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 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 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
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). |

The answer is: congruity |
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