Chinese Rings: The Puzzle You Solve by Counting

Quick answer: the Chinese rings is a set of rings on a loop that come off only in a strict order. Solving it is literally counting in binary, and the sequence of moves corresponds to a Gray code. It is the ancestor of every n-ary puzzle made since.

Best for: anyone who likes the idea that a puzzle can be an algorithm rather than an insight.

Almost every mechanical puzzle is solved by seeing something. You stare, you fail, and then you notice the one thing you had not considered, and it is over.

The Chinese rings is not like that. There is nothing to notice. The method is entirely mechanical, it never varies, and it takes a very long time.

What Is the Chinese Rings Puzzle?

A set of rings, each threaded onto a long loop or shuttle, with each ring also passing through the one before it. The goal is to remove every ring from the loop.

Only two moves are ever legal at any moment, which sounds generous and is the cruelty of the thing: one of them undoes your last move, and the other advances you. There is never a choice to agonise over, only the risk of losing track and going backwards for an hour without noticing.

It goes by several names. Baguenaudier is the French one, and it is unflattering: it translates roughly as time-waster, a name given by French peasants who used the mechanism to lock chests. Cardan's rings is the other, after Girolamo Cardano, who described it in the 1550 edition of De subtilitate.

Cardano is usually given as the first European description, and he was not. The puzzle appears around 1500 as problem number 107 in Luca Pacioli's manuscript De Viribus Quantitatis, half a century earlier. Pacioli's collection went unpublished for centuries, which is why the credit drifted to the man who got into print.

Why Is It Solved by Counting in Binary?

Because the state of the puzzle is a binary number, whether you notice or not.

Each ring is either on the loop or off it. Two states, n rings, so any position of the puzzle is a string of ones and zeros. The legal moves are exactly those that change one digit at a time, and the sequence that solves the puzzle visits every state in order without repeating any.

That sequence has a name in mathematics: a Gray code, an ordering of binary numbers where consecutive entries differ in exactly one position. The Chinese rings is a physical Gray code generator, and it was one for roughly four centuries before Gray codes were formally described.

The practical consequence is strange. If you know the method, solving it requires no thought at all, just stamina. If you do not, no amount of cleverness substitutes.

How Many Moves Does It Take?

Roughly two-thirds of two to the power n, which means the move count doubles with every ring you add.

The minimum is about ²⁄₃(2ⁿ − 1) moves for n rings, with slightly different exact formulas depending on whether n is odd or even. Five rings is a pleasant afternoon. Nine rings is several hundred moves. The commercially sold versions stop at a size that remains sane, and historical examples with many more rings exist mainly as a demonstration of patience.

Exponential growth from a trivial-looking object is the signature of this family, and it is the same property that makes the Tower of Hanoi interesting.

What Is an N-ary Puzzle?

A puzzle whose solution is counting in some base, and the Chinese rings is the binary case that started the family.

Base two means each element has two states. Make the elements have three states and you have a ternary puzzle, with a solution length that grows faster still. Designers have built puzzles in base four and beyond, and the move counts become absurd quickly, which is rather the point.

The most extreme example in the modern hobby is Goh Pit Khiam's Ternary Burr, which folds n-ary counting into an interlocking puzzle: twenty-two pieces, more than ninety notches, and seventy-five moves before the first piece comes free. There is more on him in the designer profile, and on counting-based difficulty in the guide to mechanical puzzles.

Is It Actually Chinese?

Probably, but the evidence is thinner than the name suggests, and the puzzle has been independently present in several places for a long time.

A Chinese origin is traditionally claimed and often attributed to a specific general, which is the sort of attribution that gets repeated because it is a good story rather than because anyone has a source. What is documented is that Cardano described it in Europe in 1550, and that it was in everyday use in France as a chest lock, which is where baguenaudier comes from.

That last detail is the interesting one, and it recurs across old puzzles. The Chinese rings, the Romano-Celtic puzzle padlocks and the Lu Ban lock were all security devices before they were amusements. A mechanism that only the owner knows how to operate is a password made of metal, and the step from that to a toy is short.

How Do You Solve the Chinese Rings?

By committing to a rhythm and not deviating from it.

Learn the two moves. One ring can always be taken off or put on freely, usually the one at the far end. One other ring can be moved, and only one, and it is determined by the position of its neighbours. Everything else is blocked.

Alternate. The solution is: move the free ring, then move the other legal ring, then the free ring again, and so on. That alternation is the whole method.

Never backtrack to check. This is the one that catches people. Undoing a move to see whether you were right costs you two moves and, worse, breaks the alternation, which is how an hour of progress evaporates.

Count out loud if the puzzle is large. Unglamorous and effective, for the same reason people count laps.

Can You Still Buy One?

Easily, and not from us. The design is centuries old and public domain, so versions exist in every material at every price.

We make no disentanglement puzzles of this kind, with one exception: Hedgehog in a Cage is a genuine classic of the take-apart family, and there is a full walkthrough. The broader family is covered in puzzle box mechanisms.

If the exponential-sequence idea is what appeals rather than the rings themselves, Tower of Hanoi is the other great classic of that shape and we do make it. The wider range is in mechanical puzzles.

Frequently Asked Questions

What is the Chinese rings puzzle?

A set of rings threaded on a loop, each passing through its neighbour, which can only be removed in a strict order. It is also called baguenaudier, French for time-waster, and Cardan's rings after Girolamo Cardano, who described it in 1550.

How do you solve the Chinese rings puzzle?

By alternating between the two legal moves without deviating. One ring is always free to move; exactly one other is legal at any moment. Alternate between them and the puzzle solves itself. The main risk is losing the rhythm, not failing to understand it.

Why is the Chinese rings puzzle related to binary?

Each ring is either on or off, so a position is a binary number, and legal moves change exactly one digit. The solving sequence is a Gray code, an ordering where consecutive values differ in one place. The puzzle generated Gray codes centuries before they were formally described.

How many moves does the Chinese rings take?

About two-thirds of 2 to the power n for n rings, so the count roughly doubles with each ring added. Five rings is manageable; nine runs to several hundred moves.

What is an n-ary puzzle?

One solved by counting in a given base. Chinese rings is the binary case. Ternary and higher versions exist with far longer solutions, the most extreme being burrs that fold counting logic into interlocking pieces.