Sunday, 27 June 2021

ACT II: Breitenbach’s Great 17 Variant

William Breitenbach’s Great 17 variant, as interpreted by me, channeling Calvin O. Brown.
How does he do it? Despite having massive upheaval in his life, my friend and co-conspirator Mike Desilets manages to produce not just a bit of fluff (usually how I would describe my drivel) but a major puzzling exposition. A deep analysis of a subject that completely confounds my powers of understanding and at the end leaves me with a much deeper knowledge of a subject that everyone is aware of but very few get to properly understand. I have had a tough week at work and have only just managed to solve a couple of puzzles yesterday. This guest post gives me a bit of a breather in my publishing schedule - thank you my friend.

Over to Mike...

Aloha kākou puzzlers,

Following fast on the heels of the previous Foreign Office post, and at the sufferance of my editor and patron, I present today YET MORE peg solitaire (Ed - hooray! I need to learn more). I only hope that George B., John B., and Kevin find this interesting. If anyone else does, that’s a bonus.

As you recall, we left off looking at an unproduced variation on William Breitenbach’s Great 13 puzzle design. We saw that, with only slight modification, the Great 13 can be infinitely expanded, and in either ‘direction’ to boot. Having discovered this, Mr Breitenbach took the time, effort, and expense to acquire a design patent for what I have designated the Great 17 solitaire puzzle.

But that is not the end of this story. As observed in the last post, you cannot realistically expand the Great 13 puzzle much over 17 places for practical reasons of size. It is also far from certain that progressively increasing the number of places in a solitaire puzzle enhances the puzzling experience. Quite the opposite, at least in my experience. But it is possible to modify a solitaire design in another way, and that is to tinker with the rules for allowable jumps. As you shall soon learn, this is precisely what Mr Breitenbach did.

Patent illustration for Great 17 variant.
Breitenbach’s exploration of the hidden potentialities of the Great 17 resulted in a very interesting variant, for which he applied and was granted a U.S. Patent on April 25, 1899 (downloadable from here). We will call this Great 17 variant 17v. For temporal context, Kevin, this patent was awarded one month after the Great 17 design patent was issued, and a full three months BEFORE the patent of the now-classic Great 13. It is also worth noting that the 17v patent was issued as a utility patent, whereas the Great 13 and 17 patents were design patents (you can tell this from the patent numbers, which are different sequences). Utility patents are specifically awarded for the mechanics of an invention, how it functions, whereas design patents are intended to protect the form or appearance of an invention. I don’t know exactly what to make of Breitenbach’s patent choice, but I have the nagging feeling it is meaningful.

Considering the distinctive parallelogram shape of 17v, it seems clear that more than a little thought went into the design. Stretching and squashing the puzzle, although not directly affecting play, was clearly intentional. Maybe this produces a little visual confusion. If nothing else, it certainly distinguishes the puzzle from G13 and G17. This may actually have been critical to getting the patent application past the patent inspectors/researchers. Possibly the utility patent choice also came down to this as well. Two design patents for VERY similar designs (not to mention G13) may have been a hard sell. I don’t know if the utility side of the U.S. Patent shop researched design patents when checking for originality.

Can you spot the difference between 17v and the original Great 17?

Appearance considerations aside, let's take a look at what Mr B came up with. It’s pretty straightforward actually. Taking the basic Great 17 structure, four of the diagonal movement options are removed. You can also conceptualize this as the complete removal of the center rectangle, which it is. We are left, then, with two concentric rectangles connected to the center position by eight radial lines. As George Bell rightly observes (and he of all people would), the result is a clock solitaire-like structure. If you don’t know what I mean by clock solitaire, first of all, shame on you, and second, please go to George’s solitaire website immediately and educate yourself. George’s article on clock solitaire is provided here, to save you a few clicks. I also highly recommend you purchase the only commercially available instantiation of clock solitaire from the very good folks over at Creative Crafthouse. (Ed - I have it and it is a fabulous version)

Made for George and now available to buy with quite a few challenges in the booklet
Once you have done all this, you will understand why 17v is clock-like. The fatal flaw, however, is that 17v is rectangular, not round, on the outside. Hence, you cannot communicate all the pieces continuously around the dial, only the corner pieces. From a clock’s perspective, this is intolerable. But it is possible to “clockify” 17v (see below). Rounding the outer race gives us a proper clock shape with 8 outer positions and 8 inner (as opposed to 12 outer and 6 inner for standard clock solitaire). Play around with these, if you will, and note any interesting properties in the comments section.

