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

Sunday 20 June 2021

Making 2D Patterns With 3D Objects

Or George Gets it Right Again!


Look at this...yet another easily available puzzle for you to read about. By now, most of you know that I am really rubbish at pattern forming puzzles and seldom buy them so why on earth would I add this to my collection? I saw the Blockistry puzzle described by Roxanne Miller on Facebook and she described how her husband, George was enthralled by it. Over the years I have come to realise that George and I share very similar tastes in puzzles and a similar approach to puzzling as well. Like me doesn't like exhaustive searches for solutions and has a huge enjoyment working out how to get 3D puzzles modelled in Burrtools. So if he gets deeply embedded in a puzzle that would not normally be his thing (and certainly not mine) then I sit up and pay attention. Then Michel van Ipenburg sent out his monthly newsletter describing his recent puzzling acquisitions and also seemed to like this puzzle. I went to their site and was slightly surprised at the price ($49.95) for 4 blocks of wood and a booklet - with a gulp, I purchased and trusted George.

This is a far cry from the usual craftsman made work that I buy and write about. The site describes the pieces as hand made from the native tree of Finland (Birch) giving an "authentic feel of the North". Ignore the advertising hype - this is just a plain block of a white wood (it may be Birch but it doesn't really matter) and it is nicely cut with smooth surfaces, two of which have been coated with a black layer. What this is NOT, is "finely crafted"! That doesn't actually matter - the fabulous Symmetrick puzzle that I bought from another Finnish craftsman and puzzle store owner was also made from a plain pale wood and became one of my favourite puzzles of all time. But of course, Tomas does not make any claims about it being special wood.

It arrives very nicely packaged in a square cardboard box in which the blocks are nicely arranged along with a booklet of 50 challenges and a felt cloth which is the "playing board". It is actually a very nice package (certainly very portable) and quite enticing to play with. 

The 4 identical blocks are an odd rhomboidal shape with only the top surface coated in the black film. The stated aim is to make the shapes in the booklet which get progressively tougher - so what is so special? why would this interest such a seasoned puzzler like George or someone less experienced like me? Here is where the twist comes to play - the aim is to make the shape when visible solely from above i.e. to create a 2D shape using 3D objects. To make this even more challenging all 4 blocks must be used and the shape must be self supporting. This last factor really does turn up the challenge a notch. I am not sure how to describe the shapes so here is a photo that shows them of:

4 identical "blocks"
The first few in the booklet are not too tough and are really just there to give an idea about how to think about looking at the finished shape and the colour scheme from above:

These aren't terribly tough

Number 4 looks like this when solved:

Easy peasy yes? Indeed, that one was but before long the need to think as a projection onto a 2D plane began to get quite challenging...and very enjoyable. It is not something that can be done easily on your lap - you need a flat surface and certainly having a cat in the way really does make it impossible.

A later challenge like the one below requires the blocks to be piled up and yet still be stable and self supporting and the successful shape is only obvious when viewed from directly above.

Nice easy shape
Solved it - piled up pieces
One thing that the pictures above do show off is that these are not a finely crafted puzzle. They are good enough but the edges are not sharp, the coating does not go right to the edge and it makes it look ever so slightly poorer quality. BUT once you get past this, the puzzling challenges are really quite pleasant.

Viewed from in front - it doesn't look like the diagram
I have almost finished all 50 of the challenges in the book (it took me a good few hours to work them out) and will soon be heading to their Facebook group where they have been posting extra challenges each week - this puzzle will have some decent longevity, I think.

If you do get stuck on one of the challenges or want to check your solution is correct (I did because my solution did not seem to be particularly stable) then their website has a solutions page (password protected) that has cute little animated videos showing how to assemble each puzzle. Beautifully done.

Should you buy this puzzle? Despite the craftsmanship claims not quite being met, this is actually a pretty good puzzle with a really decent set of challenges. The price of $49.95 (they will quote in a currency local to you) is supposed to be a reduced price from the usual $79.95. I do not think I could ever justify the higher price but the reduced one is acceptable. If you enjoy pattern making puzzles and like the idea of using 3D shapes to make 2D patterns then you will enjoy this. Like George, I was quite enthralled.

