The Counterfeit Coin Puzzle: Finding the Fake with a Balance Scale

Imagine you are an ancient treasurer, and you’ve just been handed a bag of gold coins. You know one of them is a counterfeit, but it looks identical to the rest. Your only clue? The fake coin is slightly lighter than the genuine ones. Your only tool? A classic two-pan balance scale. To make matters worse, you’re only allowed a limited number of weighings to find the fake.

This is the heart of the famous Counterfeit Coin Puzzle, a timeless brain teaser that has challenged logicians and puzzle enthusiasts for generations. It’s not about arithmetic; it’s about pure strategy and the power of deduction.

Welcome back to Sequentia, where today we’re putting on our thinking caps to solve this classic logic challenge!

The Classic Scenario: 8 Coins, 2 Weighings

Let’s start with a well-known version of the puzzle:

The Setup: You have 8 identical-looking coins. One of them is a counterfeit and is lighter than the others.
The Tool: A two-pan balance scale.
The Goal: Identify the counterfeit coin in a maximum of 2 weighings.

How would you approach this? The intuitive first step for many is to weigh four coins against the other four. Let’s see where that leads…

• Weighing 1 (Intuitive but Inefficient): Place 4 coins on the left pan and 4 on the right. The scale will tip, telling you which group of 4 contains the lighter coin. But now you have 4 suspicious coins and only one weighing left. If you weigh 2 of those against the other 2, you’ll find the lighter pair, but you can’t isolate the single fake coin. We’ve hit a dead end!

This shows that the key to this puzzle is maximizing the information you get from each weighing. It’s not just about what the scale does, but also what it doesn’t do.

The Winning Strategy: The Power of Three Groups

The most efficient strategy is to divide the coins not into two groups, but three.

The Setup: 8 coins.
The Groups:

• Group A: 3 coins
• Group B: 3 coins
• Group C: 2 coins (left off the scale)

Weighing 1: Place Group A on the left pan and Group B on the right pan. Now, there are three possible outcomes:

• Outcome 1: The scale balances.
  • Deduction: This is fantastic news! It means all 6 coins on the scale are genuine. The lighter, counterfeit coin must be one of the 2 coins in Group C that you didn’t weigh.
  • Weighing 2: Take the 2 coins from Group C and weigh them against each other. The one that rises is the fake. Problem solved in two weighings!
• Outcome 2: The left pan rises (meaning Group A is lighter).
  • Deduction: The counterfeit coin is one of the 3 coins in Group A.
  • Weighing 2: Take any 2 coins from the suspicious Group A. Place one on the left pan and one on the right.
    • If the scale balances, the 3rd coin from Group A (the one you didn’t weigh) is the fake.
    • If the scale tips, the coin on the pan that rises is the fake.
  • Problem solved in two weighings!
• Outcome 3: The right pan rises (meaning Group B is lighter).
  • Deduction: The counterfeit coin is one of the 3 coins in Group B.
  • Weighing 2: Follow the exact same procedure as in Outcome 2, but using the coins from Group B.
  • Problem solved in two weighings!

This puzzle brilliantly demonstrates how eliminating possibilities is just as important as identifying them. When the scale balanced in Outcome 1, it gave us the most information, instantly clearing 6 coins of suspicion!

Ready for a Bigger Challenge?

What if you had 12 coins, and you didn’t know if the fake was lighter OR heavier? That’s a famous, more advanced version of this puzzle that requires even more cunning. We might just tackle that one in a future post!

Have you encountered this puzzle before? What was your initial strategy? Let us know in the comments!