Back in 2002, my good friend Derrick Schneider noticed this nice little set of slightly strange shapes and wondered whether or not they'd make a good puzzle. He whipped up a little program to try packing the pieces into an 8x8 tray. To his delight (and later ours), there was just one way to fit in all four pieces! Many designers would have stopped there, but for some reason Derrick also tried running the program on a 7x9 rectangle: once again, incredibly, there was a unique solution!

Imagine the fun: he comes up with a simple way to define a set of pieces, the resulting set is nice and small, and that set fills both of the two most obvious tray shapes in unique ways. Believe me, such a mathematically elegant puzzle design doesn't come about every day! Add to that, the resulting puzzle falls into a real sweet spot of difficulty: harder than you might guess (those pieces are just plain tricky to get your brain around, especially the curled-up one), but easy enough to yield to a little patience.

Perhaps that explains why, when Derrick presented his puzzle at the 22nd annual International Puzzle Party, the jury for the Puzzle Design Competition awarded it an Honorable Mention, one of just three puzzles so honored.

Now, for the first time in many years, I'm happy to make Derrick's wonderful little puzzle available for sale again. Initially, I'll be selling off the remainder of Derrick's original manufacturing run; the last time I visited, I got him to dig around in the basement and pull out all of his old inventory for me. After that limited supply sells out (he could only find about 15 of them), I'll start making my own copies for you. This is simply too good a puzzle to remain unavailable for so long.

I also wanted to use a prettier material than the opaque black high-impact polystyrene I'd used before, and that ruled out pretty much any piece deformation at all. (I'd tried making Octamaze out of acrylic, since it takes etching much better than polystyrene does, but after having my acrylic prototype shatter in my hands during disassembly, I gave up and went with the much more robust and pliant polystyrene.)

I decided that the pieces would rotate into position, using some kind of interlocking, hook-shaped protrusions on the edges of each face; that would still entail fairly wide (deep?) hooks, due to the shallow angle, but avoid any piece deformation during assembly. I realized that I might run into a mechanical problem with the corners of the faces hanging up on one another as each face was rotated into position, so I started considering various piece shapes that were missing the face corners. I started playing around with actual shapes on paper, instead of just thinking about all of it, and all of this came together in a kind of practical lesson in geometric duality: the hooks would be on the ends of arms, and the puzzle would look like stars with interacting points rather than the regular polygonal faces I'd originally imagined.

The final piece of the design came from my realization, late at night while trying to fall asleep, that there was a wonderful pun opportunity in naming the puzzle, since IPP last year was held in Prague. Once I'd come up with the name "Star Czech", and stifled my giggles to avoid waking up my wife, I knew I was done.

The resulting puzzle is probably the prettiest one I've ever designed, once assembled: it would make a great Christmas ornament, or an attractive object to dangle from the rear-view mirror of your car, let alone just sitting on your desk at work. It's also quite a difficult puzzle, so keep that in mind if you buy a copy. Like all of my puzzles, it comes with the solution, though, so nobody need know if you decide to short-circuit the solving and jump to the pretty bit.

What better way to mark the occasion, they thought, than to bring together a lot of people who had enjoyed, and been influenced by, Martin's work? So they invited a bunch of people from the fields nearest to Martin's heart, from mathematics, puzzling, and stage magic, to come to Atlanta for the "Gathering for Gardner": several days of talks, performances, and exhibitions in celebration of Martin's 80th birthday. That first Gathering was such a huge success that the organizers decided to keep doing it and, every two years since then, there's been a Gathering.

I've known about the Gathering for some time now, through contacts at the International Puzzle Party, but I was pleased to be invited for the first time this year, for "Gathering for Gardner 8", or "G4G8". Similar to the Puzzle Exchange every year at IPP, the organizers of the Gathering ask that everyone provide a gift of some sort for all of the other participants. Many people fulfill this obligation by giving a talk and writing up a short article for the conference proceedings book, but many others bring puzzles, magic tricks, or other entertaining objects.

You know, of course, what I did, right?