Sunday 13 June 2021

All at Sea and Hitting the Bottle

Ship in a bottle
It would seem that every Sunday when I write my blog post I sit down with tears in my eyes and almost unable to see. Not for any emotional reason but this is the time that I have to do one of my employer mandated twice weekly lateral flow tests. 15 minutes before I start to type, I have had a good swipe at the back of my throat (pharynx for those of you keen to learn some anatomy) enough to make me significantly gag (Mrs S uses the affectionate term "boak") and then stuffed the thing far enough into my nose to feel like I am doing a brain biopsy. At the end of this with tears streaming down my face, I sit down to think about puzzles. Yep, I'm a truly depraved individual - who could think about puzzles after that?

Recently I might have received a little package from Tom Lensch and one of the puzzles that Arrived was a puzzle from last year. Ship in a bottle was designed by Goh Pit Khiam quite a few years ago and was entered into the IPP design competition in 2003 (that time made by Walter Hoppe). Tom rereleased it in 2018 and I sort of ran out of money at that time and turned it down. Of course Allard bought a copy, described it as a brilliant puzzle and immediately made me regret my decision. On top of that, at several subsequent MPP's it had been brought out and people playing with it seemed to enjoy it and further make me regret that decision - another reason for tears! Somehow I never actually got to play with it there as I am very easily distracted by other shiny things and squirrels.

Simple instructions - turn the ship around
Well, the last year has definitely encouraged me to hit the bottle - this bloody pandemic seems to have increased everyone's alcohol intake considerably. My own personal gin collection has increased enormously. When Tom let me know what he had available recently, he said he had some new copies of Ship in a bottle available and so I jumped on board and sent some PayPal. The package crossed the pond really quickly and before I had had time to warn Mrs S that yet more stuff might be arriving the postman had knocked on the door. She then knocked on me when I got home from work. It seemed like a good idea to hit the bottle as an analgesic!

Beautifully made from Walnut and acrylic with nice brass capped screws the bottle looked lovely. Tom had sent out the ship blocks (Maple I think) outside the bottle to protect it from being broken by the pieces in transit. The "cork" in the bottle is also made from Walnut and is held in place with a magnet. Pulling that cork reveals the only way to insert the pieces into the bottle. There are gaps all the way around the outside so that fingers (or as Tom suggests a pencil capped by an eraser) can be used to manoeuvre the pieces into whatever position you wish and then slide them around each other. I started on the first position and quickly achieved this:

Ok I'm hooked - definitely floating on a stable sea
Photo taken, I take the pieces out and reverse the orientation of the bottle. Now let's reassemble that ship...oh, now I see why everyone liked this puzzle. The presence of 3 vertically oriented pieces which appear to need to be inserted last (which is impossible) suddenly presents a challenge. Allard is right again (damn him!) I need to think© outside the boxbottle and this hurts (maybe I have sampled too much brain with my lateral flow swabs?)

Like most of you, I have solved the 15 puzzle many times before and initially thought that this would help but nope, not really much help at all. There is quite a lot more to this than just sliding tiles around, the presence of 2x1 tiles in both orientations really limits your options. 

In the end I solved it in about 45 minutes (probably much longer than most puzzlers) but I was distracted by TV and a cat trying to knock the 1x1 pieces off my lap. The assembly of the ship requires a fairly long sequence of moves and provides a very nice Aha! moment. After that it's time to reset to the beginning and of course, I had completely forgotten the correct sequence and had to work it out again. Really lovely!

Oh yessss! Definitely as much fun as everyone promised
Having reassembled the ship in the start position, I tried again and still couldn't do it without a struggle - this is an absolutely delightful challenge. It's not too difficult for an experienced puzzler but still fun and will be a wonderful challenge for a beginner or a child. I think I will be taking it to work to torture my colleagues with - at least they stand a chance solving this one (I have one particular orthopaedic surgeon who starts to cry whenever I threaten him with Tomas Linden's Symmetrick puzzle - if you don't have a copy then go and buy one right now!)

Having hit the bottle and completely consumed the contents (Hic burp!) I then decided to play with my balls:

Of course they are not my balls!
I just own them
My recent bunch of puzzles from Mine included a copy of Dog and Balls that he had managed to unearth. I couldn't resist adding it to my pile especially as all men like to play with their balls. The aim is to swap the green and red balls over without lifting anything off the tray. This is not as challenging as the Ship in the bottle despite needing many many more moves. Also quite fun.

I'm not very good at sliding piece puzzles but these were very enjoyable - if you find one for sale then definitely worth adding to your collection. Now it might just be time for some more gin. Cheers everyone.

