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Find the fake coin

Twelve coins, one fake, a balance: how many weighings, at worst? Derive the base-3 bounds for a heavier coin and for one that may be either.

  • Logic
  • Untimed
  • 33 levels

How it plays

Some coins look alike; one is fake. A balance tips to the heavier side or balances. The questions ask for the fewest weighings that always find the fake, however unlucky you are, or the most coins a number of weighings can handle.

The key is three: every weighing has three outcomes, so it can split the suspects three ways — two pans and the pile left off. When the fake may be heavier or lighter, you must also tell which, so there are twice as many possibilities, and a little is lost when no coin is known to be genuine.

Levels add kinds of question, then more weighings and so more coins. The answers are the known bounds (Dyson, 1946), and each one is checked against a search of every strategy for small numbers of coins.

Keys

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

What it trains

This practises reasoning about information — how much one test can tell you — on balance puzzles. It claims nothing wider.

Learn

How to get better at find the fake coin: 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 which kind it is: is the fake known to be heavier, or could it be heavier or lighter? Is there an extra coin known to be genuine? Are you asked for weighings, or for coins?
  2. Count the possibilities you must tell apart: n if the fake is known to be heavier, 2n if it could be either, since each coin could be heavy or light.
  3. Each weighing has three outcomes, so w weighings can tell apart at most 3^w possibilities. Find the smallest power of 3 that is big enough, with the small correction for the heavier-or-lighter kinds.
  4. Check the answer from the other side: one weighing fewer must not be enough.

Techniques and tricks

  • Know the powers of 3

    Learn them: 3, 9, 27, 81, 243, 729, 2,187, 6,561, 19,683, 59,049. Past that, each weighing multiplies by 3 again; 3^15 is 14,348,907 and 3^20 is 3,486,784,401.

  • Heavier coin: the first power of 3 that fits

    With a fake known to be heavier, w weighings handle up to 3^w coins: split them into three equal piles, two on the pans and one off. So the answer is the smallest w with 3^w at least n.

    e.g. 100 coins: 81 is too few and 243 is enough, so 5 weighings.

  • Heavier or lighter: double, add 3

    Without a genuine coin, w weighings handle n coins when 2n + 3 is at most 3^w. Double the count, add 3, and find the first power of 3 that reaches it.

    e.g. 12 coins: 2 × 12 + 3 = 27 = 3^3, so 3 weighings. 13 coins: 29 is past 27, so 4.

  • With a genuine coin: double, add 1

    An extra coin known to be genuine lets the first weighing use the pans fully, so the test becomes 2n + 1 ≤ 3^w.

    e.g. 13 coins and a genuine one: 2 × 13 + 1 = 27, so 3 weighings.

  • The most coins, read backwards

    For “how many coins?”, turn the tests round. Heavier: 3^w. Heavier or lighter: (3^w − 3) ÷ 2. With a genuine coin it would be (3^w − 1) ÷ 2, one more.

    e.g. 4 weighings, heavier or lighter: (81 − 3) ÷ 2 = 39 coins.

As the levels climb

  • You start with two weighings and one kind of question: a heavier coin, how many weighings. Each level then either opens the next kind or adds a weighing, in turn.
  • The kinds open in order: heavier coin, how many weighings; heavier coin, how many coins; heavier or lighter, how many weighings; the same with a genuine coin; heavier or lighter, how many coins. Questions are then drawn from every kind open.
  • Weighings grow up to 30, and the coin counts with them, into the millions and far past. Work with the powers of 3 rather than the coin count: compare digits first, then the leading figures.

Traps to avoid

  • Halving instead of thirding: the count of weighings by powers of 2. A balance has three outcomes, not two, so the pile left off the scale is part of the answer.
  • Forgetting that heavier-or-lighter doubles the work: using the heavier-coin answer when the fake could be either way. Count 2n possibilities, not n.
  • Mixing up the two corrections: (3^w − 3) ÷ 2 without a genuine coin, (3^w − 1) ÷ 2 with one. The two answers sit side by side as options, one coin apart.
  • One weighing out: the answer and its neighbours are offered together. Test both: w weighings must be enough, and w − 1 must not.
  • One coin out: on “how many coins?”, 3^w − 1 or one fewer than the bound is offered. The bound itself is reachable, so take it exactly.

How to practise

  • Tick only the two heavier-coin boxes in “Kinds of question” until powers of 3 are automatic, then add the heavier-or-lighter kinds.
  • Set “Answer by” to “Typing the number”, so “how many coins?” has to be computed exactly instead of picked from neighbours.
  • Recite the powers of 3 up to 59,049 and the matching (3^w − 3) ÷ 2 values — 3, 12, 39, 120, 363 — until you can say them without working.

Levels

It is untimed unless you turn the clock on. There are 33 levels with the default settings, and the counter stops at the real top. 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.

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