Puzzle Complexity

In the last blog, I mentioned that The Nagging Wife puzzle had a level of 3.1.2 difficulty without really explaining what it means.  If you see a string of numbers like this separated by periods, it is usually the number of moves required to remove the pieces from the puzzle.  In the case of The Nagging Wife, it takes 3 moves to remove the first piece, 1 to remove the second, and 2 to remove the third.  For many puzzles, this is used as a proxy for its difficulty.  As the number of moves increases, the puzzle is generally more complex and difficult to solve.

In reality, puzzle difficulty is completely subjective and although there may be a general consensus on the general difficulty of a puzzle, everyone is different and has their own perspective.  For instance, Hanayama puzzles come with a rating of 1 to 6 stars with 6 being the most difficult.  I particularly like the old style boxes with the die showing the level of complexity.  When a new puzzle is released, there are sometimes frequent mutterings on the miscasting of the difficulty within the puzzle community with some saying it was too low and others too high.  As I get older, it seems that the number of stars is skewing too low.

In 2017, I gave a speech on puzzle complexity and provided the following clues for gauging the difficulty of a puzzle:

Suggested Level – If the puzzle is mass marketed, it probably comes with a suggested level of difficulty.  In some cases, the level may be provided by the designer.  After all, these designers are the world's greatest puzzle experts and have a lot of experience with puzzles and how well people can solve them.  If you’re new to the puzzling world, you may find them underestimated.  If you are an experienced puzzler, you may find them overestimated.  Look for the ones with the icon of the guy pulling his hair out for the best puzzle experience.  I know quite a few bald puzzlers.

5 Piece Plate Burr
Delight by Stéphane Chomine

Number of Pieces – Piece count can be an indication of difficulty.  The more pieces that there are to work with the more difficult it usually is.  Of course, a 5 or 6 piece plate burr puzzle could easily be more difficult than a 2000 piece jigsaw puzzle.

Piece Variety – Usually puzzles where all the pieces are different are more difficult than when they are all the same.  When they are all the same, you don’t have to worry about where each piece needs to go relative to the others.  This could be a factor of N! for mathematicians.

Moves Required – In general, as the number of moves required to solve the puzzle increases, the complexity of the puzzle  increases.  However, some puzzles like N-ary puzzles may require very large numbers of moves to solve but may not necessarily be difficult.

N-ary Puzzle - Spin Out by William Keister

Piece Shapes – Everyone is really comfortable solving a cube dissected into smaller cubes.  Smash it, stretch it, skew it, and deform it in so many unspeakable ways and it becomes intimidating.  Haym Hirsh has been doing a nice job recently of distorting 3x3x3 cubes (Inelegant Cube, Inelegant Box, Somaa, Jitter Soma) by using pieces that aren’t cubic and some that leave gaps in the solved state.  It messes with your head when the pieces are no longer flush against each other when building the cube and once you start cutting pieces at non-right angles, things get much more complicated for the designers, craftsman, and puzzlers.

Burr vs. Disentanglement
Wookey Hole by Stewart Coffin
and Man-O-War

Puzzle Type – Most puzzlers have a preference for the types of puzzles that they are interested in at the moment.  Give a hard-core burr puzzler a rookie disentanglement puzzle and he may struggle like a newbie.  That’s what makes it so fun!

Rotations – For cubic dissections, the trend has been moving towards creating Turning Interlocking Cubes (TICs).  I believe that Bernhard Schweitzer coined this phrase while doing a study on TICs.  When solving a burr, knowing that a rotation is required is a significant clue.  It activates all sorts of additional puzzle solving neurons in the brain while working on a puzzle.  When you are completely stuck on solving a burr, you eventually ask whether a rotation is required.  If the answer is yes, you happily go on searching for the rotation.  If the answer is no, despair settles in.  It’s as simple as that.  Andrew Crowell is the current master of creating cubic dissection puzzles with rotations and they are all recommended.

Coordinate Motion
NOS 5 Crenel by Gregory Benedetti

Coordinate Motion – Movements with pieces simultaneously going in different directions can add to a puzzles complexity.  In many cases, getting all the pieces that have to move at the same time in position can be a real dexterity challenge like Rosebud by Stewart Coffin.  One of my favorite coordinate motion puzzle designers is Gregory Benedetti who did a spectacular job with the New Old School (NOS) Burrs.

Number of Solutions – Many packing puzzles have multiple solutions.  In general, it is expected that as the number of solutions increases, the chances of finding one becomes easier.  For instance, the Soma Cube by Piet Hein has 240 ways to create a 3x3x3 cube and it is not that difficult to find one.  The Half Hour puzzle by Stewart Coffin, however, has only one way to create a 3x3x3 cube and is much more difficult.  I’m guessing that the Half Hour puzzle was named for how long it took Stewart to make the first copy of the puzzle given how much longer it took me to solve it.

Two 3x3x3 Cubes - They Don't Look That Different!
Soma by Piet Hein (Left) & Half Hour by Stewart Coffin (Right)

Cord Length – For disentanglement puzzles, the length of the cord is usually just long enough to accomplish the solution.  The longer it is, the higher probability that you can tie it into a knot.  Beware if it is long enough to hang yourself with.

I’m much more careful now about giving out information on puzzle difficulty after an extremely embarrassing faux pas.  While attending my first International Puzzle Party, hobnobbing with the cognoscenti of puzzledom, I put some of my puzzles out in the hotel lobby for others to play with.  One attendee asked me if one of the puzzles was difficult to solve.  It's easy I told him and then watched him struggle for hours trying to solve that puzzle.  I think that he may have been shamed by my comment into trying to find the solution and instead of enjoying the puzzle, it may have become a burdensome task.  Of course, I like to think that, like myself, he enjoys being frustrated.  Why else would we do it!

If you are interested in the Man-O-War puzzle, you can find it at PuzzleMaster along with Hanayama Puzzles, the Soma Cube, and the Half Hour puzzle.