The seven-piece tray packing puzzle City Block first entered my radar back in July when I emailed David Litwin to enquire if he had any copies of his previous tray packing collaboration with Bram Cohen, Breadbox, available for sale. He did not, but mentioned a new packing puzzle was in the works and asked me to check back around August.
The email half-slipped my mind until I saw Dave’s post on City Block in the Twisty Puzzles forum and shot out an email. A week or two later it arrived on my desk at work. I had a quick peek and took it home for further investigation.
The puzzle contains seven acrylic pieces with small holes in them, resembling the lighted windows of a darkened office block late at night. Most tray packing puzzles I have encountered do not have a theme. It’s a unique aspect of City Block and Breadbox that Dave has come up with a theme for each puzzle related to the shape of their pieces.
City Block poses two challenges for the puzzler. The first, described by Dave as a warm up, is to create the skyline printed on the puzzle’s frame with the pieces available. The next challenge is far more difficult. My first attempts had all the pieces fitting into the frame but looking suspiciously like the unintended solutions that I’d encountered with tray packing puzzles. I emailed a photo of my attempt to Dave. He confirmed that it was indeed wrong. I would know the solution when I saw it.
One aspect of tray packing puzzles I like is that, outwardly, they are so simple. Puzzles of this sort often work around “blind spots”, that is, counter-intuitive ways of combining the pieces that are not a solution most people would naturally arrive at.
City Block is a collaboration between David Litwin and Bram Cohen, although Dave holds his role in this puzzle is smaller than in Breadbox. In late September-October 2013 I spoke with them via email about City Block to get some insight into its creation.
Saul Symonds: This is the second puzzle you two have worked on together. Can you tell me a little bit about your collaborative process and how it works?
David Litwin: Bram and I both work in Downtown San Francisco and have met at many puzzle parties (IPP and local). A few months before IPP31 I was walking to lunch and bumped into him on the street. I lamented how most of the puzzles I build take far too long to build 100 for the IPP puzzle exchange, and he offered two of his ideas that might be suitable. He later emailed me with the details. The description of the Breadbox idea was only 12 words: pack eight AxB rectangles plus eight CxD rectangles into a WxW square. The actual dimensions helped lead me to the solution so I have removed them.
Iterations of the design continued over email where I suggested the bread theme, added curves and various different shape iterations. When I cut a prototype I brought it over to his office to show him over lunch.
City Block was similar, but the conversation was a mix of phone messages and email, and I was able to show him a prototype at a local puzzle party.
SS: You two seem to have mostly worked on twisties and tray packing puzzles, is there any connection, in your eyes, between these two sorts of puzzles?
DL: Not a lot. I focus much of my collection on twisties but we both have wide interests in puzzles. Tray packing puzzles happen to be far better suited to the constraints of an IPP exchange puzzle (small, easily packable, not horribly expensive to make) so that has been our collaboration so far.
Bram Cohen: To my mind tray-packing and twisty puzzles are very different, in that one is completely 2D and the other is (usually) very 3D. I normally stick to 3D, because my brain has this odd quirk of being much better at 3D than 2D, but sometimes stray into the simpler worlds of 2D.
SS: This is a question for Bram: what was the genesis of the puzzle? Did you have an idea of what the final pieces/puzzle would look like or did it take shape over time after playing around with the tiles and a tray?
BC: I can’t answer this without giving massive spoilers, so be forewarned.
The general inspiration is that there was a Stewart Coffin packing puzzle involving funny angles included in the IPP exchange, and I spent some time messing with that. My idea was to make a puzzle with a rectangular boundary where the only solution was to have all pieces at a funny angle, and where each of the pieces only touch the boundary at one point. I’d previously made an antislide puzzle in 3D which has the same property, although that one is twelve identical pieces, because there are symmetries in 3D which make it easier to do this than in 2D.
The first question is how many points of contact are necessary. Since an object in 2D can move up or down, right or left, and clockwise or counterclockwise, that’s three degrees of freedom each of which need two points to block it, for a total of six points of contact. In 3D the similar calculation is that an object can move up/down, right/left, forward/back, and the axis of rotation can be at any latitude and longitude, and exists in the first place, so that’s three degrees of freedom for translation, one for rotation being there at all, and two more for variants in the rotation, which is a total of six, each of which has to be blocked in either direction, so that’s a total of twelve points of contacts. (I am of course short circuiting a whole bunch of math here, but the counting of degrees of freedom is accurate.)
To keep things simple I decided to maintain 180 degree symmetry. It turns out that things hold together a lot better if there’s a seventh piece in the middle, because that can block the existence of a single cut which runs through the entire puzzle. The central piece isn’t necessary for my main criterion, and come to think of it glueing it to one of the other pieces could probably solve the unintended solutions problem discussed below.
In order to prevent the whole solved assembly from rotating in place it’s roughly necessary to have two points of contact on the top and bottom, one on the left and right, and have the ones on the top and bottom be one on the left half and one on the right half. After a bunch of iteration I found that it’s not only possible to make a puzzle which meets all this criteria but it’s possible for the cage to be square as well, so I decided to keep that.