Some Breiten-clocks.
Setko-style Breitenbach 17v.
As you know by now, when I find an obscure historical puzzle, I am compelled to make a version. Predictably, I modeled it on the standard Setko 5.5 x 5.5 inch board in American black walnut. Note the Eye of Sauron, which is a little menacing and probably not the best choice (Ed - the whole thing is stunning!). I also could not think of a good way to inscribe ‘guidelines’ to show the allowable moves. Thus it’s somewhat abstract and probably not suitable for general consumption; but hardly a problem for the seasoned solitairian, of course. I did make the effort to retain the precise proportions and angles of the patent illustration, so on that count at least, it is a faithful instance. It may well be the only instance, in existence.

Pretty grain and perfect pegs.



George Bell and John Beasley have given this little puzzle some thought, and also some computational treatment for good measure. From George’s work, we now know a few things about the 17 variant (Thanks George! Let me know if I don’t get this right). In addition to the standard center compliment problem, other compliments are also solvable. It is important to first observe, however, that the puzzle has only five unique positions (1, 2, 4, 5, 9 can represent them). Of these, compliments for positions 1, 2, 5, and 9 are solvable. Position 4, somewhat inexplicably, cannot be solved for compliment. If you remove the position 4 peg, you can only end at positions 1 or 17. Beyond compliment problems, there is a broad range of ending position solutions for starting positions 2 and 5. From either of these, you may end at 7, 8, 10, 11, 13, or 16.

Place numbering schema for George Bell’s analysis
The larger point, Kevin, is that there is always more to a solitaire puzzle than the conventional center compliment (as we already saw with the Great 13 in Act I). One of the most enjoyable and original challenges, for me at least, comes from the good Doctor John Beasley. John proposes removing the center peg and ending with only the four corners in place. The corners can (and must) move, some of them at least, but they must end back at the corners. This is a very clever challenge the discovering the unique dynamics involved is hugely satisfying. It is comparable to learning an entirely new board. (Ed - for some reason, when I thought about that challenge my whole body went cold and I shuddered! It sounds impossible!)


I’m sure there are many solutions to the central vacancy/four corner problem, so it won’t be a great spoiler to present one of them here. I show this one because I really like it, and I really like it because it demonstrates the beauty possible in a solitaire solution. That’s important, because when you find a beautiful solution, you tend to remember it. And if you remember in a manner that doesn’t involve rote memorisation, one could even say you ‘understand’ the solution (Ed - hahaha! Me? Understand one of these? NEVER!). Consistent with the highest PuzzleMad standard, you have solved the puzzle.

It takes a little work to follow the solution movements in the illustration above. Sorry, that just the nature of trying to show the solution in a reasonably compressed manner. You can actually just look at the final movement sequence and its beautiful symmetry, that’s all you need to understand its attraction. And thanks to the mysterious properties of the Breitenbach Singularity, the final two movements can even be made simultaneously! (Ed - I noticed that and find it somewhat unbelievable. Quite beautiful!)


Since you all seem to enjoy cryptic solitaire movement diagrams as much as I do, I’ll treat you to a couple more. These are intended to beat to death the point made above and also way back in Act I, to wit, solutions to peg solitaire puzzles are ideally more than arbitrary movement sequences. For Breitenbach’s Great 17 variant, I happen to have found a couple interesting solutions. Check out the image below. This is a pleasing clock-work solution. The beauty of a solution like this one is that once you discover the basic concept, you will always and forever know how to solve the puzzle. An arbitrary, unstructured solution, by contrast, will generally need to be memorized, which I suppose counts for something. But under the Puzzlemad solving hierarchy, it is unquestionably a lesser form. A memorized solution, like a random string of numbers, is unlikely to stick in your long-term memory. A concept-based solution, like the password you use on all your accounts (with minor variations), generally will.