Every Gathering for Gardner has a theme; I think you may be able to guess what all of the previous themes were when I tell you that this year's was "8, or the crazy lazy 8 (infinity)". I wanted to bring a new puzzle design that would strongly incorporate the theme, of course, but I also wanted to continue down the path I'd forged with my Ooo Tray puzzle last year. I wanted to design another multi-stage puzzle, with each solving stage revealing clues to the next stage, culminating in a "final answer" that was somehow satisfying.

I'd been idly thinking about three-dimensional edge-matching puzzles for a while (What? Don't try to tell me you don't think about such things, too.), and I'd wondered if it would work to make a polyhedron where the sides fit together with tabs and slots. With a theme like "8", this was a perfect (some might say Platonic) opportunity!

A little software work and many design iterations later, Octamaze was born. There are at least four stages in solving this puzzle, providing a good hour or two of "play time". If you buy a copy of Octamaze and get stuck, I've created a sequence of web pages giving a progression of hints for solving it. (Don't worry: clicking on that link won't immediately reveal any spoilers. If you keep clicking on the links at the bottom of each page, though, you'll eventually see all of Octamaze's secrets, so be careful.)

I originally blogged about this puzzle a couple of years ago, shortly after I got my first copy of it, six years in the making. A few days after that, I got email from my friend George Miller, telling me that he'd laser-cut his own copy and liked it a lot. He showed that copy to long-time IPP attendee Stan Isaacs, who asked me for permission to use it in the IPP 26 Exchange, in Boston. At the time, I had a different puzzle in mind for my own exchange gift, so I agreed. Unfortunately, that other idea fell through (sometimes that happens with puzzle designs), so I ended up not exchanging that year. Instead, I was Stan's exchange assistant, which was fun in its own way.

The version that Stan exchanged was somewhat different from my original copy: he and George reshaped the pieces from rectangles to half circles, making the completed puzzle into a sphere instead of a cube; they called it the "Fan-Way Park Ball", following the Bostonian theme. They also used laser-cut maple instead of plastic and reduced the size to about 1-5/8 inches. It was cute in its own way, and I was happy to see the puzzle exchanged, but I still preferred the clear Lucite look of my original version; it looks a bit more stylish sitting on your desk.

I'm now happy to announce that I can offer copies of my version for sale here. The puzzle is shipped disassembled, flat, and putting it together provides a very satisfying but accessible solving experience.

Flush with the successful design of the "Perkinson Guest Bathroom Tile" puzzle, the obvious next step was to consider dominoes instead of trominoes. This time, I allowed the pieces to be flipped over, and I also counted pieces whose two tiles had their tilted edges at right angles to each other. This leads to a complete set with 13 members, some of which are slightly strange looking, and one of which (with the two tilted edges joined together) is a quite boring perfect rectangle.

Leaving out the boring rectangle, we get twelve roughly one-by-two-unit dominoes; the obvious tray shape is a six-by-four-unit rectangle. Can the (nearly) complete set fill that tray? Sadly, my packing program said no, and this time it also didn't work to change the top and bottom edges into the angular "sine wave" pattern from the bathroom-tile puzzle. I had my local laser cutter, Joe Pelonio, make me up a set of the pieces anyway, so that I could play with them and try to get a feeling about why the rectangular tiling wouldn't work. The first thing I noticed when I got the pieces in my hands was that one of them looked a lot like the profile of one of the famous "moai" heads from Easter Island; thus the eventual name of the puzzle.

When I presented Easter Island Dominoes at the 2007 IPP Exchange in Australia, I told the following story:

It's not well known (especially to archaeologists), but many, many sets of these 12 pieces have been discovered in excavations on Easter Island. Never, though, have they come across a copy of that elusive 13th piece, the perfect rectangle. From this, we can infer that the ancient Easter Island culture, now long lost to us, did not approve of straight lines and perfect rectangles. Being a culturally sensitive fellow, I've created a tray that has one tilted tile edge exposed on each edge of the tray, thereby avoiding violating the islanders' taboos.

