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### Solutions for Act IV Scene 3 - Changing Sides

The answer is: elements This puzzle seems to be an encoding of some kind, turning words or letters into rows of numbers. Judging by lengths and general appearance, along with punctuation, it would seem each number is a converted letter, each line a word, and each stanza a sentence. But the encoding seems difficult to grasp since there are certainly more than 26 different numbers present. Nevertheless a general cyptographic approach can prove useful. For example, some lines like "4 35 36 1 246 21" appear more than once throughout the two pages, implying a repeated word, and probably asserting that a given word only has one encoding. Other strings appear very similar to each other, for instance "-2 51 17 165 1 60", "-2 51 17 165 1 364 60", and "-2 49 51 17 165 1 60" in the sixth stanza. The likelihood of three words in the one sentence having the same spelling except for one (inserted) letter is unlikely, so this seems to imply a word's letters aren't necessarily provided in their original order. We can also observe many small (1-3 letters) words that might provide some insight, as well as the general pattern that there is a 1 in every line. The other hint we can use is provided by the title. "Changing Sides" can mean a few different things, but in the context of numbers/mathematics, one might expect it to be referring to geometry, namely sides of polygons. A few brief searches can confirm that the numbers on display in the puzzle are all polygonal numbers, or numbers that can be represented as a number of dots together forming a regular polygon. Most people will recognise these in square numbers or maybe triangle numbers, but in general any n-gon has an associated number sequence. For example the pentagonal number sequence goes 1, 5, 12, 22, 35... Some solvers may also have recognised many of the polygonal numbers on sight. The question still remains how polygonal numbers are being used to encode words though. The aforementioned word analysis along with knowledge of certain polygonal numbers (e.g. "35 is the fifth pentagonal number") helps eventually reach the explanation. What's been going on here is that in any given word, each letter's alphanumeric value n is converted to its corresponding n-gon number sequence, and the rth entry in that sequence is taken, where r is the position each letter has in its word. This is more easily explained with an example: NUMBER has N=14 as its 1st letter, so we take the 1st number in the 14-gon sequence. U=21 comes second in the word, so we take the 2nd number in the 21-gon sequence. Similarly we want the 3rd 13-gon number, the 4th 2-gon number, the 5th 5-gon (pentagonal) number, and the 6th 18-gon number. At this point you might ask what a 2-gon number is, or indeed what happens if we have the letter A in a word. As it turns out, the general formula for the rth n-gon number can be given by r*((n-2)*r-(n-4))/2, so we can extend this formula to 1- and 2-sided "polygons" in this way. It is due to this that sometimes negative numbers appear in certain words, affirming they contain the letter A. The final thing to note is that the letters of NUMBER have also been rearranged after being converted to numbers. Once noticed this can be easily explained as following alphabetisation of the letters. That is, we have the mapping N->1, U->21, M->36, B->2, E->35, R->246, but the numbers need to be presented in the alphabetised order B, E, M, N, R, U. This finally gives us our mysterious line "4 35 36 1 246 21". The alphabetising can be explained as being used to encourage understanding of the "rth position trick", and to help the solver work out what the 1 stands for in each case, since by the nature of polygonal numbers, the first letter will always become a 1. With the code now finally sorted out, we can go about deciphering the puzzle proper. This can still be quite challenging even knowing the mechanic, but solving enough will eventually clue the solver in on certain tricks to use in decoding. Translated, we get the sentences: Number of different convex regular polytopes in four dimensions. Interior angle in degrees of one of the outer points of a regular pentagram. The Euler characteristic of the real projective plane. Number of uniform polyhedra, excluding infinite sets and not allowing edges to coincide. Number of vertices of a hexeract, also known as a cube in six dimensions. Least number of smaller squares needed to make a simple perfect squared square. Number of platonic solids that are their own dual. Using Miller's rules, the number of stellations of the icosahedron (not including itself). Total number of Archimedean solids including mirror images. Magic constant of a normal three by three by three magic cube. These are all somewhat geometric questions with integer answers. A bit of research will uncover all these answers, giving us the new numbers 6 36 1 75 64 21 and 1 58 15 42. This now finally translates to Euclid book using the same encoding, which implies our answer is Elements, the seminal treatise known amongst other things for introducing Euclidean geometry.