The Hidden Dragon

I hope you’ll forgive me blowing my own horn a bit.  After all, it’s not every day you discover something new about something invented nearly two-thousand years ago.

The photo above is of an ancient puzzle variously called the “Chinese Puzzle Lock”, “The Prisoner’s Lock”, “Meleda”, “Ryou-Kaik-Tiyo,” “Baguenaudier”, and the “Devil’s Needle.”   According to the Puzzle Grail website, it was invented by the Chinese general Hung Ming in the 2nd century AD, and its unique mathematical properties were noted as early as the late 1600s by the British mathematician John Wallis.

The puzzle is to remove the long loop from the rings.  The solution is similar to the pattern of increasing binary numbers, where the digit that changes as you count in binary corresponds with the next ring that must be moved.  But, as far as I know, no one else has noted one of the most mystical and mysterious things about the rings.

There are basically two moves for the rings, “on” or “off” (just like a computer byte).  If you chart a perfectly efficient solution to the problem of starting with the loop completely free, and ending with it held only by the furthestmost rung, and if you notate each “on” as the letter “R” and each “off” as the letter “L”, you will end with a long sequence of Rs and Ls.  The exact length will depend on the number of rings.  With one ring, the sequence is “R”.  With two rings, the sequence is “RRL.”  With three rings, “RRLRRLL.”  In general, the sequences is of length 2n – 1, where n is the number of rings.  Also note that the sequence can be generated in the following manner.  For sequence n, take sequence n-1, add an additional R, and then add the same sequence, in the reverse order, and with all Rs changed to Ls and vice versa.

Here’s where it gets spooky.  What happens if you take the sequence as a set of instructions.  Draw a line segment of length 1 in any direction.  Turn 90 degrees right for an R, and left for an L and then draw another unit segment.  What shape will you be drawing?

The answer, amazingly enough, is a famous fractal called the Harter-Heighway dragon, notorious for also being constructable using a folded strip of paper.  If you add the number of rings, you’ll soon get this pathway:

An infinite number of rings would yield this figure, a strange demonstration of the ubiquity of fractals in real life:


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