NACLO 2026 - Problem BGame: Set Match

There are 4 disjoint groups of sets, so we first want to put the phrases into 4 unrelated bins.

A alleged spies, H future spies, K happy spies, Q spies
B berries, D blueberries, G fruits, O small blueberries, R strawberries
C big elephants, F elephants, N small big elephants, P small elephants
E cars, I green cars, J green Italian cars, L Italian cars, M purple Italian cars

I'm going to name the 4 groups of sets in the Venn diagram as groups A, B, C, D, and the 4 bins above as bins 1, 2, 3, 4. We want to match each bin to a group, and then match each phrase to a set in the Venn diagram.

Note how 3 out of the 4 groups have 1 set enclosing all others, except group C. We have "fruits" in bin 2 (everything here is a fruit), "elephants" in bin 3 (everything here is an elephant), and "cars" in bin 4 (everything here is a car) that enclose all others. However, this doesn't hold for bin 1: "alleged spies" and "future spies" are not "spies"! So group C must be the "spies" group.

Let's start with group C. 14 is fully enclosed in 12, so 14 = K happy spies and 12 = Q spies. 11 partially overlaps with 12 but 13 doesn't at all; "future spies" are never "spies" (otherwise they would be "current spies"), while "alleged spies" can be "spies" (if the allegation is true), so 13 = H future spies and 11 = A alleged spies.

Group D has 4 items, so it must be bin 3. So 15 = F elephants. Inside, we have 2 disjoint sets: 16 and 17, so they are C big elephants and P small elephants, in some order. Because 18 is contained in 17, and N small big elephants are still "big elephants" (as the notes say, "small big elephants" are still "big elephants", just smaller relative to other big elephants; you can think of "big elephants" and "small elephants" as two species, like "black bears" and "brown bears"), so 16 = P small elephants, 17 = C big elephants, and 18 = N small big elephants.

Group A has 1 containing 2 containing all others. Bin 2 has fruits containing berries containing all others, so group A is bin 2, with 1 = G fruits, 2 = B berries. "Small blueberries" are still "blueberries", but "strawberries" are not "blueberries", so 3 = R strawberries, 4 = D blueberries, and 5 = O small blueberries.

Finally, group B is bin 4. The enclosing set 6 = E cars. Inside we have two overlapping sets 7 and 10, with 10 fully containing another set 9, so 7 = I green cars, 10 = L Italian cars, 9 = M purple Italian cars. The intersection 8 = J green Italian cars (cars that are both green and Italian).