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### Solutions for Act V Scene 2 - Hexagrams

These big-noded symmetric graphs are reminiscent of "magic shape" puzzles, like the magic star. The goal in these puzzles is to have each straight line sum to the same total.

Instead of being provided numbers to enter into these magic hexagrams though, we have letters. Letters can be naturally mapped to numbers based on their position in the alphabet (A->1, B->2, etc.) so we can still fill out the shapes with a little effort.

The presentation implies we want to write these letters into the diagrams, but it's not clear what orientation they should be in. For any given solution there are 12 possible ways to write it into the grid if rotations and reflections are considered distinct.

The trick here is to notice that in every solved hexagram, there is a pair of parallel words hidden somewhere amongst the lines. Depending on the orientation you used, they may be written backwards. For example, the words ASTI and POEM are parallel in the first hexagram, and the words KEPT and MANX are parallel in the second. A natural way then to orientate the hexagram would be by having these parallel words lie in the two horizontal lines of the hexagrams as presented. This leaves just one ambiguity: which order should the words be written? The most logical ordering system is alphabetical (as hinted at by the way the letter pools are given), so let's always write the first word alphabetically in the top row, and the second in the bottom. Thus for example the first hexagram looks like:

What can we do once all the hexagrams have been filled out? It should become apparent that in each row, the top nodes and the bottom nodes spell words or word fragments. For instance in the first row, the top letters of each hexagram read FIVE, while the bottoms read YOUD. In the second row, the top reads IDNT and the bottom reads PALM. While some of these words are less common than others, they should all be recognisable on sight, and at least one of each pair should be a very common word. Altogether we have in order:

Top rowBottom rowTop letterBottom letter
astipoemfy
keptmanxio
meowslipvu
anteflogip
peltskinda
haystomenl
apexdubsom
sureyolkht
fatswicken
kipsmungoe
moleragsdw
vansyoketi
expohymnik
aeropickhs
parswokeib
manspoketi
briofuelst
slitwrenom
holymugsfe
knitslaveo
dosetickaw
mapsoinklf
holyrubsei
furljinxgw
lawsmoveuh
tapsyokeri
fundgimper

It seems the top letters of the second row continue from the bottom letters of the first, and so on, alternating between tops and bottoms. This message says YOU DID NOT NEED THIS BIT OF EACH FIGURE. This seems a bit useless at first, but it's what the message isn't telling us that's useful - that is, that the other halves of each hexagram (the tops when we took bottoms, or vice versa) are the words we should care about. This is promising since the opposite halves are all four-letter words. The first row provides FIVE with its top nodes and the second gives PALM with its bottom nodes.

It seems likely that these two alphabetically-ordered four-letter words are themselves meant to sit in the top and bottom rows of a new hexagram. Indeed if we try constructing such a hexagram, there is a unique such construction assuming no letter may be repeated (as has been the case every time so far). The same is true for the other three pairs of words. We get:

 five palm c u hero wick a l hits meow l u tule whir c s

This time the tops spell CALC and the bottoms ULUS, which combined give our answer, CALCULUS. This ties into the mathematical theme of this puzzle and the original meaning of calculus as adding together small stones, which the nodes might visually represent.