Puzzler in Paradise

I attended my first IPP earlier this month and had a terrific time. First and foremost, the people are just fantastic: generous, imaginative, super smart, just great people. The huge support and welcome I received as a newbie just floored me with outright puzzle gifts, advice on how to get the most out of the event, suggestions on who to talk to for help on my puzzle ideas, encouragement to participate in the Puzzle Exchange (more on that in a bit). What a wonderful way to be introduced to a whole new level of puzzling!

Second, the events were mind-bogglingly great: a puzzle competition, a puzzle exchange, and a puzzle “bazaar”. There were over five dozen puzzles in the puzzle competition and I had a chance to play with all of them. I liked one from VIN&CO so much that later I bought one. I will make a nice icosahedron out of these some day, I swear:

Next came the puzzle exchange in which one hundred participants all bring one hundred copies of a never-before released puzzle and exchange it with the other participants. I was blown away not just by the cleverness of the puzzles but also by the professional quality of their manufacture and finish. The craftsmanship was unbelievable and there were several I hope will reach the marketplace so I can buy them. Stan Isaacs’ was based on a tiling problem posed in the as-yet-unpublished Volume 4 of Donald Knuth’s “The Art of Computer Programming” series: “Find seven different rectangles of area 1/7 that can be assembled into a square of area 1, and prove that the answer is unique.” I was able to buy a copy of it later:

On the last day there’s a kind of puzzle “bazaar” in which puzzle sellers and puzzle buyers go completely nuts. Errm, I plead no contest. There were puzzles there that I’ve been looking for for ages, like Wil Strijbos’ Lotus Puzzle. (Even cooler than finally procuring the Lotus was having an extremely informative conversation with Wil about manufacturing puzzles and he gave me some very helpful advice – thanks Wil!) Here’s a picture of the puzzle along with the reason for its name:

At dinner, I had the amazing good fortune to sit at a table with Sven Baeck who runs Mallorca Puzzles and he had brought a super rare, super cool Roger D puzzle, “Gartenschlauch”. Given its rarity and its price I figured this was probably the last time I would actually be able to work on solving one so I got to work, dinner be damned! I have to admit, I’m pretty proud of this:

In summary: Pure. Puzzle. Euphoria.

Model #808 is Done!

Surf’s up! Dinner is served! Lift-off!

Well, the day has finally arrived and Model #808 is now available for purchase. There are only 106 of them – that’s all I made – so if you’re interested please head over to my Etsy store. And here it is:

I have to say, everything about this process took longer than I expected so it’s a great relief to have at last reached the finish line. I never would have guessed that I would go through fifteen design revisions and three machine shops over the course of almost three years. Well, I guess that just means Model #360 will be a cakewalk!

First Batch of Puzzle #808

Since my last post on Model #808, I have learned a lot about machining and manufacturing. For one, I realized I needed a professional mechanical engineer to do the tolerancing of my design and prepare proper design drawings. I had previously tried to get by with the .STL files generated by my 3D design tools. For another, I learned that cost is not the only factor to consider when choosing a machine shop. I am now working with J&J Machine Company and these guys are outstanding!

But here’s the important news: the first batch of puzzles came back last week and I have assembled two of them so far. Drum roll please! Ta Dum:


I have to admit, I am really pleased with how the engraved logo looks: red ink on the matte black (bead-blasted) anodized aluminum. The bottleneck will soon be me. Next week I expect the remaining puzzles to arrive of this limited edition run of 100. Once I get the machined pieces, I have several manual steps to go through: adding the red ink, adding the internal pieces (I won’t spoil the fun by saying what they are!), hand numbering and signing the puzzle, and doing the packaging.

It has taken a long time to reach this point but it won’t be much longer before Model #808 is ready for sale so check back soon!

The Partridge Puzzle

I recently stumbled across the Partridge Puzzle invented by mathematician Robert T. Wainwright. Consider a collection of square tiles, the smallest of which is 1 unit by 1 unit in size (and there is one of them) and the largest of which is N units by N units in size (and there are N of them). The total area of all the tiles would be 1x(1×1) + 2x(2×2) + … + Nx(NxN). That sum of cubes turns out to be [(N x (N+1) / 2]^2. This means that the area covered by the tiles is the same as the area of a single large square whose length and width are [N x (N+1)]/2. So wouldn’t it be cool if you could find an arrangement for all those tiles that allows them to fit inside that larger square? That’s the Partridge Puzzle.