My favourite and most memorable center compliment solution is not clock-style, however, with all due respect to George and John. There is actually a solution so simple in concept that I guarantee, once learned, it will stick. It is relentless and rather gruesome actually. It is based on the fact that pegs on a radial line, any radial line, can be formed into a type of buzz-saw. It takes just two moves to set up (if using the vertical and horizontal lines in the diagram, the moves are actually mandatory). After this, simply transport all the outer pegs to the center position, one-by-one, and dispose of using the reciprocating peg. Be sure that the final pair is adjacent to your set pegs, and then perform a clever little clean-up operation. This works with the original Great 17 as well, of course, but I didn’t seem to see it on that board. The diagonals beg your attention on the Great 17, but without them the radial structure dominates and I think facilitated discovery of the cross-cutting solution.

I’m sure there are many other interesting concept-based solutions for you to find. I wanted to share these two mainly to illustrate and drive home the point that peg solitaire can and should be far more than a series of trial and error attempts ending in an arbitrary solution. The sophisticated puzzler should aim higher, and solitaire boards will reward those who do. I suspect most casual puzzlers don’t know this about solitaire. I didn’t until I finally started to THINK©.

This wraps up the trio of Breitenbach solitaire puzzles, one of which is well known and has echoed forth to the present day. The other two were apparently never produced for one reason or another. Circling all the way back to my original declaration, I think the Great 13 was the right choice to produce, and indeed its seems to have hit the sweet spot for both the turn-of-the-century and the modern puzzle consumer. Once the Great 13 was on the market, there was likely no need for a slightly larger and very similar looking Great 17, nor indeed its bastard variant, 17v. For the 21st Century metagrobologist, however, they sure are fun to toy around with. (Ed - I'm not so sure - my brain hurts now. It is probably better with a physical puzzle in your hands.)

I suppose that is QUITE enough peg solitaire for one day. After suffering through this article, you are probably all desperate to get back to Kevin’s beautiful puzzle porn. Me too. But please stay tuned for Act III, which will be delivered to PuzzleMad HQ whenever I get around to it. No headache-inducing diagrams next time, I promise. Now, as always, a closing word from the most tolerant and beneficent editor you will ever have the pleasure to write for...


Beneficent? Wow! I have never been described as that even by the present Mrs S! Thank you so much Mike! What an exposition! I am truly amazed at your skill and knowledge. You certainly made me Think and even made me go back to George's peg solitaire board and play - yes, I failed to solve almost all the challenges yet again. I love the idea of these and love that they can be properly analysed but I wish I had the ability to look at a board and see an approach/solution.

Mike is right that next week will be back to the usual semi-incoherent rubbish that I usually post on the internet! I have been trying to acquire and solve some new interesting stuff for you.website

8 comments:

  1. Mike, you should have placed "The Eye of Sauron" in the exact center! I can see why you didn't do that--you would have been unable to resist Sauron's mighty grip and would be forever lost to the dark side.

    Incidentally, the link to my peg solitaire web site is the old version on recmath.org which hasn't been updated in years. The latest version is at http://www.gibell.net/pegsolitaire/ The paper under that link has been updated to include the boards here.

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    1. The wrong link is my fault George. I just googled it and that’s what I found. I will correct the link shortly.

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    2. Lord Sauron made me write that article, actually. But I'm keeping this shiny ring I found in the puzzlemad basement.

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    3. The precious! You have found it! I knew I misplaced it somewhere.

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    4. Mike! Please don’t put it in my basement! Wait…I don’t have a basement!

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  2. One note about "clockification" (a new word!). It is very important in Clock Solitaire that the number of holes around the circumference is divisible by 3. If not, then the "coloring argument" (as described in my paper) fails. Not a huge disaster, but if the argument fails it often means you can begin with a single peg missing anywhere and finish with one peg at any hole. I suspect this might be the case for the fully clockified 17v board.

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    Replies
    1. I have verified using my solving program that the Clockified 17v board is universal, adding only 4 jumps around the outer rim. Universal means you can begin with a single peg missing anywhere and finish with one peg anywhere else.

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    2. The computer might be able to do it but I very much doubt I could!

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