Your first challenge in solving this puzzle is simply to lay all twelve pieces flat in the tray; there are 250 ways to do that, and it's not very difficult if you just have a bit of patience. You'll find, though, that almost every such packing has at least one blemish (as least from the point of view of ancient Easter Island culture): there will either be (a) a straight-line crack all the way across or down the puzzle, or (b) a subset of the pieces that form a perfect rectangle or square, or (c) both!

There are just 83 ways to pack the pieces without a straight-line crack, and only six ways to do so without forming a perfect square or rectangle. Your real challenge is to find one of merely three solutions that have neither "blemish". That'll take you a little bit longer to achieve, I think.

In the fall of 2004, I was playing around with a deceptively simple little tray puzzle designed by Stewart Coffin and called Four Sleazy Pieces. The eponymous pieces are polyominoes with between five and seven squares each, and the tray is a perfect square whose size is not an integral number of units; it's about 5.8 units on a side. I won't spoil Coffin's fine puzzle here, but suffice to say that, at the time, I hadn't solved it very recently so I'd temporarily forgotten just how "sleazy" the solution is.

While failing to solve it, I stumbled across an interesting property of such a puzzle, and I was excited to note that what I'd found was probably a good psychological "blind spot". These blind spots usually take the form of an assumption most people make that's so "obvious" that it's never questioned, even though it isn't in fact true. Armed with such a blind spot, you can often create a puzzle to exploit it, a puzzle whose solution violates that implicit assumption. That kind of a puzzle has a very pleasant "ah-ha" feeling to it; once you've solved it, you can't figure out how it could possibly have taken you so long.

My resulting puzzle, Sleazier, has that property, and I'm inordinately pleased with it: I think that it's probably the best puzzle I've yet designed. I presented it in the 2005 IPP Exchange, in Helsinki, and it's gotten great feedback in the years since then.

Like Coffin's puzzle, Sleazier has four polyomino pieces ranging from five to seven squares each, and a square tray of a suspiciously odd size; your goal is simply to fit all four pieces flat in the tray. I won't say why, here, but when you compare the "trick" of my puzzle to that of Coffin's, mine definitely deserves its name.

In my very first IPP Exchange, in 2000, I presented a puzzle designed by Bill Darrah, called Raft 5. It consisted of 10 sticks, each with a dovetail notch cut across it and a matching dovetail tab glued on along it. There were five different positions on the stick where a notch or tab could be located, for a total of 10 different pieces, and the puzzle contained one of each. Somewhat surprisingly, it's actually possible to assemble these 10 pieces in a raft-like arrangement, with five sticks going one way laid across five going at a right angle.

Raft 5 is a good puzzle, and for the 2003 Exchange in Chicago, I decided to take inspiration from it. The raft is essentially two dimensional, and I wanted to somehow extend the idea into 3-D. To keep the number of possible pieces down, I made my sticks shorter, with only three positions where a notch or tab could be placed, but I also made the sticks square in cross-section (as opposed to rectangular, as in the raft). This allowed tabs and notches to appear on different sides of the stick, even at right angles to one another.

If you leave out the cases where a notch and tab appear in the same position along the stick (unless they're on opposite sides of the stick), then you get a total of 14 possible pieces. That seemed like too many for a good puzzle, so I picked just half of them; I was perversely tickled by the idea of using seven pieces in an interlocking puzzle, instead of the traditional six. Of course, seven pieces can't make a very symmetric shape, but I made a virtue of that, and designed the puzzle so that the final shape isn't particularly important. Instead, the goal is simply to arrange the pieces so that every piece's tab is inserted into some other piece's notch; that forms the pieces into a kind of folded-up loop, each inserted into the next, like a snake eating its own tail.

There are four solutions to the puzzle, two of which have the fun property that the resulting assembly will balance nicely on the end of one of the sticks. Those two solutions look a bit like a figure standing on one foot, which I like quite a lot; at some point, I want to make a very large set of the pieces to use as a bit of artwork for our yard.