Now it turns out there are zero solutions for N=1 through N=7 and there are 18,656 solutions for N=8 (so far, solutions have been found for N=8 through N=33).  You can buy an N=8 one from Kadon Enterprises:


It’s really nicely made; here’s a picture of mine:

Although there are 18,656 solutions to this thing, after several frustrating days I was able to find precisely zero of them. So I wrote a program to solve it, which I suppose is either winning at the meta-level or cheating depending on how you want to look at it. I (or if you prefer, my program) found all the solutions to the N=8 puzzle and I let it run long enough to generate a solution or two for N=9 through N=13. Here’s my GitHub repository where I’ve parked my source code, solutions, and some analysis:


This was a fun diversion and gave me an excuse to learn Python, Java, GitHub, and AWS EC2 management. I will leave you with a picture of my favorite of the 18,656 solutions:

It’s my favorite because it is the most fragmented of all the solutions. That is, this solution has the fewest number of same-sized tiles touching each other. If you want to learn more, here are some excellent links to explore:





Constant Negative Curvature

One of my favorite events is the local annual Celebration of Mind gathering inspired by Martin Gardner’s immense contribution to recreational mathematics. I went to one last night in Brookline, MA sponsored by Eureka Puzzles and gave a short presentation on my interest in what I’m calling (probably incorrectly) a “unit hyperbola”. My talk was called “Horsing Around With Z=X*Y” and if this kind of thing interests you, you can download the PDF of it here.

Basically, I used OpenSCAD to model Z=X*Y for X and Y values ranging from -1 to 1. That generates a saddle-shaped half-cube (with Z value from -1 to 1). I found that you can take those half-cubes (or unit hyperbolae) and compose them into some very weird-looking shapes. Here are some examples:













I’ve convinced myself that every point on the curved surface sees constant negative curvature in all directions – hence the title of this post – but I’m no mathematician and it wouldn’t completely surprise me to learn I’m wrong. The basis of my conviction is empirical: I can slide the curved surface of one of the half-cubes over the curved surface of the more convoluted shapes while keeping the surfaces in full contact with each other at all times. Since I know the curvature is fixed for the half-cube, I conclude that the curvature must be fixed for the convoluted shapes too.

In any event, it’s clear that the “holey” shape repeats its pattern in all three dimensions so it should be relatively straightforward to define the minimal subset cube that “tiles” space accordingly – but I ran out of ideas when I was playing around in OpenSCAD. Now that I’ve had to organize my thoughts for the presentation, I think I’ll take another shot at it.


Puzzle Collection Extravaganza

I was recently asked how many puzzles I had in my collection and I realized I had no idea. So I figured if I was going to go to the trouble of taking them all out of their boxes and actually count them, I might as well take some pictures too.

Here’s my kitchen table over-run by puzzles:

Puzzle Collection

As you can see, there are some old classics in there, like an original 1980 Ernő Rubik cube and an IMP puzzle (now known as the “Fifteen Puzzle”) from the 1930’s. The most recent addition is Wil Stribos’ “First Box” puzzle – the blue cube in the upper left. That was a fun sequential discovery puzzle, beautifully made from anodized aluminum.

Strictly speaking, not all of those are puzzles. The three black and white toys on the bottom left are just fun to manipulate or make shapes with. And the white rectangular puzzle on the right, sitting next to the black calculator looking thing (a super cool Vulcan Electronics XL 25!), is a Rubik’s Magic puzzle that I modified. Normally, it has paper inserts printed with a pattern of rings; I replaced those with a pattern of semi-circles that I think makes it look more interesting even when it isn’t solved.

OK, coffee break’s over! Back to working on Model #808.

Progress on Model #808

Model #808 continues to make steady (if slow) progress. I had some prototypes machined from aluminum by a fantastic machine shop, Cantabrigian Mechanics. They are gorgeous, if I may say so myself. Take a look:

808 Preview

The prototype at the top of the picture was bead-blasted, then anodized in black. The one at the bottom of the picture was anodized in black and engraved with lettering, which I then infilled with red. Actually, only the top piece in the bottom prototype was anodized black; if you look carefully you can see the un-anodized bottom piece. I wanted to see what the different finishes and combinations looked like.

This is Rev. 12 and it’s been an amazing journey from Rev. 1. Along the way I’ve learned two CAD tools, OpenSCAD and Onshape; prototyped in ABS, PLA, nylon, and aluminum; built an FDM 3D printer; and tried a bizillion different internal mechanism ideas.

I also dove deep into the world of fonts to find what I was looking for: the machine shop typeface that we’re all used to seeing engraved on heavy equipment. It turns out that almost everyone (e.g., NASA, Boeing) used engraving machines from the Gorton and Graham Machine Co. Here’s a picture from one of their brochures. To my immense good fortune, the typeface has been recreated by a fellow named Josh Kraemer and it can be seen on his website here.

