Sunday 9 May 2021

The Breitenbach Singularity

Setko’s Bullseye, grandchild of W.C. Breitenbach’s Great 13.
Hooray! I am finally feeling better! Thank you to those of you who sent well wishes, I really appreciated it. I was contacted by Mike apologising that he had let me down but he really hadn't...he had sent today's post the day before I got sick and it had been my intention to edit and post it last Sunday but could not get out of bed or off the loo for long enough to even log on. I have lost a considerable amount of weight and Mrs S has said that this confirms that I am "full of $&*t" as it all left me over a torrid 5 days. So, today is the long awaited next post in the Setko series from my wonderful friend and PuzzleMad foreign correspondent, Mike Desilets:

Before I let Mike take over, I must point you all to the TwoBrassMonkeys store where Big Steve and Ali have released Number 4 in their Brass Monkey series. Promising to be different to the previous iterations, they claim it a "puzzler's puzzle". How could any of us resist? I haven't and it should be posted to me tomorrow. I will no doubt let you know my thoughts but if it is anywhere near as good as number 1, number 2 or number 3 then I will not be disappointed. Go buy it whilst stocks last.

Now over to Mr D...

Aloha kākou puzzlers,

Since the last communique from the PuzzleMad Foreign Office, Hawaii Branch, I have enjoyed a lively correspondence with noted peg solitaire analyst George Bell, and at a degree of separation, even more noted mathematician, computer scientist, and peg solitairist John D Beasely. What follows is largely inspired by those exchanges, but of course the foolishness, misrepresentations, and errors are all my own. It is a fairly small subset of the community that take a serious interest in peg solitaire puzzles, which is unfortunate, but hopefully this post will encourage many of you to give them a try. The cost to value ratio is exemplary. (Ed - I have asort of fascination for them but really do not have the skills to solve them).

Now, I have to confess right off, the English 33-peg swiss cross and its many variants, of which you are all painfully aware, have never been my thing. These DO deserve your attention, and George Bell has given the musty old cross some especially serious attention, (article here) which you should make it a point to study (yes, you will always receive a homework assignment from the Foreign Office). But working a 33-peg board is often a real pain in the arse (Ed - I am sooo pleased that you have used the correct expression and not the American donkey!), especially the constant board resetting needed when, a good 25 jumps in, you discover an irretrievable peg. Let’s put it this way for now, those standard commercial boards are really just gateway drugs; there exist better, more potent drugs, if you are willing to search them out. 

So then, how does one improve on a classic puzzle? If it involves 33 moving pieces, a good first tactic would be simplification. The well-worn truism applies: the most elegant puzzles are those wherein a minimal elements produce maximal challenge. Geometric symmetry doesn't hurt either. The 33-peg cross board has a very pleasing shape, to be sure, but it does not shout elegance (which should never be shouted, Kevin) (Ed - as an almost cockney, I have nothing to do with elegance!). This was noticed well over a century ago during puzzling’s Golden Age, and many great solitaire innovations were the result. We will focus in on one particular designer from that era and see if we can’t expand our consciousnesses ever so slightly.

Original 1899 design patent drawing, The Great 13.
Kevin Sadler, Eggman to my Leper Messiah, you know exactly where we are heading now don’t you? Yes! We are going to look at the brilliant work of late Victorian era puzzle designer William C Breitenbach. The cultured among you will instantly know Breitenbach from his most influential and long-lived peg solitaire puzzle design, The Great 13 Puzzle. The design for Great 13 was patented by Breitenbach on July 25, 1899 (Patent). You will find a nice picture of it in your copy of Slocum and Bottermans, New Book of Puzzles (turn to page 99). Regrettably, I cannot show you a picture in this post because I am writing several thousand miles away from my stuff. I’ll include it in Act 2, at a later date. 

