# Reverse Puzzling 4

[This continues prior Reverse Puzzling here, here and here]

George rubbed his hands together with glee. "So this is Andrea's puzzle, is it?"

"Yes," I answered. "It's the first of them, anyway."

"It's not very big, so it shouldn't be too hard. How does it work?"

I explained the concept to him, and after a few false starts a frown furrowed his brow.

"Let me see," he muttered to himself, as he got out a pen and paper.

After a little while, following his usual systematic (and somewhat laborious) approach, he had drawn the following:

I knew what he was drawing, so I think I can say that the lines are all meant to be arrows (either single or double-headed). He occasionally didn't draw them in properly, but you shouldn't let that put you off.

It was remarkable to me how complicated it looked, when the original puzzle looked so simple and elegant.

What was the original puzzle?

Edit: After George saw the puzzle, he was embarrassed and insisted I update the diagram to show all the arrows more clearly.

• Hmmm, interesting. – Gareth McCaughan May 24 '16 at 22:30
• George has wonderful handwriting. – paste May 25 '16 at 2:40
• Observation: red/green seems to be red herring as clearly the correct path start to end goes through red nodes only. To me, this nearly seems to be a simple 'maze' puzzle, but that would contradict the assumption of a 'well known puzzle'. – BmyGuest May 27 '16 at 22:13
• @azgreentea, the other's were all solved too quickly. So I chose a less famous one that would need to be solved by deduction! – Dr Xorile May 31 '16 at 18:16
• Great puzzle, and great puzzle behind it. What a perfect way to introduce us to "somebody else's puzzle" without copying the puzzle. Very enjoyable, but I've given you my +1 already... – BmyGuest Jun 6 '16 at 13:01

This is Andrea Gilbert's first handmade

Orientation Maze (http://clickmazes.com/orient/g4g5.htm), made for Gathering For Gardner 5.

The rules are simple:

You enter the maze on the lowest tile facing north. You may follow any direction on the tiles in front of you. The goal is the top right tile.

How it relates to the diagram:

Each group of 4 (or 1) in the diagram represents a square, and the positions in those groups represent directions that you face. Arrows represent valid moves.

• Maybe I don't follow correctly. Wouldn't you hit an arrow pointing south that would send you right back into your starting position? – Engineer Toast May 31 '16 at 19:30
• More important than this answer is how you came to find it. – LeppyR64 May 31 '16 at 19:45
• @LeppyR64: I was already familiar with Andrea Gilbert's website. The grid looked very similar to something I distinctly remembered seeing several years back, so I went to the website and looked around until I recognized it. – Deusovi May 31 '16 at 19:47
• I hoped that someone might be able to figure it out based on the arrows. After a week, I would have added an extra clue. For example, people had already noticed the 4x4 grid, and the possibility that the direction was the third dimension. If you noticed that facing outwards gave you no options, and options were only added as you moved back from the edge, it seemed possible that you'd get it. I included Andrea's name in there as a big clue too. Congratulations, @Deusovi! Hope everyone enjoyed wrestling with it! – Dr Xorile May 31 '16 at 21:06
• Huh. It never occurred to me that the name Andrea in the puzzle might be the name of the actual creator of the actual puzzle. D'oh! – Gareth McCaughan Jun 2 '16 at 20:42

So, some observations. (Nothing close to a solution.)

Obviously the diagram shows the state space of a puzzle and (some of?) the available moves from state to state.

The state space seems to be four-dimensional: 4x4x2x2. (Two size-4 dimensions for which group-of-four. Two size-2 dimensions for which blob within a group-of-four.) Every move changes just one of the four components, by just 1. This feels a bit like a sliding-block puzzle -- perhaps the four components of the state are two pairs of coordinates, or something. (But see below for one reason to dislike the sliding-block-puzzle hypothesis.)

There are a few places where a sequence of moves has no overall effect, but startlingly few. On the whole, any move you make is a permanent commitment and if you make a wrong one you will eventually find yourself out of moves and having to start again. I find this odd in combination with the absence of any sort of "directionality" apparent in the state-graph: it suggests that the nearly-one-way-ness of the puzzle is just a matter of clever state-graph design, rather than emerging from some physical feature like an ever-tightening screw or a gradually-dropping ball or whatever.

There appear to be separate start and end states. E.g., if this is a sliding-block puzzle then maybe it has gaps at the sides and the challenge is to get one piece in at one side and out at the other.

Obviously red is being used to mark the shortest path between start and end. It looks initially as if there's some redundancy in the northeast corner, but if those arrowheads are correctly drawn then indeed we need to use all the red states. Note that this means that the available moves really truly are often irreversible, which is strong evidence that this is not a sliding-block puzzle.

The "short" edges -- within the 2x2 clusters -- have no arrowheads. This may just be because there isn't space for them, but another possibility is that whatever process changes state within a cluster is always reversible. [EDITED to add:] After Dr Xorile's update with the clearer diagram, it's apparent that actually the short edges are frequently unidirectional. So ignore this paragraph.

Earlier "Reverse Puzzling" puzzles have featured specific rather famous puzzles. This one isn't ringing any bells for me yet, though.

• A good start. This puzzle is certainly not as famous as the previous ones, but hopefully the format can be deduced from the evidence – Dr Xorile May 25 '16 at 15:06
• I keep thinking that those clusters of four represent some sort of rotation. Irreversible moves may imply gravity is in play? Also note that we are seemingly working with 3 degrees of freedom, which may imply a 3-dimensional puzzle. – paste May 25 '16 at 21:01
• Rotation is a nice idea, yes. At least as plausible as my claimed 2x2 possibility. I'm not so sure about gravity -- I'd expect the directions of the arrows to be more consistent, somehow, in that case. – Gareth McCaughan May 25 '16 at 21:13
• Note that the group of 4 in the end is the exact opposite of the group of 4 at the start (colors, position in the grid, position of the start/end in the group of four) – Fabich May 25 '16 at 21:40
• Another note: Since there seems to be a number of irreversible moves, it's possible that there's some sort of "reset" function that puts you back to the start, or at least some way to start over without traversing the state machine back to the beginning. That may imply that this puzzle works on soft rules (disallowing moves) rather than hard rules (e.g. you can't physically swap two corners of a Rubik's cube). – paste May 26 '16 at 3:40

More items of note:

All outside edges of the state map are either untranslated (no action) or a dead state (acted upon but no further translations). Hence, none of them are red (red must have a continuous line of uninterrupted action).

The path of the red circle is as follows with (1,1) being the upper left set (increasing to the left and down) and labeling the positions as A-D from the left going clockwise:

(2,4) B [ENTRY]
(2,3) B
(1,3) B,C,D
(1,2) D
(1,1) D,C
(2,1) C
(3,1) C
(3,2) C,B
(4,2) B
(4,3) B,A,D
(4,2) D,A
(4,1) A
(3,1) A,D [EXIT]

After working hard on this 3-D directed graph, I guess it's the:

Hat-guessing problem, with $4$ logicians and $6$ hats, 3 red and 3 green.

The reason that it feels that way is,

At each level (or say row), there are two arrows emanating from one of the spheres(or simply circles) which represent the two possibilities of the colour of hat on his head. And the single arrow at the same level represents that one logician is either looking at their hats or thinking from their perspective(trying to deduce the color of his hat using inductive logic).