So all the pieces have fallen into place. I think this may be the final design and at this point I’m working with Cantabrigian on getting the dimensions exactly right so that the fit and feel are perfect. Getting closer, almost there!

More Prototypes

Though I haven’t blogged in about a year, at least I have been making progress on my puzzles. Pictured here are Model #808 (the rectangular one with two holes), Model #518 (the multi-colored cube), Model #921 (the circular one), and Model #360 (the thicker rectangular one with the red logo).


The internal design of Model #808 (formerly known as Model #873) continues to evolve and I’m hoping to get a prototype today that will confirm the reliability of the new design. I guess it’s no surprise but getting a puzzle mechanism to work in theory is easy, getting it to work once or twice in an actual mechanism is doable, but getting it to work dependably and reproducibly is really hard. I’m on Rev. 9 on this one.

Model #360 turned out to be easier to get working than I expected and it may be the first one to reach production in aluminum. Models #518 and #921 still have a ways to go.

In other news, I now have an FDM type 3D printer (the i3PRO from MakerFront) to shorten the cycle of revise design / make new prototype / test new prototype. I’m still learning how to get good quality prints from it but I look forward to being able to get a physical prototype of a puzzle design within hours instead of days. I’ll post some videos of my experiences with it in case anyone else is contemplating buying one of these things.

On the Asymptote to Perfection!

I just received my latest prototype back from the 3-D print service I use (i.materialise – whom I recommend highly) and I’m very happy with this latest revision (the sixth I think!)

The puzzle has two major pieces, the top and bottom, and the biggest design challenge I’ve had so far has been to keep the two pieces aligned while the puzzle solver is manipulating the puzzle. If the pieces move out of alignment, it can put the puzzle into an unsolvable state or it can keep the puzzle in a partially solved state when it oughtn’t. In a previous puzzle I used Vlier pins to hold everything in place and they worked beautifully. Unfortunately, they’re much too expensive so I’m experimenting with some other approaches. Here’re what the top looks like:


I’ve never used this material before. It’s nylon (polyamide) mixed with aluminum powder and it’s called “alumide”. I like the metallic color but overall the finish isn’t as nice as with the matte white nylon so I’ll probably go back to that. Here’s a picture of the bottom:


If you squint you can see the model number embossed on the right end: “Model #873”. If you really squint, you can make out “www.pyrigan.com” embossed on the left. The low print resolution makes it hard to read the text but this is just a prototype. The final version will likely be made from machined aluminum and engraved with the text on the back and the logo on the front.

The good news is that the puzzle mechanism works very reliably so I will press ahead with plans to manufacture a couple of dozen and see if anyone enjoys these things as much as I do.

Puzzle Prototyping with 3D Printing

The summer was way too busy but I nonetheless was able to design and prototype my first mechanical puzzle. It even works!

Building a working prototype turned out to be much easier than I expected now that 3D printing has become so affordable and so widespread. All I had to do was create .STL files (I used OpenSCAD) and take them to a local outfit called Einstein’s Workshop whose fantastically helpful staff (thank you Katy and Matt!) printed them for me.

I tried prototyping a puzzle about twenty years ago by taking my design to a machine shop and that cost me a couple of hundred dollars. The 3D printed prototype, on the other hand, was about a twentieth the cost and took a quarter the time, thanks to the amazing march of technology. Also, even though I made a mistake in the first version of my design it only cost me the wasted 3D print, not the cost of a machine shop’s time and materials.

So here is a picture (the pen is there for scale):


As you can see, the surface finish is pretty rough since it was made using FDM (Fused Deposition Modeling). I’m currently waiting to get back a higher tolerance version that was made using SLS (Selective Laser Sintering) and hopefully that will compare favorably with the smooth finish of machined aluminum. My plan is to iterate my design with 3D printed prototypes until the puzzle is working perfectly; then I will decide whether it’s worth the extra cost of having them machined from aluminum or delrin or something.

Most important is that the puzzle mechanism is working dependably and as it was designed to. I’ve shown it to a couple of friends and the feedback has been positive: it’s very hard to solve but not unfairly so. (At some point I want to write an opinion piece about puzzle “fairness” but that’s a post for another day.) For some reason I’ve always liked dovetail joints on puzzles – I guess that’s why I have all three (1, 2, and 3) of Wil Strijbos’ beautiful cube puzzles (great review here) – and so I couldn’t resist the temptation of using them for my first puzzle.

So anyhow, I have reached the prototype milestone and hopefully I will be able to finish the puzzle design and put it into production before the end of the year. Feel free to comment or email me at kay en oh see kay at-sign pea why are eye gee aye en dot see oh em.