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One Quick Try

Chinese Rings Puzzle

Rings hang on a bar and can only come off in a strict order. Take every one off.

Type
Logic
Takes
about 2-5 min
Result in
moves

Chinese Rings

~2-5 min

  1. Every ring starts on the bar. Take them all off.
  2. Ring 1 can always come off or go on. Any other ring can move only when the ring before it is on and every ring before that is off.
  3. Tap a ring to move it. With 5 rings the fewest possible is 21 moves.
Rings

Sometimes you have to put a ring back on to get further.

Your result: — moves

How it's measured

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How does the Chinese rings puzzle work?

A row of rings hangs on a bar, all of them on at the start, and the goal is to take every ring off. Ring 1, at the free end, can come off or go back on at any time. Any other ring can move only when the ring just before it is on the bar and every ring before that one is off. Tap a ring to move it; a tap that breaks the rule is refused and not counted. You can play with 4, 5, 6 or 7 rings.

How is your result calculated?

Every ring taken off or put back is one move. Your result is your number of moves, compared with the fewest possible for that many rings, and your time from your first move.

How many moves does it take?

The fewest moves for n rings is (2ⁿ⁺¹ − 2) ÷ 3 when n is even and (2ⁿ⁺¹ − 1) ÷ 3 when n is odd. That is 10 for 4 rings, 21 for 5 rings, 42 for 6 rings, 85 for 7 rings, and 341 for the traditional nine. Each extra ring roughly doubles the work. The site's tests check the formula against a search through every position for up to nine rings.

Because at most two moves are ever possible, the puzzle isn't about choosing well among many options. It is about not undoing your own progress: half the time the move that looks right puts you back where you were.

How do you solve the Chinese rings?

  • Alternate. The solution switches between the two legal moves: move ring 1, then the one other ring that can move, then ring 1 again, and so on.
  • Start with the right first move. With an odd number of rings, take ring 1 off first. With an even number, take ring 2 off first. Starting the wrong way leads you away from the goal.
  • Expect to put rings back. To free a ring far along the bar, rings before it often have to go back on. That is not a mistake.

Where does the puzzle come from?

It is old: the Italian mathematician Gerolamo Cardano described it in 1550, and it is known in China as the nine linked rings. In 1872 the French magistrate Louis Gros showed how its positions follow the pattern now called the binary Gray code, where each step changes exactly one digit.

For another puzzle whose moves double with every piece, try the Tower of Hanoi; for one where each move removes a piece, Peg Solitaire.

Questions

What is the rule for moving a ring?
Ring 1 can always move. Any other ring can move only when the ring just before it is on the bar and every ring before that is off.
How many moves does the nine linked rings puzzle take?
At least 341 single-ring moves. In general it is (2ⁿ⁺¹ − 1) ÷ 3 for an odd number of rings n, and (2ⁿ⁺¹ − 2) ÷ 3 for an even number.
Why do I have to put rings back on?
Because a ring far along the bar can only move when the ring before it is on. Clearing the way often means putting earlier rings back first.
What is the Chinese rings puzzle called in other languages?
In China it is the nine linked rings. In French it is the baguenaudier, and in English it is also called the Chinese rings or the tiring irons.
Is it related to the Tower of Hanoi?
Yes: both are solved by a recursion where each extra piece roughly doubles the moves, and both are closely linked to binary numbers.