Circle third counter: rules and printable materials

Track repeated counting around a shrinking circle and predict the final survivor.

Players

1–6 (cooperative)

Plan for

10–15 minutes

Bring

12 numbered counters, circular mat and removal log

Arrange a fixed circle

Place counters 1–12 clockwise, with 1 at the top. Counters represent positions, not people: no guest is eliminated. Start counting at counter 1 and move clockwise. Say one, two, three over three currently present counters and remove the third. The first removal is therefore 3. Keep removed pieces in order on the log instead of returning them to the circle.

Continue from the next survivor

After each removal, restart the count at one on the next remaining clockwise counter. Never count a removed position or restart at label 1 unless it really is the next survivor. After removing 3, counts 1, 2, 3 land on 4, 5, 6, so remove 6. The following removals are 9 and 12. Close the counting gap without moving the printed number labels.

Predict and test

Before continuing, each participant may privately predict the last remaining label. No one has to explain a prediction immediately or speak quickly. Then execute the same deterministic rule until one counter remains. A wrong prediction is not an error in counting; distinguish it from removing the wrong counter. If a physical piece is moved accidentally, restore the last logged state together before proceeding.

Check the complete sequence

With twelve starting counters and every third removed, the removal order is 3, 6, 9, 12, 4, 8, 1, 7, 2, 11, 5. Counter 10 remains. The host key lists both this order and the changing counts. Do not switch to removing every second counter mid-game. A new circle size or step is a separate puzzle and needs a separately checked answer.

Finish with a reconstruction

Ask a different participant to replay the last four survivors using the log, or explain why counting resumes after the removed piece. Stop after the group can reproduce the result, without ranking guests by speed. A helper may read the next labels or point to counters. Use large tokens or pencil crosses, with suitable care around small children. This is a paper tracking puzzle, not a random draw or a social elimination game.

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.