You did this in solving the Prisoner’s Dilemma problem. The 2-person decision-trees depict the state space for all the possibilities there, and it is easy to see the solution once you represent the problem to yourself like this.

However, the state space can be too large and complex to represent easily (certainly in your head). For this reason, a better mindware strategy is to reason backwards from the goal state and planning sub-goals.

For example, you may figure out that to get to the goal  (7), you need to get the largest disk onto C (a subgoal), and to get to that subgoal, you need to move the top 2 to B. Thus, you might make (3) as your first subgoal and (4) as your next subgoal. From there it is relatively easy to see how to get to your final goal (7)

Planned subgoals for solving the Tower of Hanoi Problem

1. Move the top 2 to B

2. Move the largest to C

3. Then Move the top 2 from B to C

Mindware strategy: Means-ends analysis

Essentially we are breaking the problem down into smaller steps through what is called means-ends analysis. By breaking the problem down into simpler, achievable goals (subgoals) and then achieving each sub-goal in turn, the problem can be solved.

Means-ends analysis can be done with many problems you are faced with and it can help to schematize sub-goals and their possible solutions on paper.