The rules of Great 13 are just as you would suspect: jump peg-over-peg (marble-over-marble if you are lucky enough to have an original), in any direction, removing the jumped peg. The center starts empty and the puzzle is solved when you discover a series of jumps which place the final peg back in the center. This is known in the industry as the center compliment problem. With only 13 pegs, and a beautifully symmetrical design, I consider this puzzle as among the most elegant of the entire class. Mr Beasley observes that the allowance of diagonal moves “does tend to make the game too easy.” Fair point. However, I think for the average person-on-the-street, it’s a fine and fun challenge. The general public was, after all, the intended audience for the Great 13. There is a replay-value angle as well. Solving Great 13 once does not guarantee success on the next try. As we will explore in Act 2, the ability to solve a puzzle is one thing, the ability to solve it every time is quite another. The later is the gold ring and the PuzzleMad standard. (Ed - if only I could solve things more than once!)

For some very useful analytics on Breitenbach’s Great 13 design, please now flip to page 10 of John Beasley’s 2014 IPP paper An update to the history of peg solitaire (Download it from here). Mr Beasley has generously provided some hard boundaries for your exploration, regarding especially what is and is not possible to achieve on a 13-hole diamond board. In so doing, he also provides the motivated puzzler with many fun challenges beyond the classic center complement. Quoting directly from the article:

With the help of diagonal moves, the following single-vacancy single-survivor problems can be solved:

  • Initial vacancy at 1: the final survivor may be in any hole except 6 or 8.
  • Initial vacancy at 2: the final survivor may be in any of the eight holes around the edge.
  • Initial vacancy at 3: the final survivor may only be in hole 1 or 13.
  • Initial vacancy at 7: the final survivor may be in hole 7 itself or in any of the four corner holes.

Position assignments for Beasley’s analysis. 

Based on this, you can generate quite a number of unique challenges, even considering the redundancies inherent in a symmetrical board. The latent potential of the Great 13 as a multi-challenge puzzle, so critical to modern commercial puzzle design, was put to its greatest effect by none other than Nob Yoshigahara in his Hoppers. Hoppers was produced by Thinkfun starting in 1999 and is still going strong. Don’t be fooled by the juvenile design, this is a puzzle you should pay attention to. Someday I will do a post about the gems known only to the 5 to 10 demographic. You are walking past fantastic and noteworthy designs every time you take your kids to the toy store (or section). 

Early Bullseye from Setko.
Your instructions.
I don’t know how many versions of Great 13 have been made over the years, but it’s a lot. According to the patent text presented further down, design patents were good for 7 years at the time Breitenbach was working. By 1906 then, the Great 13 design had entered the public domain. Since it was a design patent and not a utility patent, the basic scheme could have been used by others much earlier. It was actually, but that is a story for Act 3. (Ed - soo many acts in this play!)

At any rate, I’m not aware of many produced examples during this early period, based only on a quick survey of the Slocum and Hordern-Dalgety online resources. I do know that by the mid-1960s numerous versions had been produced in the US, most of which are still available and relatively cheap on auction sites. My favorite version, and I think one of the earliest from this resurgence, is Calvin Brown’s Setko version, called Bullseye. That’s clearly a better name, by the way, than Great 13; far more expressive. Bullseye was offered in walnut and oak, with characteristic Setko brass, aluminum, copper, or nickel-plated-something pegs. No other version comes close in terms of raw quality. (Ed - I will need to find one of these in Walnut with nice pegs for my own collection)

Setko means business with these ‘pegs’.

A little back-of-the-box ad work.
I’m not going to focus on the handsome versions just mentioned, of course. The images you see here are instead of Calvin’s earliest issue of Bullseye (as far as I can determine), consisting of a modest pine board with authentic #¼-28 x ½ heat treated socket head cap screws. Prior to producing puzzles as a serious business venture, Calvin Brown’s peg puzzles were in the service of his actual business, manufacturing and selling set screws. The text on the box is pretty clear, and if you’ve handled the Setko ‘pegs’, pretty convincing as well. This is definitely at the top end of the “advertising puzzle” quality hierarchy of the time, and certainly a significant investment for a hand-out. But then recall (from previous post), that Mr Brown had, around this time, patented several new puzzles of his own invention. He was a puzzle guy, so this was probably a completely natural course of action, regardless of cost. It was the first important step on the road to building his independent puzzle business. 

