This article appeared in Cubism For Fun, issue 88, 2012 (possibly with minor editing)

Novel Puzzle Rings By Bram Cohen and M. Oskar van

Deventer Introduction Puzzle rings [1] are a classical puzzle concept in which a ring which can be worn will, once removed, easily scramble and become hard to get back into its wearable state. Nearly all puzzle rings made for the last several hundred years have followed the exact same design; see Figure 1. We will throughout this article describe puzzle rings with diagrams showing how the bands are braided on the front, with the assumption that all bands go straight around the rest of the ring. The quality of the traditional design indicates that considerable experimentation must have occurred before that design was settled on as 'best', but then the experimentation was forgotten and only that one design was passed along between generations. So far no one who is interested in puzzle rings as puzzles has gone over histori-cal examples to trace back the history of the design; hope-fully someone will do that in the future. In the later 20th century Jose Grant [2] did some experimen-tation with puzzle ring designs, mostly variants on the tradi-tional design consisting of either splitting existing bands in half or adding a fifth band. Several years ago Bram Cohen begun experimenting with puzzle ring designs, and this arti-cle will explain his way of thinking about them and give sev-eral example designs. The methodology of building these has been that Bram designs them using pen and paper, then sends them to Oskar van Deventer in ASCII art (see Figure 2), who then builds them using a CAD system and prints them on a 3d printer, usually Shapeways [3]. It turns out that puzzle ring designs are fairly forgiving, with lots of small braid variations gen-erally still producing enjoyable puzzles, albeit usually inferior ones. The trick is to find the most interesting and difficult designs involving the fewest number of bands. Analysis The first part of the analysis of a puzzle is which bands are linked to which other ones. This can be done by just looking at each pair of bands without including the others; see Figure 3. If a band is only linked to one other, that generally makes the puzzle very easy. To maximize difficulty each band should be linked to two

Figure 1. Classic four band design

Figure 2. Sixth Sense in ASCII art

Figure 3. Linked rings (left) and unlinked (right)

This article appeared in Cubism For Fun, issue 88, 2012 (possibly with minor editing) others, and bands shouldn't be linked to the ones which lie next to them in the solved state. It may be that for puzzles with large numbers of bands difficulty is increased if each band is linked to three others, but that hasn't been experimented with much, and it becomes difficult to make the puzzle scramble-able if the criterion that bands should not be linked to the ones next to them is kept. The other main part of puzzle difficulty is whether for each band the two bands linked to it would hang in the orientation they need to be in the solution if you hold the first band up and let everything else fall by gravity; see Figure 4. Puzzles are most difficult when all bands make the two they're linked to go the wrong way. Ideally a puzzle ring should be so easy to scramble that simply dropping it from an inch in the air will scramble it. One way that bands can come apart very easily is if the outermost bands have the pattern of first going over some bands, then under others, and finally over. That allows them to hinge outwards. In the Sixth Sense design, the outer bands both hinge out, and once those come apart the next ones hinge out, making the puzzle want to scramble very badly; see Figure 5. Another way in which bands can scramble very easily is if a band first goes over a bunch of bands and then under several more, which allows in to swivel in place even if it's an interior band. In the Weave Six design (Figure 11) the outer bands swivel out, and then the next in bands can swivel out, resulting in the puzzle scrambling by a single big corkscrew motion. Designs Given the above criteria we can analyse the tradi-tional four-band-ed puzzle ring design. Each band is linked to exactly two oth-ers, with the two middle bands not linked to each other. Every single band tries to make the two it’s linked to go to the opposite orientation relative to each other that they need to be in the solution. One might wonder if the two edge bands could be made to not link to the ones next to them and have the two middle bands be linked to each other instead. There is a simple design which has that property and also the orientations property, but it doesn't scramble; see Figure 6. In terms of scrambling, both of the outer bands swivel out and the two remaining aren't even linked to each other, so it scrambles very easily. It

Figure 4. Directly linked (left) and misleadingly linked rings (right)

