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### Solutions for Act III Scene 4 - Composites

We are presented with a long list of numbers followed by an unusual diagram. Ignoring the image for now and looking only at the numbers, most of them seem to be huge, and many end in a lot of zeroes. One of the first things any mathematician should do when presented with such numbers is factorise them into their prime components (which is hinted at by the title, Composites), and doing so here helps make the lists far more manageable. In particular, it should become clear that the only primes represented are less than or equal to 23. Also of note is that 17 only ever turns up once as a factor.

This combined with the fact some numbers are separated by punctuation implies each number represents a word, and its factors somehow represent letters (all prime factors fall between 1 and 26 inclusive - the range of the alphabet - and the 17th letter is Q, a fairly rare letter). So how does this encoding work exactly? Evidently we can match up primes with their corresponding letters, so 2 is B, 3 is C, 5 is E, etc. But this leaves out the letters corresponding to composite numbers. Since there's no good reason to assign the sixth letter F, say, to any one particular prime, the logical conclusion is that F would be associated with both 2 and 3, its own prime factorisation. This mapping is supported by the fact there are usually many instances of each prime per factorisation (i.e. the indices are very high), so it's likely primes are representing more than one letter each. Also supportive of this mapping is the fact 1 appears by itself a few times, which makes sense as the number representative of A.

This still hasn't quite solved the problem though, as our indices are still far too high. The trick here is to realise that each index is in fact being used to index in the letter it's attached to. This can be determined by looking at small numbers like 196, which is 14^2, and likely represents the word AN since N is the 14th letter. To make this make sense, we need to write AN as 1^1 * 14^2 = 196, which can be read as putting the first letter to the first power and second letter to the second. Another repeat offender is 540, which we can now deduce with a little more work is 15^1 * 6^2 which implies the word OF.

Now that we have the encoding rule, decoding the given numbers is a pretty fun logical exercise (assuming you didn't brute force it with a program, which is of course acceptable but far less enjoyable!). Some of the longer numbers aren't feasible to decode by hand, but luckily almost all of them are predictable and then checkable based on context. The lines when decoded give clues which seem to describe distinct words, and working out the synonyms here is aided by realising the fact they're arranged in alphabetical order. Putting this all together we get:

 Decoded clue Meaning one who lives north of france and south of the netherlands Belgian what sharks have instead of bones cartilage in an icy manner coldly sweet meal eaten after dinner dessert fade away, or go extinct die off show; item in a museum exhibit completing a golf course; hiding (with 'up') holing a quantity of odds and ends; british sitcom following 'the' job-lot of or resembling rock or stone lithoid mother; one swallowed a fly old lady a lot, like the amount of gold at the end of a rainbow? potful release from distress; carved sculpture relief of the back of the eye; orange pigment retinal item of furniture for cup and saucer; hoodoo rock formation tea-table bathrooms; loos toilets eject or expel; remove grass? turf out

Some of these answers may still be too difficult to work out despite their clue and relative alphabetical position, but we should be able to rectify this upon discovering the next step. So far all we have done with this new code is decypher numbers with it; perhaps now we should try encyphering the new words we've extracted. Doing so should give a good idea of what's going on:

 Word Encoded number Belgian 1291262977998840000000 cartilage 5557418920397500000000000000000000 coldly 18139852800000000000000 dessert 12161026325655000000000000000000 die off 743847338880000000 exhibit 210594726147627000000000000 holing 1291262977998840000000 job-lot 18139852800000000000000 lithoid 210594726147627000000000000 old lady 71663616000000000000000 potful 455178184460284000000000 relief 743847338880000000 retinal 455178184460284000000000 tea-table 71663616000000000000000 toilets 12161026325655000000000000000000 turf out 5557418920397500000000000000000000

So in fact we have found eight pairs of words that share the same encoding. Promisingly, there are eight pairs of squares in the supplied image. There doesn't seem to be any good way to distribute the words/numbers though, so it seems we might still need to use the words somehow beforehand. The numbers have linked phrases like Belgian/holing, cartilage/turf out, and coldly/job-lot, but is there any other property that links them?

The trick here is to spot that for each pair, exactly one letter appears in the same position in both words (ignoring punctuation). So BELGIAN and HOLING both have third letters L, while COLDLY and JOBLOT both have second letters O. This gives us the following letters:

 Number Word Pair/Letter 1291262977998840000000 BELGIAN/HOLING 5557418920397500000000000000000000 CARTILAGE/TURFOUT 18139852800000000000000 COLDLY/JOBLOT 12161026325655000000000000000000 DESSERT/TOILETS 743847338880000000 DIEOFF/RELIEF 210594726147627000000000000 EXHIBIT/LITHOID 71663616000000000000000 OLDLADY/TEATABLE 455178184460284000000000 POTFUL/RETINAL

These letters seem like they could anagram to something, and indeed putting the numbers in ascending order gives us the word FLOATIER. Now we can finally use the image at the bottom of the puzzle. It stands to reason that we want to apply the previous step again, and find a second word whose encoding is the same as FLOATIER's, since this word itself is not a particularly compelling answer. While this step is uniquely solvable without use of the image, the image is provided to narrow down the search space of words with the encoding 4269358515114641664000000000000000. If we fill FLOATIER into the top line of eight squares, the symbols beneath them imply the size of each letter to be placed in the final line. For instance, the first column contains an F at the top, and implies the first letter of the second word must be a letter "less than F", i.e. with associated number less than 6. Using these hints helps us readily find a fitting answer:

Design Notes: When we first saw Gareth's idea of encoding words into numbers and discovered how decoding said numbers is almost a puzzle in itself, we knew we had to use it for a hunt. The overlapping letters step resulted from brainstorming loose definitions of 'composite' (amongst other words), and a quick check revealed there were enough viable pairs out there to create a solid puzzle.

Some of the words and phrases were a little more obscure than we would have liked, but hopefully the clues and backsolving were enough to get them all, or at least the majority. In some cases nicer pairs could have been used (inheriting/roadrunner was my favourite), but this would have made the puzzle too tricky or sacrificed the ordering-by-numbers step, which we believed was important for justifying the puzzle theme.

Another reason the image was added was so that teams that wrote up a code to solve this puzzle were not given too much of an advantage over those that didn't. We were very aware that if teams decided to program something to solve this puzzle, they could solve it faster than expected, and so while we tried to mitigate this as much as possible, this was essentially the only reason Composites wasn't labelled 5-stars.

The answer is: cofactor