Hoyle/Stancraft marketed version of Bullseye (Setko manufactured). Copper pegs on walnut.

Again, for emphasis, I am without access to my stuff. Otherwise I would show you the pretty versions. The best I can do right now is the above Hoyle (Stancraft) version of Bullseye gleaned from the web. More sophisticated packaging, but still made by Setko. If you don’t have some version of Bullseye in your collection, you really should. If you’ve already spent too much on puzzles this month, as my editor surely has (Ed - how did you know?), you can just make one with a piece of paper and some farthings. I have it on good authority that Mr George Bell does this regularly, it’s perfectly legitimate.

Cool as they are, I wouldn’t highlight the Great 13 and Bullseye unless there were something more to the story. Kevin wouldn’t keep me on staff if I couldn’t tell you something you didn’t already know. So, for the next part, I’ll limbo under that bar. (Ed - so flexible! I'd just fall in a heap)

The Great 17 (13+4=17).
The Great 13 puzzle (and its' later reproductions) gets all the attention from collectors and analysts, and it is no wonder, given its simple elegance and great longevity. It has the advantage that it was actually produced and sold. But, as most designers tend to do, William Breitenbach also explored other variants, none of which appear to have made it to market. One example is what can only be called The Great 17. This design was patented March 28, 1899, four months before the Great 13 patent issue (Patent). 

Looking at the Great 17, we see that it is an extension of the Great 13, simply adding an outer square perpendicular to the last. Rules of play are not listed in the patent, but we can assume they are precisely the same. This extension, adding only four new peg places, obviously complicates the puzzle quite a bit. It may just be the answer to the easiness issue dispensed with above. 

Unfortunately, you will have to look elsewhere for any kind of technical analysis of the Great 17; I don’t have anything whatsoever to offer. I can only hope that the recreational maths portion of the readership become intrigued (Ed - George, please feel free to send me stuff to append or even create an extra guest post). The questions here are potentially much broader than simply establishing parameters, maxima, minima, etc for a larger version. What we should be concerned about are Great N problems. Thirteen is apparently the limiting minimum size for this puzzle, but it can be extended infinitely thereafter. At every iteration, you simply add an outer square. It's in the picture below.

Infinite possibilities, of a single type.
Great N analysis would look at what this growth entails. Take something like the fewest moves for center compliment. What is the growth curve of that quantity? Is there a formula for determining the fewest moves for arbitrary n? Would a brute force calculation end up NP-complete? Or, from a puzzling perspective, is there an elegant way to always achieve fewest moves, some kind of deep pattern that remains consistent as n grows? There are dozens of questions of this kind. (Ed - I wish I understood what you were talking about! George?)

This could take forever to solve.
Alas, you will never get the answers from Kevin or I, that is safe to say. This requires the services of the gentlemen mentioned earlier. Let’s shelve the technical stuff for now. In a practical sense, as a mechanical puzzle, you can’t really grow the Great 13 much past 17, because each new growth doubles the area of the puzzle. It isn’t obvious at first, but looking at the progressive G-13 to G-21 images above, you can see it better and measure off the changes. Visualize folding. The newly added area for each iteration equals the previous (now inner) total area. With this doubling rule in effect, a Great 13 puzzle, beginning at a very modest 2-square-centimeters will after 26 iterations be the length of George Bell’s great state of Colorado. It gets big fast. It also gets small fast. I wasn’t quite right previously when I suggested that 13 was the bottom. As it grows outward infinitely, so too does it grow infinitely inward. There is no bottom. The Great 13 puzzle is simply a slice two layers thick (so to speak) of an endless fractal.