Figure 5. Sixth Sense design scrambles very easily

Figure 6. Impossible to

Scramble

This article appeared in Cubism For Fun, issue 88, 2012 (possibly with minor editing) also has a subtle asymmetry, which Oskar likes. The traditional puzzle ring is proba-bly the absolute best design for four-banded puzzle ring. The Holistic Ring (see Figure 7) is another four-banded design. In it, unfortunately, each band is connected to the ones next to it, but they are each linked to two others, and the orientations property holds for all bands. What's unique is that none of the bands individually hinge or swivel, but the puzzle as a whole can scramble in one big holistic move for a very subtle reason, hence the name. When you get to more than four bands there's an additional design requirement to look out for. If you design things wrong, the bands can get stuck around a linkage be-tween two other bands; see Figure 8. There is a tendency for the bands to be able to pass through each other and get out of this state, but that has a tendency to feel like it was accidental or cheating, and isn't very aesthetic. Ideally the bands passing through each other should be either clearly al-lowed or disallowed. If you make the bands thick enough the passing through can be clearly disallowed, but that has a tendency to make the puzzle's motion feel very restricted. If the puzzle were made sufficiently elliptical as a bracelet instead of a ring then it would be very easy to pass bands through each other and one could go the opposite direction of designing a puzzle where that phenomenon was forced to happen as much as possible, but that has not yet been explored. The Weave Five design (Figure 9) has the property that each band is linked to exactly two others, no band is linked to either of the ones next to it, and the orientations property holds for all but the middle band. Because there are an odd number of bands, it was impossible to achieve the orientations property for all of them. It has a subtle asymmetry. The outer two bands hinge out, and the rest of the puzzle then scrambles easily. It also has none of the bands passing through each other phenomenon. It is much harder to solve than the traditional design. The Sixth Sense (Figure 5) is a gener-alization of the tradi-tional design. In ret-rospect it is surpris-ing that no one found it before, al-though it was discov-

Figure 7. Holistic Ring

Figure 8. Bands passing through each other; middle band stuck (left), bottom

band stuck (right)

Figure 9. Weave Five

This article appeared in Cubism For Fun, issue 88, 2012 (possibly with minor editing) ered via a painstaking iteration process with the obvious pattern only being noticed at the end. If you remove the outer pair of bands, or the middle pair of bands, or the other two bands, the remaining four bands form a traditional puzzle ring. This pattern can be generalized to any number of bands, with an interesting difference in flavour between even and odd numbers; see Figure 10. Four produces the traditional design, three results in a design which several people have found independ-ently, and five has not been made yet although is undoubtedly fun. Odd numbers have rotational symmetry, while the even numbers are asymmetric. Larger numbers work as well although it seems likely that at some point the puzzle won't be able to scramble well any more. The Sixth Sense is an extremely difficult puzzle, for which I still don't have a completely reliable set of steps to solve it even after solving it hundreds of times. The Weave Six is a puzzle where each band links to exactly three others, no bands link to the ones next to them in the solution, and there is no bands passing through each other; see Figure 11. Unfortunately to get those criteria to be met and make it easy to scramble it was nec-essary to drop the orientations require-ment, which is fol-lowed by exactly zero of the bands. It has a clear asymmetry. It scrambles easily by swivelling. It's a fairly easy puzzle to solve once you know to start at one edge and work your way to the other, and that the next band to place is always the one not linked to the last one positioned, but nearly impossible otherwise. This leads to a pleasant and insightful solving experience. The weave pattern obviously generalizes to any number of bands, but the same solving approach applies to more bands, so they don't really add anything to it, and smaller numbers of bands can be solved in much easier ways, so the version with six bands is clearly the best one. References [1] Wikipedia, ”Puzzle Ring”, http://en.wikipedia.org/wiki/Puzzle_ring [2] Jose Grant, “Puzzle Rings”,http://www.josegrant.com/ [3] Oskar van Deventer, Bram Cohen, “Puzzle Rings”, Shapeways,

http://www.shapeways.com/shops/oskarpuzzles?section=Puzzle+Rings [4] Oskar van Deventer, “Puzzle Rings”, YouTube playlist,

http://www.youtube.com/view_play_list?p=PL3147B5F553435B98

Figure 10. Canonical

3-banded (top) traditional 4-banded (middle) and generalized 5-banded

(bottom)

Figure 11. Weave Six