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Solutions for Act II Scene 4 - Sums Puzzle

We are presented with three lists of sums here. It seems natural to want to add them together, although we should also expect to use the individual summands at some point else we'll be ignoring a big chunk of information.

The first intended step is to notice that the second and third sums in the first line differ by 26. Puzzlers should immediately become suspicious of this (the number of letters of alphabet), and so checking the difference between either of these two sums with the first in the line reveals that they also differ by a multiple of 26. People who found the sums in other lines further down could also occasionally notice more obvious differences of 26, 52, 260, and other small/obvious multiples of 26.

As mentioned, 26 being the number of letters in the alphabet implies there might be a letter encoding going on here. A natural approach is to look at the sums modulo 26, which would mean the three sums in each line should return the same letter. Indeed doing this tells us the first line's sums' values mod 26 are all 5, which maps to the letter E, while the second's are all 1, mapping to A, etc. All up this spells out the message EACH LINE NEEDS TO RAISE SUPERSCRIPTS.

This seems to be describing indices, so what we probably want to do here is identify digits in the summands that need to be raised/converted to superscripts. It's not clear which digits exactly should be raised though, but it's probable we need to use the fact they are sums again. With enough experimentation it can be shown that it's always possible to raise the indices in a sum such that the new total is unchanged from the original, index-less sum. For example, 24 + 35 + 206 = 265 = 24 + 35 + 206. Note raising an index in the middle of a number implies the first digit is raised to the power of the second digit, then the result multiplied by the third digit.

Once every sum correction has been worked out, we should have raised up to 3 digits per sum. At this point the leap to make is to realise we can interpret these new sets of digits as new concatenated numbers. From the previous example, we can read off just the indices as the number 450 (we can interpret the indices as having been raised to a new line of their own). The presentation somewhat supports this, since we'll be left with three numbers less than 1000 per line, which looks very similar to the original entries in each column. So we can now imagine the three new numbers per line being connected by plus signs, giving a new list of sums to apply the same mechanic to. These are:

450 + 312 + 205
200 + 30 + 28
705 + 611 + 286
920 + 910 + 359
902 + 620 + 405
41 + 132 + 341
620 + 510 + 405
196 + 212 + 230
370 + 412 + 901
620 + 12 + 405
901 + 611 + 287
508 + 725 + 735
231 + 151 + 35
103 + 711 + 284
94 + 306 + 348
332 + 444 + 841
640 + 671 + 293
507 + 480 + 45
81 + 304 + 348
312 + 650 + 992
626 + 212 + 35
302 + 321 + 348
30 + 259 + 323
243 + 361 + 443
330 + 271 + 451
312 + 650 + 885
780 + 208 + 45
370 + 412 + 902
400 + 292 + 159
61 + 265 + 604
415 + 231 + 35
902 + 381 + 293

To confirm this approach, one might have originally checked these new sums mod 26, which offers an encouraging message: EXPECT ANSWER AFTER REDOING LAST STEP. So if we look at the first new line, 450 + 312 + 205, we can again raise certain indices so that the sum remains unchanged - in this case,
450 + 312 + 205 = 967 = 450 + 312 + 205.

Again concatenating indices per line, we get a new string of numbers, this time all promisingly less than 27 (e.g. the previous example gives 020). Mapping these to their corresponding letters from alphabetic position gives the message THREE-LETTER WORD USED TO DENOTE TYPOS. This describes the answer SIC, which is what you might be writing after a copy-editor erroneously forgets to raise an equation's superscripts (which happens more often than you'd think!).

The answer is: sic