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Overlapping sets

Seventy like tea, fifty like coffee, thirty like both — how many like neither? Count overlapping groups without counting anyone twice.

  • Logic
  • Timed
  • Endless levels

How it plays

A group is surveyed: how many like one thing, how many another, how many both, how many neither. One count is missing; work it out.

The trap is the overlap. Adding “likes tea” and “likes coffee” counts the people who like both twice, so take them away once. With three sets, taking away every pair takes the people in all three away once too often, so add them back.

Levels move from two sets to three and grow the numbers without end. Answer by choosing or by typing; the wrong options are the overlap counted twice, forgotten, or the neither left out.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises counting overlapping groups — inclusion and exclusion — in survey questions like these. It claims nothing wider.

Learn

How to get better at overlapping sets: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the question first: neither, both, the total, exactly one, all three, one thing only, or exactly two. Each has its own short formula.
  2. Sort the counts you are given: single counts (“70 like tea”), pair counts (“30 like tea and coffee”), all three, none, and the size of the survey.
  3. Remember that a count is everyone in it, overlaps included. “70 like tea” includes the ones who also like coffee, and a pair count includes anyone in all three.
  4. Either use the formula for the question, or draw the circles and fill the regions from the middle outwards — all three first, then the pairs, then the singles, then outside.

Techniques and tricks

  • At least one: add, then take the overlap off once

    With two sets, the number in at least one is A + B − both. Everyone else is in neither.

    e.g. 120 surveyed, 70 tea, 50 coffee, 30 both: at least one is 70 + 50 − 30 = 90, so neither is 120 − 90 = 30.

  • Both from the total

    When you are given neither instead of both, first find at least one: the total minus neither. Then both is A + B minus that.

  • Exactly one: take the overlap off twice

    The people in both are inside A and inside B, so to leave them out entirely, subtract them from each: A + B − 2 × both.

    e.g. 70 tea, 50 coffee, 30 both: exactly one is 70 + 50 − 60 = 60.

  • Three sets: add, subtract the pairs, add the middle back

    At least one = the three singles − the three pairs + all three. Taking away every pair removes the middle three times after counting it three times, so it goes back once.

    e.g. 40, 35, 30; pairs 12, 10, 8; all three 5: 105 − 30 + 5 = 80.

  • One thing only

    Only A = A − the two pairs with A + all three. Each pair includes the middle, so subtracting both pairs takes the middle away once too often.

    e.g. A 40, pairs 12 and 10, all three 5: 40 − 12 − 10 + 5 = 23.

  • Exactly two

    Add the three pair counts, then take off three times all three: the middle is inside every pair.

    e.g. Pairs 12, 10 and 8, all three 5: 30 − 15 = 15.

  • Fill from the middle

    When formulas tangle, draw the circles and write all three in the centre, then each pair minus the centre, then each single minus what is already in its circle. Every question is then a sum of regions.

As the levels climb

  • You start with two sets and small counts. Each level changes one thing in turn: the counts grow, the clock tightens, and, with both boxes ticked in “Sets”, three sets take over early and stay.
  • Three sets bring the “add the middle back” step and many more given lines. Sort the lines into singles, pairs and the rest before you add anything.
  • After that the counts grow without end and the clock closes to 80%, 65% and then 55%. The formulas never change; only the arithmetic gets longer. Add the singles, then the pairs, then subtract one total from the other, rather than juggling seven numbers at once.

Traps to avoid

  • The overlap counted twice: with two sets, A + B taken as everyone. That gives a neither too small or a total too big by exactly the overlap.
  • Neither forgotten: “How many were surveyed?” answered with the number in at least one. Always add the people outside the circles.
  • The middle not added back: with three sets, singles minus pairs, then stop. That makes “at least one” short by exactly the number in all three, and “none” too big by the same.
  • The middle taken off too few times: for exactly two, the pairs minus all three once or twice. It must go three times, once per pair.
  • Exactly one taken as at least one: the people in both are excluded from “exactly one”, so subtract them twice, not once.

How to practise

  • Untick “Three sets” in “Sets” until the two-set formulas are instant, then let three sets back in.
  • Set “Answer by” to “Typing the number”, so a slip of one or of the middle count cannot be rescued by spotting the option.
  • For three sets, draw the circles on paper for the first few, fill the regions, and check each formula against the drawing until you trust it.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More logic

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