In order to minimise the chances of unintended solutions it’s very important for the size of the cage to be just slightly less than an integer number of voxels rather than just slightly more. Burr Tools was able to find completely rectilinear unintended solutions to a bunch of variants until I managed to get it just barely under an integer.
Dave came up with a nice idea for making the central piece a little more misleading, which we had to nix because it lead to some unintended solutions. Since then some unintended solutions based on much deeper geometric properties have been found, basically direct variants on the intended solution. With the glueing the central piece to another one idea I just came up with, it might be possible to resurrect Dave’s idea.
Breadbox was inspired by the Hoffman packing puzzle, with the idea being to make a 2D equivalent. I spent several days messing around with it in Burr Tools. Despite its symmetry, the solution wasn’t something I came up with by hand, it was something I noticed was in a solution which Burr Tools found, and I then massaged the dimensions until that was the only solution, again with a lot of help from Burr Tools.
SS: The pieces that were chosen are very unusual in that they have many sides and corners, which adds to the difficultly of fitting them in the tray. Can you tell me about you choice of pieces? It seems as if you tried to do the exact opposite of your previous tray packing puzzle Breadbox, in which the pieces have a lot of curved edges.
DL: Breadbox was originally a rectangular piece concept. The bread theme came later and the curves from that. But those rectangles were simpler shapes. The shapes of the pieces in both of these designs come from beauty of the solution. The solution of Breadbox was a much denser packing than City Block so full rectangles made sense. The solution of City Block has much more space and those particular shapes serve a different purpose.
SS: What do you both think of the final puzzle when compared to the initial idea?
DL: Unlike Breadbox I don’t feel I added much to the core puzzle of City Block. I dressed it up with nice materials, a theme and a name, but the puzzle is unchanged (other than some iterative tweaks) from the original concept. This particular design had a few unintended solutions not thought of in the initial idea, which is always a challenge with packing puzzles. The “true” solution has a very elegant property that is a bit hard to spot due to the inherent inaccuracies of the production process (laser cut acrylic). It is a shame mentioning them would hint at the solution.
BC: I like Dave’s theming in both puzzles, it adds a lot to the puzzle’s appeal and I myself am incapable of doing such themes. The curves Dave added to Breadbox are a functional part of the puzzle, and both make possible (and require) the 17th piece and make it a little different from other rectilinear tray packing puzzles, which are nice properties.
SS: Dave, you mentioned that your mother found an unintended solution to the puzzle and it was adjusted so that this solution would no longer be possible. Can you tell me about the testing phase and how you went about attempting to eliminate unintended solutions?
DL: Mostly it involved letting people play with the prototype. Each one came as a surprise. One was solved before the final puzzle was cut, but two more have been found with the final puzzle. It turns out to be difficult to remove an unintended solution and keep the desired solution.
BC: Puzzles in general should be paradoxical, and the paradox in City Block is that it should have a lot more solutions than it does. I found many, many, unintended solutions to variants while doing them by hand, and then while checking with Burr Tools, and finally being playtested by other people. Hopefully we’ll be able to massage it a bit and get rid of the last of the problems.
SS: Dave, you said that the City Block theme came out of your desire to reduce the weight of shipping 85 puzzles from America to Japan. Did the skyline warm up puzzle follow cutting small holes in the pieces or did it come first? Can you talk me through the process of adding this warm up to the puzzle.
DL: Weight was a consideration as I brought all my exchanges puzzles in my carry-on luggage. The nature of this design meant that only the outside shape of the piece was important, so I saw an opportunity to remove a portion of the inside of each piece. Originally this was going to be done in a way that would make them structurally sound but minimise material, but the idea of cutting holes out of the interior (and the shape of the pieces) quickly led me to a “building” theme. A collection of buildings is a City, and a square of buildings is a block.
Once the theme was established I wanted to add a bit more, so I thought to come up with other combinations of pieces (outside the block) that might be interesting. Nothing clever came to mind quickly but the idea of a non-trivial skyline seemed reasonable. A bit simple perhaps but it gives the puzzle some appeal to entry level puzzlers.
SS: Breadbox had quite a large number of pieces for a tray packing puzzle (17!) and City Block has seven. Increasing the number of pieces is one way to increase the difficultly level of a tray packing puzzle, was this a factor in deciding how many pieces to include?
DL: The original Breadbox concept was 16 pieces but revisions and curves allowed for the 17th. This is a huge number of pieces and probably a bit outside of the norm. If the solution weren’t so pretty and elegant I don’t know that many would have patience for so many. Bram can confirm, but I don’t believe the number of pieces was ever chosen with difficulty in mind, the number fell out of the solution.
BC: I always try to go for having a unique solution and elegant pieces, which generally means either a limited number of pieces or a lot of identical pieces, otherwise there tends to be too much room for unintended solutions. Using identical pieces is sort of cheating, because permutations of the identical pieces are counted as the same, but people still accept the solution as unique. People tend to find the number of pieces in Breadbox intimidating, which is too bad because it’s a very rewarding solve experience and nowhere near as hard as it appears, but there’s no way to keep the same concept with a smaller number of pieces.
David Litwin’s puzzles can be purchased directly through his website