Solution: Flexibility

Answer: A MANIFOLD MAKER

Written by Thomas Gordon

This is the metapuzzle for Floor 2. It uses the answers from the floor, which are:

As teams solve puzzles in this round, symbols will be listed with the puzzles on this Floor; additionally, these same symbols will appear in the Shape section of the meta.

As brief experimentation (or some insightful knowledge about recreational mathematics-related toy-things) will tell you, the Shape is a net for a hexaflexagon. Folding it up correctly will give you a functioning one. Additionally, if you write the letters of the answers onto the triangles of the net (with the first letter of the answer being placed on the symbol for that puzzle, and writing left-to-right with one letter in each triangle), that will spell words on the faces of the hexaflexagon that begin with the letters A through F. These words are:

• ARDOUR
• BARKER
• CASINO
• DRAGON
• EMOTES
• FRUGAL

These faces can be "moved between" by folding down and flexing out the hexagon. (Note that the direction which the word is read around the face can vary, depending on the state of the hexaflexagon.)

The next part of the puzzle involves understanding the Notation section of the puzzle. Associate each of the flexagon faces with a letter from A-F based on the word that is spelled in it. As the diagram in the Notation section explains, if you flex the flexagon and the letter goes up alphabetically (eg B to D, A to E, C to F), this is notated as a “+”; if the letter goes down alphabetically, it is notated as a “-”. This provides us the givens that we'll use in the Directions section.

Finally, you will need to follow the directions. The shape has a certain face shaded (BARKER); this is the starting face, and the configuration made immediately after the folding is finished is the starting configuration. There is a unique way to fold through the faces, following all of the directions exactly and fulfilling the global constraint of visiting every face twice. The easiest way to work this out is by drawing out a diagram of all the states that the hexaflexagon can reach, and what faces are available from each of these different states; this is known as a Tuckerman diagram.

Using the Tuckerman diagram above, a path can be devised that follows all of the directions exactly, and visits every face twice, starting from the configuration that the hexaflexagon is in when it is first folded (the central B face on the diagram). The logic can be resolved by considering several moves ahead, and where you will need to fold in order to be able to execute future instructions as well as current ones.

The opening break-in is potentially the most difficult, because it begins with an ambiguity which will need to be resolved at the end by the overall constraints. After the + move, you will be on either the C or E face; after the - move, you will be on either the inner A or outer A face. However, since this is followed by two + moves in a row, regardless of which face is visited, after the next move, the face will have to be B (since going from the inner A to the F forces a - move next). From here, while C and E are both forced, the next + move locks it into being C (since going to E forces the next move to be a -). After that, faces are determinable using just the next move for a while:

BC/E A B C D B C D B C A

At this center A, consider the next three moves: after a + move, we will need to make two subsequent - moves. The only place we can do that anywhere in the diagram is by passing through F! Hence, we can make more progress:

BC/E A B C D B C D B C A F C AF C A

The next move must go to B, because otherwise we would be forced to make a - move:

BC/E A B C D B C D B C A F C AF C A B

Next is the same ambiguous C/E pattern, and the same forced move back to B afterwards. We'll put down both options for now.

BC/E A B C D B C D B C A F C AF C A B C/E A B

To uniquely determine the next step, we'll need to look five steps ahead, at that double - move. We know that the only place to do that is after passing through F; hence, we'll need to make our way directly over there in order to complete the - - moves:

BC/E A B C D B C D B C A F C AF C A B C/E A B C A F C A

And the rest is determinable from using just the next move!

BC/E A B C D B C D B C A F C AF C A B C/E A B C A F C A B C D B C A

Time to use that global constraint. Since we know that every face is visited at least twice, and E can only be visited twice (in those two C/E cases), both times must be visits to E. Thus, the complete moveset is:

BE A B C D B C D B C A F C AF C A B E A B C A F C A B C D B C A

Finally, indexing into the words on the faces at the indicated points, we can extract the necessary letters.

Which indicates what is menacing the library: A MANIFOLD MAKER.

Author’s Notes

As is the manner of things, I remember precisely where I was when I first formulated the very beginning of this puzzle. I was waiting for a train...

Metas are written first in a hunt situation, and as such their details are sometimes lost to time. But I would be lying if I didn't say that I remember the absolute ringer that this puzzle put me through. Poring through endless qat queries, balancing good words and bad words and stranger words besides; all activites which will drive you mad, if you have not steeled your heart. Despite the usual crew not being credited on this puzzle, I thank them anyway for their input on this puzzle, its endless adjustments, and which of the bad options for answer sets would ultimately be most palatable.

For Shardhunt, the story was written meta-puns first. (The metas were also written in 1-2-3 order, for this and other reasons.) So when I originally pitched this puzzle and its answer - which I believe back then was shorter - that ended up significantly redirecting the story that we had been formulating for the hunt, much more than any of the other meta answers! It's now hard to imagine Shardhunt without the various things that this meta answer implies, but back in those heady and halcyon days we had really no idea what the story would even be. Strange to think about.

Yes, I folded a lot of hexaflexagons in the process of writing this puzzle. Yes, it's inspired by those ViHart videos.