Sets, Groups & Overlaps

A pets question gives three lists and asks which pets Ari keeps that NEITHER Bo nor Cal keeps. The method is a checklist, not memory: walk Ari's list item by item, crossing out anything that appears in EITHER other list. Geckos — only Ari, keep. Fish — Cal has fish, cross. Turtles — only Ari, keep. Mice and frogs — both others have them, cross. Answer: geckos and turtles. The trap options pair one right item with one wrong ('fish and geckos') so a half-checked list picks a wrong answer. Check every word.

Overlap questions live in four zones: only-cats, only-dogs, BOTH, and NEITHER (outside both circles). The words map exactly: 'has a cat AND a dog' is the overlap; 'a cat OR a dog' is anywhere inside either circle — including the overlap; 'only a cat' is the cat circle minus the overlap; 'neither' is the outside. The trap: treating 'has a cat' as 'has ONLY a cat'. Someone in the overlap has a cat — and a dog too. When counting, decide zone by zone and add the zones the question actually names.

Constraint puzzles: pick one subject from each of four lists, and you already want maths, drama and tech. Start with the want that has only ONE home: drama appears only in List 1 — List 1 is now used up. Maths lives in Lists 2 and 4; tech in Lists 3 and 4; they can't both use List 4, so if maths takes List 4, tech takes List 3 (or maths takes 2 freeing 4). Chase the forced moves and the free slots reveal which fourth subjects remain possible — and which option 'can NOT be chosen' because its only list is already taken. Rarest item first; the rest falls into place.

Counting puzzle: 20 students, 12 play soccer, 9 play tennis, everyone plays at least one — how many play both? Add the lists: 12 + 9 = 21 names, but there are only 20 people. The extra name is someone counted twice — a both-player. Both = 21 − 20 = 1. The rule: overlap = (listA + listB) − total, whenever everyone belongs to at least one list. It also runs in reverse: 'total = A + B − both' finds the class size, and 'neither' people appear when the lists add to LESS than the total.

The set toolkit: CHECKLIST every item against every list (half-checked lists walk straight into trap options); know the FOUR ZONES and which words name them ('or' spans the circles, 'only' subtracts the overlap); in constraint puzzles place the RAREST want first; and for counting, both = A + B − total when everyone's in at least one group. These questions are pure carefulness — almost every lost mark is a skimmed word ('neither', 'only', 'not'), not a missing idea. Read twice, tick every box. Quiz and exercises!