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Nine Gold Coins, Two Weighings (illustration)
FILE No.20Unsolved

Nine Gold Coins, Two Weighings

Difficulty ★★☆about 5 minLogic PuzzleBalance scaleCase analysis
One of nine gold coins in a safe was swapped for a fake that is slightly lighter. What is the fewest weighings that always finds it?

Case Record

One of nine identical-looking gold coins in a safe was swapped for a fake. The fake is slightly lighter than a real coin. There is no other difference in weight.

All you have is a balance scale that compares two pans. You cannot read any weight; you only learn one of three results: "left is heavier, right is heavier, or equal."

Your question

What is the fewest weighings that is always enough to identify the fake for certain?

Case Files

Evidence

Diagram

What the scale tells youItemDetailCoins9, one is a lighter fakeScale resultsLeft / right / equal (3 outcomes)QuestionHow many weighings are always enough?
Diagram

Timeline

  1. One gold coin turns out to be a fake.
  2. The fake is identified with a balance scale.

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

Try splitting the nine coins into three groups of three.

Open hint 2

A weighing has three outcomes. How far can the first weighing reduce the candidates?

Open hint 3

How many more weighings does it take to find one coin among three?

Solve It

Your question

What is the fewest weighings that is always enough to identify the fake for certain?

Choose the answer closest to the truth.

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The Truth

2 times

The answer is 2 times.

Weighing 1: split the nine coins into three groups of three (A, B, C) and weigh A against B. If one side is lighter, the fake is in it; if they balance, the fake is in C. Either way, the candidates are narrowed to three coins.

Weighing 2: put one of the three candidates on each pan. The lighter pan holds the fake. If they balance, the coin left off is the fake.

Since there are three possible results, one weighing cuts the candidates to a third. Doing that twice always narrows 9 to 3 to 1.

9 coins, 3 coins, 1 coinWeighing 1: 3 vs 3 (keep 3 aside)The lighter side, or the 3 aside, holds the fake (down to 3)Weighing 2: one coin on each pan from those 3The lighter coin, or the one left off, is the fakeTwo weighings are enough.
Diagram of the solution

Key Points

  • A measurement with three outcomes can cut the candidates to a third
  • Splitting into three equal parts narrows things with no waste
  • See that 9 = 3 x 3 and build a two-step measurement

Why people get fooled

You tend to compare two coins at a time or test them one by one. Unless you think of using all three outcomes, you overestimate the number of weighings.

Similar tricks

  • The "fewest weighings" balance scale type
  • The type that narrows by splitting into three

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