Nine-coin balance: rules and printable materials

Use two equal-pan comparisons to find one known-heavy coin among nine.

Players

2–6 (cooperative)

Plan for

10–15 minutes

Bring

9 labeled cards and a host-only heavy-coin choice

State the promise

There are nine labeled coins, 1–9, and exactly one is heavier. All other coins weigh the same. The heavier coin cannot be lighter, and there are no two defective coins. These promises are essential: changing them makes the printed two-comparison method invalid. A host privately chooses a number before play and writes it on the separate key. Use cards and truthful simulated results; no actual precision scale is required.

Make equal-sized groups

For the first comparison, put 1, 2, 3 on the left and 4, 5, 6 on the right. Keep 7, 8, 9 aside. Equal numbers of coins go on both pans, so a heavier pan identifies the group containing the heavy coin. If level, the heavy coin is in the group not weighed. The host announces only left heavy, level or right heavy, not the secret number.

Use the second comparison

Take the three remaining suspects in ascending order. Compare the lowest against the middle, leaving the highest aside. A heavy left pan identifies the lowest; heavy right identifies the middle; level identifies the highest. Thus first-level then second-right means coin 8. The same second-step rule applies whichever group survives the first comparison. Record both outcomes before naming the coin.

Worked host round

For secret coin 5, the first comparison is right heavy because 5 is among 4–6. Comparing 4 against 5 is then right heavy, so the answer is 5. Reveal the host record after the group's answer. If the host accidentally gives an outcome inconsistent with the chosen number, void the attempt and replay with a new secret; do not pretend the group made a reasoning error.

Rotate the information role

Play enough rounds for each willing participant to host once, or stop after everyone understands the decision tree. No elimination or speed scoring is needed. A helper may read the numbers and outcomes. The key page lists all nine cases for checking, but players should not see the selected secret. This is a reasoning exercise under stated assumptions, not advice for detecting real counterfeit money.

Printable materials

Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.