This is all very interesting, says Kevin, but what does it all mean? Well, my friend, the Great N perspective answers the age-old question of why there is a point in the middle of Mr Breitenbach’s puzzle, and why the center compliment is the most beautiful form of play. As the internal squares become smaller and smaller, in an infinite progression, they converge on the precise center of the puzzle, ending in a dimensionless point. Notice that all the other peg places are on vertices of the squares. The Great N center peg, however, is on a special point which can be thought of as encompassing all vertices and all lines (the dimensionless point of collapse). That is why it is appropriate to place a hole there, why it is especially appropriate that it be vacant at start of play, and why it is surely most appropriate that the final peg end in the center, effectively swallowed by the singularity. Kevin, you dullard, Great N theory allows us to finally construct a solitaire puzzle in which NO PEGS REMAIN! (Ed - I am definitely a dullard - you lost me several paragraphs ago!)

That concludes Act 1. We have not even gotten to the good stuff yet. The third Breitenbach design is a real kicker and I can’t wait to tell you about it. And Kevin, thanks for the very kind mention a few posts back. It’s always a great honor to have my musings on PuzzleMad. You’ve built something really special here and I’m thrilled to be a part of it. I’ve just consulted the staff and we agree that the Foreign Office is committed to another 10 years of unscheduled, infrequent submittals. Take us home my friend...

Wow!!! What a tour-de-force! Thank you so much, my friend. I am in awe at your puzzle writing skills. You make me look like a rank amateur. You have even committed yourself to another two articles on a similar subject. These puzzles look so amazing and I really must try and obtain some tice copies for my own puzzling and collection. I have very few vintage puzzles. My words from my tenth anniversary post were very sincere, you have considerably helped me in this journey and I am very grateful for everything you do. I am looking forward to another 10 years of guest posts from you.


  1. Great stuff!! I find it particularly interesting that one can go on to larger and larger boards (Great 13, 17, 21, 25, etc). I am pretty sure Great 17 and beyond are universal. This just means one can begin from a full board with a single peg missing, and finish with one peg at any board location. I have a program which should be able to demonstrate this, as well as find shortest solutions.

    Breitenbach (note spelling, ei not ie) sure had a lot of patents! While reading this column, I realized that the "Great 17" is not the same as what I call "Breitenbach's Board" (patent 623876). The holes are the same but some of the jumps have been removed.

    1. George, you’rea font of all knowledge. These things fascinated me because they are such a simple idea and yet so terribly difficult to solve (at least for me they are). Has there been a huge rigorous analysis of these in mathematical or rec math journals?

    2. I have not seen any analysis of these boards, they are too obscure. I've only just seen them myself! I checked and the Great 17 board is NOT universal. I can easily check the Great 21, and beyond, just need to specify all the jumps.

  2. Although the area of these "Fractal boards" doubles with each iteration, the number of holes only goes up by 4. Thus, although they are getting large in area, they may not actually be that hard to solve.

    For example, the board with caption "This could take forever to solve" looks complex but has only 29 holes, which is quite easy for a computer to solve. Granted it may be a bit tricky to solve by hand, since the jumps are a tricky to understand, with their lengths being so different.

    1. You say it is easy for a computer to solve but is that really solving? Or is it just performing an exhaustive trial of all possibilities and therefore only “easy” because of the speed of a processor rather than an actual analysis or understanding?

    2. The latter of course, but contrast that with solving peg solitaire on a square board of arbitrary size. This is very easy for a human to solve one peg missing to one peg, even 100x100. The reason is that you can break it down into chunks, and clear one section at a time. Similar to a twisty puzzle but even easier because one part does not affect other sections. On the other hand a 100x100 will be really tough for a computer since it is considering all possibilities, which quickly grows horrendous.

      So anyway, these "Great 13, 17, 21, 25, ..." are easy for a computer but hard for a human. But square boards of increasing size is the opposite.

  3. Thank Goerge for all that info and effort. And the spelling correction as well. I think I did a wrong universal find/replace at some point. Better yet, I'll just blame my lackadaisical editor!

    1. Yep! It’s my fault! I trusted my foreign correspondent. In truth, the only editing I do is to make smart arse comments and to ensure the layout works!

  4. Can you solve it or not?

    1. I’ve never tried but Mike has solved it after extensive study.