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### Solutions for Act V Scene 4 - Hexagons

There's not much to go off to begin with in this puzzle. We have two lists of words with a strange image between them, and six small shaded hexagons on the left. It's likely that we want to go from words on the left to words on the right in some way, probably involving the gear image. But to be able to do this we need to know how to use this gear, which in turn implies we want to understand how the shading of the hexagons works.

It's reasonable to assume then that the six hexagons on the left are meant to be enough for us to determine an encoding rule. The first two having the "first" (clockwise) two segments of the hexagons shaded seems promising, though the remaining hexagons don't follow this trend. Still, it seems safe to assume these hexagons represent the numbers 1-6, hopefully in order, and running with this assumption helps confirm the correct encoding. In particular, if the third hexagon represents 3, that coloured section must represent 3, and since 6 colours in this and the adjacent section, it's likely both sections represent 3. At this point we can extrapolate that the encoding is a form of ternary - a number is converted to ternary, and then the cells corresponding to position shaded. Note this means the cells have the values shown below, and also that whenever a 1 appears as a digit, we have a choice of two cells to shade (and the given numbers 1-6 don't seem to enforce any particular ruling on which ought be chosen).

So we have a way to convert numbers/letters to shaded hexagons, as well as a natural cell order (clockwise from top). Writing a six-letter word into the gear then should probably also go clockwise from the top.

The next step here is to expect that writing a word from the left column into the gear and turning the gear will give a word from the right column at some point. Because of the ambiguity of shading for digits that are 1 in any letter's ternary representation, this can prove difficult to check for, but a few tricks can be used to eliminate possible pairings. First of all, certainly the number of black/white cells shouldn't change, so calculating the number of black cells per word helps narrow things down a bit. Another thing to focus on is letters whose representations include only, or mostly, 0s and 2s in ternary. Looking at numbers of cells in individual letters will also help eliminate some pairs, as well as confirm how far a gear should be turned to create the new word.

For example, testing this with DEGREE, which one might focus on considering it's the first labelled word, shows we can write DEGREE in the gear in such a way that after turning 180o, it becomes DUMDUM, a word on the right.

In fact we can find 18 pairs (half the total) that have this property. At this point we should expect the other pairs to behave similarly but under different degrees of rotation, and indeed looking a bit closer, 9 of the remaining words turn into right-column words after a 60o clockwise rotation, as read starting from the third position - and the other 9 pairs are found from rotating 60o anticlockwise and reading clockwise from the fifth position.

The niceness of the partitioning of these pairs seems important, and indeed thinking of the fact these words are written on gears, it makes sense that there should be as many that can be turned a fixed amount in one direction as those that can be turned the other. It strongly implies we want to interlock certain gears together. At this point an intuitive leap to make is to suspect the gears belong in sets of six, which can also be arranged in a interlocking hexagon configuration.

The question now is how these sets of gears should be defined. What we haven't yet used are the arrows pointing to a central, faded but larger hexagon in the gear. They seem to imply the outer hexagons inform the centre one somehow, and the most straightforward interpretation is that each cell of the centre hexagon adopts the colour of the outer hexagon's cell that's pointing to it. So in the DEGREE->DUMDUM example, the centre letter is a J in the first instance, and a D in the second.

Continuing with our hunch that the gears should be placed in hexagonal sets, it seems likely the informing rule is that the six centres should spell out a new word, again clockwise from the top, likely both before and after rotation. After a little trial and error, one can quickly form the three sets of six gears that rotate 180o to create common words each time. These sets are (noting if two pairs produce the same centre letters, they can obviously be interchanged):

Left wordCentre letterRight wordCentre letter
DEGREEJDUMDUMD
KUMARAAIDIOMSI
SURFERMJUDGEDS
PAMPERMNUDNIKE
Left wordCentre letterRight wordCentre letter
TAPPEDRTURPIND
KAMIKSICOMMIEA
PLUGINGMANTLEM
DETAILOCALL UPM
KEDGEDUMURMURE
JAMMEDRSURFIED
Left wordCentre letterRight wordCentre letter
CANIDSAALMAINC
CURL UPMLEVIEDU
LEONIDOVALLUMM
PULPITETAPPEDU
MINIMSBAEGEANL
SIGLUMALEMMASI

