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### Solutions for Act V Scene 2 - Sets

The answer is: congruity What we are looking at is essentially a disguised set of Set cards. To this end, one of the typical cards is provided as the image behind the card marked 0, whose description can be Googled to reveal the game for solvers who haven't come across it before. The title is also an obvious hint here. A Set deck contains 81 cards representing all possible combinations of three different choices each of four different features (colour, number, shape, and shading). For instance, the provided card is uniquely described as "purple, double, squiggle, half-shaded". In a game of set, one wants to find allowable sets, which are groups of exactly three cards characterised by having, for each feature, all its properties the same, or all different. Another way to interpret this rule is that a set of three cards is NOT a set if and only if there exists some feature for which there are exactly two of one property amongst the three cards - for example, if two of the three cards are red and the third purple, they can't possibly form a set. The leap to make here is to realise that the cards numbered 0-80 can be rewritten in base 3 notation and thence mapped to Set cards. This can be intuited from the fact a defining feature of the game of Set is the fact each card feature has exactly three possibilities. If we write each base 3 number with four digits, we can treat each digit as a different feature, and each digit value as a different feature's property - for instance, a 0 in the first digit could represent the colour purple, a 1 in the first digit, red, and a 2 in the first digit, green. It follows that in the example given, 0000 corresponds to [purple][double][squiggle][half-shaded] in some order. It seems though that we don't have anywhere near enough information to know which digit represents each feature, nor which feature property 1 and 2 represent in each case. The key here is to realise that in a game of Set, it's only the relative characteristics that matter when trying to find sets. Whether all the cards in a three-card set are red or green does not change the fact it is a set. With this in mind, we can attribute an arbitrary mapping to determine the entire set. We had better need to find sets for the next step, but we certainly can't find sets without some restriction else the entire deck will be covered and its ordering rendered useless. It turns out the presentation is useful here: Playing Set with each of the 3x3 card groups as marked out seems far more plausible. The last step is to shade in all the sets that can be found per group, which is really one of the only things we can do with overlapping sets. This results in the following: These shapes seem to form letters spelling out CONGRUITY, a fitting answer which describes both the set-finding nature of this puzzle as well as the fact the arbitrary card assignment gives rise to multiple possible solution sets. Design Notes: It turns out it's incredibly difficult to find an arrangement of the entire Set deck satisfying a property such as that exhibited above. As such, I wanted the first step of this puzzle to be similarly difficult so that solvers would find discovering the solution as rewarding as the writer found it. This took the form of a crazy nonogram, where the values for each property's feature were only given once per three columns/rows. While this puzzle WAS solvable, none of our testers (including the writer and a brute-force program) could complete it in under 24 hours. Thus the first step was rewritten to the far simpler version presented, but it was hopefully still enjoyable to solve. Also, an alternative way to find the sets in each 3x3 group is to work directly from the base 3 representations. For any three cards, look at one of the digits (e.g. the first digit of each card). If its sum is congruent to 0 modulo 3, and if this holds for all four digit places, then and only then do the three cards form a set. However hopefully most teams still performed the card-attribution step, since it's far easier to detect sets when the cards are presented pictorially (and in fact this property was the reason the game was invented in the first place).