A technique in solving puzzles

In the 1960s when pentominoes were popular, a challenge was to fit the 12 pentominoes into a rectangular frame. I formed the view that it was helpful to deal with the awkward* ones first, the compliant ones would be more likely to fit into the last few spaces. This wasn't a great heuristic, but it was better than nothing. The picture to the right shows a set of pentominoes, with what I perceived as the more awkward ones further to the left.

This page is an attempt to refine and clarify the heuristic "deal with the awkward ones first".

Below are some specimen puzzles, which I thought might help me refine the heuristic. I hope you will look at them, try to solve them, and maybe look at my answers.

Puzzles

  1. Two-dimensional jigsaw
  2. Three-dimensional jigsaw
  3. Joining points on a disc
  4. Permutation of a set of integers

Conclusions

Before I started writing this page, I had vague thoughts about how to refine the concept "deal with the awkward ones first". Some ideas were

Now that I've thought about these four puzzles, the advice is

This is simple and obvious, once stated. It leads directly to answers for puzzles 2, 3 and 4. No thinking, no trying the more promising looking things first and backtracking when they don't work. This advice reduces three tricky-looking puzzles to non-puzzles, where you can go direct to the answer. I haven't just refined a heuristic, as I had hoped; I've found a powerful technique.

Ok, these four puzzles were cherry-picked to be susceptible to advice like "Deal with the more constrained pieces first". But I was surprised how thoroughly they succumbed to "make forced choices first".

Problem 1 is meant to be easy. It seems so obvious to place the "snakes" before the smaller pieces. Is it obvious to everyone? Why is it obvious?

But I've seen smart people fooled by puzzles 2, 3 and 4. Ok, if you're not aware that there are any forced choices, the advice to make those first isn't helpful. But if there might be some forced choices, identifying and making them is a big step forward.

All four puzzles can be regarded as finding a route through a search space. The advice "make forced choices first" can be interpreted as "if you approach your search space in the right way, it might turn out to have no choice points".

Footnote

In the course of writing this page, I've changed my views on which pentominoes are most and least "awkward". The X pentomino, column 1 row 2, is the most awkward. This is because, if you have space for one pentomino still to fill, there's only one way the X could fit into it. Whereas if your one remaining pentomino is the P, column 4 row 3, there are eight ways it might fit. Pentominoes with few symmetries are more compliant than those with many.

This is one of several miscellaneous pages listed at http://www.weddslist.com/writing/.