For the remaining sets, we want to alternate clockwise and counterclockwise rotations so that the gears behave normally, so this should help limit the potential word pool of centres. It soon becomes clear though that there are too many inconvenient letter conversions in the case of these gears, and less common letters appearing as centres for right-column words, for these sets to behave exactly as the 180o-turn ones do. Still, it is possible to partition the 18 letters provided by centres of the left-column words into three somewhat common words in only a few ways, so we can at least set these gears up and see what happens under all possible rotations. Indeed one set gives us three configurations whose top gears all rotate 60o clockwise to turn produce the individual gears' right-column words, and 60o counterclockwise to produce a new gear-centres word. These sets are:

Left wordCentre letterRight wordCentre letter
YEANEDMGANGLYO
SHRANKOSUNLITU
MUTISMNSNAKEST
INDIUMAPLASMAA
GALAGOYMEALIEW
Left wordCentre letterRight wordCentre letter
DYNAMONGLYNISN
MEANIELMANAGEL
CLAMPSIMALIGNI
YUPONSNSPUNKYN
GRAMMEKKELIMSE
Left wordCentre letterRight wordCentre letter
MALICEACLAIMSI
COIN-OPPPAGINGP
YANKEEPCHEMMYP
AVISEDYBIVIUMY

The confirmation that these sets in particular are correct is that in the individual gears here, rotation in one direction gave us the required right-column words, but they didn't start from the top, and were instead offset by an amount rectified by turning the same amount in the other direction. This corroborates with how the 180o turns worked, since in both cases we're turning one direction to find the word and the other to position it - but this gives the same result in both cases if you're turning 180 degrees either way. The motivation behind this construction is more easily explained by the GIF used in the third hint for this puzzle:

Since we've now constructed more hexagons-of-hexagons, this time where the outer hexagons are represented by the gears' centres, it seems reasonable that we should do the same as the original gear suggested, and extrapolate a centre hexagon from the touching cells of the outer ones (also seen in part in the GIF above). Doing this for the original gear sets gives the letters UWAMOT, which disappointingly don't anagram to give a new final word. We do have an inherent ordering on these though, which we haven't used since the very start - the numbers 1-6 written on the left of the puzzle can now finally be used to order something, namely our six sets of six gears. Each number is associated with a unique letter pair that belongs to only one gear-set's two words (e.g. J->D from DEGREE->DUMDUM only happens in JAMMED->DISEUR), so this is how we can associate an ordering with the final sets.

The only thing left to do is turn the gears and see what letters we get in the centres. In order the letters turn out to spell the answer hex key, a hexagonal tool used to rotate things, and also a description of the hexagonal encoding this puzzle hinges on.

Design notes:
This puzzle was inspired by the idea to pictorially represent three-digit ternary numbers, themselves nice since they run up to exactly 26, the number of English letters. Hexagons were an obvious contender, and their rotational symmetry seemed a good thing to exploit. When early testing revealed there were two natural ways to interpret letter rotation - either individually, or all together as a word, we realised we could try using both interpretations and up the difficulty of this puzzle. At some point it was decided this was a good contender for the final 5-star.

The original plan for this puzzle was to not provide the list of words, but rather crossword-style clues for them, and furthermore to convert all the letters to their hexagon counterparts beforehand. This had the benefit of giving many more examples of shaded hexagons for the solver to deduce the encoding, but was messier, more dangerous (in that people might derive the wrong word), and required nicer words in general. It also meant that we couldn't as easily provide the ordering that's finally used at the end, which meant a less intuitive ordering had to be constructed - the ideal version of this puzzle would have the final six hexagons spell out ANSWER before being turned to reveal HEXKEY, but this proved to be an impossibility.

In fact, many parts of the puzzle had to be reworked because there simply weren't any pairs of normal-ish words that produced the desired results.

Later into testing, a program was put together to exhaustively find word pairs that worked. This turned out to have an error in it that produced words that don't always start from the top position of gears, which tried to rectify the problem by turning the gears back the same amount they'd been turned forwards before extracting the central letter. When this error was soon spotted and fixed, the new list of words we got meant certain required letters were no longer possible to produce from normal/common words. The two ways to deal with this were either to come up with a new answer or even mechanic from scratch, or adopt this two-way rotation approach and cement the puzzle as a 5-star difficulty one.

As we've seen statistically, the average difficulty of this year's puzzles was likely higher than usual, so deciding to stack this last trickier step onto the final puzzle was in retrospect not a great idea. We've now seen one unfortunately promising alternative way to interpret the 60o-turn gears via feedback from team The New Plugh, who realised that reinterpreting the hexagon encoding as behaving counterclockwise allowed for a neater gear conversion that required only the one rotation. So apologies to those teams who were led down the wrong path, and we hope you still found some fun in extracting the answer either during the hunt or while reading through this explanation.