The challenge
Nine identical-looking coins include one heavier coin. Using a balance scale only twice, how can you always find it?
Nine Coins and Two Weighings Puzzle
Solve this easy logic challenge, then compare your reasoning with the hint, checked answer, verification, and variation.
Hint
Each weighing has three possible outcomes.
Split nine candidates across three outcomes
Split the coins into three groups of three. Weigh two groups. If they balance, the heavy coin is in the unweighed group; otherwise it is in the heavier group. From the suspect group, weigh one coin against another. The heavier coin is the answer, or the third coin is the answer if they balance.
Outcome-count proof
The first weighing reduces nine candidates to three. The second reduces three candidates to one for each of its three outcomes.
Two-level search tree
- Divide the nine coins into three groups of three and weigh two groups.
- Keep the heavier group, or the unweighed group if the scale balances.
- Weigh two coins from the remaining group to identify the heavy coin.
Balance the branches of the search
Design tests whose outcomes divide the remaining possibilities as evenly as possible.
An uneven split wastes information
Splitting the coins into two unequal groups wastes the balance outcome and can leave too many candidates for the final weighing.
How confident were you?
Design around the scale's three outcomes
A balance weighing has three outcomes: left heavy, right heavy, or balanced. Design each test so those outcomes leave groups of equal size whenever possible.
Find the two-weighing capacity
How many coins could the same two-weighing strategy handle if one coin is known to be heavier? Justify the limit using outcome counts.
Study how to solve logic puzzles for the reusable method behind this challenge. Then explore the logic puzzles and deductive reasoning learning path for connected examples and practice.