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### Solutions for Act V Scene 3 - Digital Division

The layout of this puzzle can immediately be recognised as the shape of the keys on a keyboard, and the subset of keys it shows seems to be all the letters and numbers of a standard keyboard (the top row being the numbers and the rest, letters). Following on from this, those keys with zeroes on them can be identified as the 0, Q, W, Y, J, K, Z, V, and B keys, and these letters certainly seem to be amongst the rarer ones used in English. Also of note is that those letter keys with numbers greater than one (E, R, T, I, A, C, N) are in general more common letters than those with numbers less than one (U, O, P, S, D, F, G, H, L, X, M).

Since each of the keys sport a fraction of some sort, it seems natural to want to homogenise the format somehow so that the keys might be more easily compared. One form that every key can be converted to is the form x/y, i.e. an integer over another integer, since this is simply a matter of converting every mixed numeral to an improper fraction (e.g. the first key, 8 144/779, can be converted to 6376/779). Doing so should result in the following list:

 Key (Improper) Fraction Key (Improper) Fraction Key (Improper) Fraction A 5862314 M 23/386842 Y 0 B 0 N 961/614 Z 0 C 4909/734 O 53/84898 0 0 D 29/7504 P 37/17126 1 6376/779 E 14985/88 Q 0 2 4511/748 F 1/72485 R 782263/17 3 24704/73 G 2/98737 S 689/1642 4 61/32114 H 61/66224 T 1962574/83 5 10319/515 I 359624/7 U 289/4031 6 658/11927 J 0 V 0 7 5947/848 K 0 W 0 8 6428/709 L 31/49589 X 1/328823 9 2/486403

Of these fractions, there is still no apparent link that connects all the keys even in this more similar format. There is one fraction in particular that stands out however - T's fraction, 1962574/83, is the only one to use nine digits altogether (all other fractions use less than nine), and furthermore it appears that in writing this fraction, every digit from 1 to 9 appears exactly once.

Could this property apply to the other fractions? An obvious key to test this on is A, which in its current form is an integer (5862314), and thus undesirable in so far as it is not of the form x/y which we have turned all the other key's numbers into. Clearly in order to make it this form without changing the value we must express it as an unsimplified fraction. 5862314/1 does not satisfy our digits condition, and nor does the equivalent 11724628/2. But using the next multiplier seems to do the trick - the number on the A key is equivalent to 17586942/3, and this fraction does indeed use every digit from 1 to 9 exactly once.

The title should also confirm the conversion step we've discovered; Digital Division suggests fractions ("division") that in some way use digits, and here we are making every fraction use all the decimal digits excepting 0. Thus our next step is to "unsimplify" the provided fractions, by multiplying each fraction's numerator and denominator by a common number, and checking if the new fraction contains all the digits 1 to 9 exactly once. This task can be made easier by obtaining bounds on the multipliers for each fraction through considering the total number of digits of the unsimplified fraction. Furthermore, for all but one case in which it is 37, the multiplier for each fraction is less than 20 (and indeed the 37 case involves a denominator ending in 5, so even multipliers automatically need not be checked for this key anyway).

Ultimately the following list of fractions should be found, with each fraction being the unique equivalent form of the key's original fraction that uses exactly one of each digit from 1 to 9 (and no zeroes):

 Key New Fraction Key New Fraction Key New Fraction A 17586942/3 M 92/1547368 Y 0 B 0 N 15376/9824 Z 0 C 63817/9542 O 371/594286 0 0 D 493/127568 P 518/239764 1 76512/9348 E 134865/792 Q 0 2 54132/8976 F 37/2681945 R 2346789/51 3 197632/584 G 34/1678529 S 8957/21346 4 976/513824 H 854/927136 T 1962574/83 5 92871/4635 I 2517368/49 U 4913/68527 6 3948/71562 J 0 V 0 7 65417/9328 K 0 W 0 8 83564/9217 L 186/297534 X 12/3945876 9 38/9241657

Next we must work out what to do with these fractions; the title helps again here, suggesting the fractions have broken the digits into two divisions, or sets. Regarding the observation made earlier that larger numbers are associated with more common letters, we can expect the numerators of the fractions on each key correlate somehow with the general frequency of the key's letter. In fact, the unsimplified numerators for each key may refer to a completely different kind of digital division - namely a division between when and when not one's finger ("digit") should hit a given key.

Following this train of thought, or otherwise just listing the information obtained in a simpler way, we can list each key in accordance with when a particular digit appears in its numerator. So A, L, X, and 3, for instance, are all examples of keys that are pressed during the first round of typing, since their associated fractions each contain a 1 in their numerator in unsimplified form.

 Digit Keys with Digit in Unsimplified Numerator 1 A,C,E,I,L,N,O,P,T,U,X,1,2,3,5,7 2 A,I,M,R,T,X,1,2,3,5 3 C,D,E,F,G,I,N,O,R,U,2,3,6,8,9 4 A,D,E,G,H,R,T,U,2,6,7,8 5 A,E,H,I,N,P,S,T,1,2,7,8 6 A,C,E,I,L,N,R,T,1,3,4,7,8 7 A,C,F,I,N,O,R,S,T,1,3,4,5,7 8 A,C,E,H,I,L,P,R,S,5,6,8,9 9 A,D,M,R,S,T,U,3,4,5,6

It appears that for each digit/'round', the letter keys pressed can uniquely form a word when ordered correctly. In order, these words are EXCULPATION, MATRIX, CONFIGURED, DAUGHTER, THESPIAN, CLARINET, FRACTIONS, SPHERICAL, MUSTARD. As for the numbers associated with each set, of note is that the greatest number never exceeds the length of the word created. The most logical thing to do with the numbers would seem to be to take those letters from the words indexed by their associated numbers - for instance the numbers 1, 2, 3, 5, 7 associated with the word EXCULPATION (set 1) would suggest taking the 1st, 2nd, 3rd, 5th, and 7th letters from this word, giving EXCLA. Continuing in this way gives:

 Word Numbers Indexed Letters EXCULPATION 1,2,3,5,7 EXCLA MATRIX 1,2,3,5 MATI CONFIGURED 2,3,6,8,9 ONGRE DAUGHTER 2,6,7,8 ATER THESPIAN 1,2,7,8 THAN CLARINET 1,3,4,7,8 CARET FRACTIONS 1,3,4,5,7 FACTO SPHERICAL 5,6,8,9 RIAL MUSTARD 3,4,5,6 STAR

Thus we derive the message EXCLAMATION GREATER THAN CARET FACTORIAL STAR. These are all references to symbols that can be found on a standard keyboard, specifically the characters !, >, ^, !, *. Each of these symbols is obtained by holding down Shift while hitting another key on the keyboard, and can often be found printed on said keys. If we imagine pressing the corresponding keys without holding down Shift, then, the following keys would be pressed: 1, ., 6, 1, 8. This sequence gives the number 1.618, a four-digit approximation of the famous golden ratio, which amongst many other things represents the ratio of lengths of adjacent sections of one's fingers, yet another interpretation of "digital division".