Sunday 6 June 2021

A Cube Made of Cubes - Easy? Nope!

Key and Keyway Cube
Today's post may be a little less coherent than usual - I have spent the lsat 45 minutes trying to provide computer support to the mother out-law who lives 250 miles away and things weren't working. Eventually got it all working but I think we both left the telephone conversation rather frazzled! 

I am all too aware that I do spend a lot of time writing about puzzles that are extremely hard to get hold of for most people so every now and then when something hits my radar that might be good and is available to order then I try to get a copy to write about and encourage you, my patient readers. I was alerted to this puzzle by Michel van Ipenburg, a very prolific and hugely talented puzzler who certainly knows a good puzzle when he sees one. Our interests don't completely overlap even if he is a huge advocate of the N-ary puzzle group like me but if he advises people to buy something then I sit up and pay attention.

Nicely presented.
He showed off the Key and Keyway cube designed and produced by John Kelly in Ireland. I have not heard of John before but dealing with his site ( was delightful with quick acknowledgement of the order and updates when manufactured and delivered. At the moment there is only this one puzzle on his store but after this experience, I look forward to more. The puzzle is only 20€ plus postage and was manufactured and posted within a couple of days. When it arrived it was nicely presented in a cloth bag with a label giving description and instructions strung onto it. This string didn't last very long in my house! Unfortunately I had left the label and string on the kitchen work surface after I took the puzzle to work (I had hidden the label reasonably well but nothing can keep my pussy-boys from finding string). The following morning the cats followed me into the bathroom for our habitual ablutions and to my disgust a very loud puking noise occured behind me followed by a large amount of bright green liquid and said piece of string in the middle (completely intact). Down in the kitchen, the label was waiting to be found in the middle of the floor! Doh!

Later that day, I got to have a proper look. It has been 3D printed in a vibrant matt purple plastic and very nicely done too. The pieces are pretty solid too and don't feel completely hollow. I guess that is important so that the screws can be screwed in securely. Each cube has 3 sides which are smooth and obviously expected to be the outer faces and 3 which will have either screws in various positions or keyways pointing in different directions. The objective is obviously to build a 2x2x2 cube by sliding the screws into the keyways and pushing them into locked positions (not allowing them to just sit in the entry hole). 

I began to play whilst at work (waiting for a surgeon to decide which urgent case took priority). Quite quickly it becomes apparent that this could get quite confusing and also it becomes obvious that there is a temptation to set it up and engage the screws using rotational moves of pairs of cubes. This is NOT allowed - everything should be done with linear motions and the trick is to find which pieces engage with each other and the correct sequence to make it work. Quite often I found that I wanted to slide two cubes together but they were blocked and I needed to start again. Clever!

Still waiting for the bloody surgeon, I actually solved it and surprised everyone waiting with me. They thought it might be impossible. It had taken me about 20 minutes in all. Quite proud of myself, I dismantled it (that took a bit of doing after losing track of which face was which) and handed it to the surgeon to play with whilst I anaesthetised our first victim. He handed it back to me in pieces with the simple word "NO!" Oh well - maybe not one for the novice? Later in the week, I gave it to one of my colleagues who expressed an interest in the pieces lying on our office desk - he had a fiddle and made a shape and then handed it back with a similar expression of discontent:

It's going to be easy isn't it?
This is the best that Dr Moll can do!
I took it home that day and tried again only to find that I seriously struggled - there was obviously a lot more to it than I initially understood. The sequence must be exactly correct. This time it took me about an hour and required multiple restarts. Interesting! I had better do it a few more times. No matter what, it always seemed to take me about 45 minutes. There is definitely something complex with this that requires proper exploration and thought to solve - I have tried not to just memorise the sequence from dismantling it. It locks together and looks quite attractive - John has also provided a stand to display the puzzle once solved:

Assembled with stand
Looks nice on display
Like Michel, I have really enjoyed this puzzle. It is not made from gorgeous wood but it is very nicely printed. It is a nice challenge that requires some thought and planning rather than loads of trial and error and is a great price. I highly recommend it - you can either buy it ready made from John or can buy the stl files to print your own (obviously you will need the right screws too). I think this is suitable for experienced puzzlers and beginners alike but probably not ideal for surgeons or less than bright anaesthetists - most of you out there should be fine! Go get it here.

Enjoy the rest of the weekend guys - I now need a large gin to calm my nerves after my computer support experience earlier!