The Nature of Puzzles

I was thinking about how puzzles work and the nature of solving puzzles and I came to a couple of conclusions.  Generally, the way you go about solving any sort of puzzle is looking for a logical way to make some sort of progress, whether that be figuring out one number in a Sudoku puzzle or putting two physical puzzle pieces together.  If a puzzle allows all logical conclusions to yield significant progress then it is not a good puzzle because it fails to give the solver any sort of mental challenge.  There are many ways that a puzzle could be good.  One of them is having the first few logical conclusions be easily discovered, and then the final few would come with much more difficulty.  This would give the solver some hope before sprinkling in a little frustration.  Another could be a sort of middle of the road approach, giving relatively equal difficulty to any logical conclusions that could be necessary.  A third way would be to not give any logical conclusions that are easily discovered at first, but once a few are found, the rest would unravel somewhat easily.  An example of the first would be where you are given a square that has been cut up into many pieces and are asked to re-assemble it.  An example of the second would be a traditional jigsaw puzzle (even though the very end gets a little easier).  An example of the third would be a difficult Sudoku puzzle.  These, of course, aren't the only variations, but they provide some sort of basis upon which I can further discuss the topic.

I am particularly intrigued by the seemingly short puzzle that turns out to be equally frustrating and perplexing (especially when it involves a significant amount of mathematics).  These puzzles can oftentimes involve making an "illogical conclusion" that becomes logical within the given parameters.  Another solution could result in combining unrelated logical conclusions to make a leap that will push the solver in the right direction.  A puzzle that fits the "illogical conclusion" category could look like: if half of a half of 5 is 2, what is a third of a quarter of 8.  This isn't a particularly frustrating or perplexing problem, but the "illogical conclusion" lies in equating numbers to other numbers which clearly have no business being equal.  Through proportions, the answer is 16/15.  

A type of puzzle that I haven't really seen (not that I am actively solving a large number of puzzles) could involve embedding or threading two puzzles that traditionally have no involvement.  An easy to see example of this would be putting sequences with a missing term in each square of a Sudoku puzzle.  My question is how far could this embedding and threading go?  Let's say we take 10 kinds of puzzles and find a way to embed and/or thread them.  Then solving the 10th could lead to the solution of the 9th and then the 8th until the 1st.  Or, the 10th could lead to part of the 5th which leads to the full solution of the 9th and then to another partial solutions which allows the 8th, 7th, 6th and so on.  Or, (even worse... or better...) the part of the first 9 could lead to the 10th, a different part of the first 8 could lead to the 9th, and a part of the first 7 could lead to the 8th until the final solution is achieved. 

The next thing to consider would be how to make the combined puzzles be good puzzles.  Even with the numerous steps involved, it could still not be challenging enough if one of the embedded puzzles is very simple, giving too much help on the other puzzles.  But if all are too difficult and give no starting point, then that could surely make a poor puzzle as well.   An ideal scenario could look something like this for 5 intertwined puzzles with 10 sections each:

You start with A1 and that leads to B2, then C1, D2, E2, E1, D1, C2, B1, A2-A5, B5, ..., D7.  Such a puzzle as this could provide difficulty since there is only one possible path, but never leaves the solver without a possible logical next step.  If the puzzle maker wanted to make this even more difficult, the line could be discontinued at one point and multiple other steps would have to be figured out simultaneously to make any advancement.  This could involve taking guesses at some parts of the puzzle and working with those guesses until you discover you guessed correctly or have to try again.  There are many more possible puzzle formats that could be explored and there will always be ways to change parts of them or make them more complicated.

I feel like puzzles matter because they are an excellent method for one person to give another person a mental challenge that could be rooted in abstract or concrete concepts or any combination of the two.  They are, in a sense, a rope taken by one person who tangles it up fiercely and then says to another, "Here, solve this."  They can teach the solver to think out of the box and can teach the creator to think inside out of the box.  They can be a form of communication, frustration, and eventually (hopefully) elation.