# DIGITS IN A BOX

## A puzzle designed by Eric Harshbarger.

Available from Amazon, and other retailers.

[How To Play] - [More Details] - [More Challenges] - [Gallery] - [History]

## Elemental Digits

I recently completed a second set of "Elemental Digits" for myself and documented the changes from the first set with a few more videos:

Some readers of this page have asked if I could make similar sets for people besides myself. Short answer: sure. But they wouldn't be cheap. If you think you might like one (or more), read this webpage I have written about custom sets.

Here's a Digits In A Box project that I started working on in the Spring of 2020, over ten years after I first designed the puzzle. It is something I have wanted to do since those very early days: I am attempting to create a custom version of Digits such that each piece is cast from a different elemental (non-alloy) metal. To do this I am having to teach my self the basics of DIY metal melting and casting. The videos below summarize my progress.

I finished the above set (shown in the final video segment in the playlist above) on 22 May 2020. Here are the data on the puzzle pieces:

 MELT DIGIT ELEMENT SYMBOL °C °F DENSITY (g/cm³) FINAL MASS (g) 0 lead Pb 327 621 11.340 227 1 silver Ag 961 1761 10.490 126 2 tin Sn 232 449 7.310 130 3 bismuth Bi 272 521 9.780 182 4 cadmium Cd 321 610 8.650 127 5 antimony Sb 631 1167 6.697 116 6 copper Cu 1084 1983 8.920 170 7 indium In 157 314 7.310 96 8 aluminum Al 660 1220 2.700 58 9 zinc Zn 419 787 7.140 137 Total Mass Of Puzzle: 1370 grams (just over 3 pounds)

How To Play. DIGITS IN A BOX is one of those puzzles for which it is very easy to explain what you are supposed to do; actually doing it is the hard part. There are ten plastic pieces (each shaped as one of the digits 0-9). You simply take them out of the box (don't memorize the configuration as it come out -- that's no fun!), separate them (maybe put them in numerical order), and then try to get them back into the box so that the lid closes shut.

More Details. There will be spaces in the box when you find a solution (18 unit gaps), you are not trying to build a solid cube. Each piece is designed to be bounded by a 1×3×5 volume, and the box is a 5×5×5 volume (so each number is as tall as each of the box's dimensions).

The instructions on the box read that there are "over 4000 possible solutions"; there are actually 4239 solutions. That is the number of distinct solutions; it does not include repetitions due to symmetries or answers in which like-shaped pieces are exchanged (the "2" and "5" are identical, as are the "6" and "9"). However it does include many answers which are almost identical (for example, often the "8" piece can simply be swapped with another of the pieces -- but these are considered separate configurations).

The size of the puzzle, as manufactured by Popular Playthings, is based on a unit cube of 12mm (just under half an inch), with the interior of the box being just bigger than 60mm (about 2½ inches) per side. Each piece is a different color (though which pieces are which colors varies from set to set).

More Challenges. If you have bested the puzzle (or maybe it has bested you), you might be looking for alternate challenges to toy with. Here are some that I've thought of or that have been suggested to me:

• 16×8×1 Box. Essentially, keep the pieces flat on a tabletop and try to fit them into a 16×8 unit rectangle. This is not as hard as the original puzzle challenge, but it may still take some work. All of the solutions I have seen to this challenge have required at least one of the pieces to be flipped over, but I have not ruled out the possibility that it could be done with all of the numbers "solid face up".
• 7×6×3 Box. With a volume of 126 (one larger than the 5×5×5 original), this challenge should not be too tricky.
• 6×5×4 Box. Similar to the original challenge, but the target volume has slightly altered dimensions. Surprisingly, even though this volume is smaller than the original 5×5×5 one, this is an easier challenge to solve (you'll probably quickly figure out why when you get the answer).
• 5×5×5, Sequentially Adjacent. This is a much harder variation of the original task. Try to find a configuration that fits within the 5×5×5 volume such that the "0" touches the "1", the "1" touches the "2", the "2" touches the "3", and so on, all the way, so that the "9" even touches the "0". It can be done, but I do not know in how many distinct ways.
• 5×5×5, One-At-A-Time. There a good chance that if you found a solution to the original challenge, you did it by assembling most of the pieces outside of the box first, and then placing them in the container when complete. Try to find a solution whereby you place one piece in the box at a time; once in the box you cannot take them back out. Can you still find a way to get all ten pieces inside so that the lid closes?
• Largest Contiguous Hole. Every solution of the 5×5×5 challenge will necessarily have 18 unit holes or gaps present. Can you find a solution in which 13 of those unit gaps are adjacent (make a contiguous hole)? That is the largest such hole possible (there are only three ways out the the 4239 solutions that yield this best answer).
• Spelling Words. Often, if other people see you manipulating the DIGITS-pieces, they might think that the shapes are supposed to represent letters rather than numbers (the "4" upside-down looks like a lowercase "h", the "0" could obviously be a "O", and so on). This leads one to a completely different type of task: what words can you spell with the ten pieces you get in a DIGITS set? A little imagination in translation between numbers and letters might be necessary, but quite a few words are possible. For example, here are images of "SHOES", "SIGH", and "JOBLESS" (with the "1" now used as a "J" rather than an "I" as earlier). "BOGGLES" is also possible, but "GOGGLES" would only be possible if more than one set of pieces is used. What other words can you spell with the pieces?
If you think of a new, creative way to play with the puzzle, let me know.

Gallery. If you have found one or more solutions to DIGITS IN A BOX, and you are so ecstatic about it that you want to send me a picture, please do. I will create a seperate page that displays happy solvers. You're also welcome to send photos of frustration if you can't find an answer, I guess... I'll post those too.

History. I first conceived of DIGITS IN A BOX while driving home from Atlanta after the seventh "Gathering for Gardner" (a bi-annual conference of mathematicians, magicians, puzzles, and other eccentrics held in honor of the later writer). I was in even more of a puzzle designing mood than usual, and for whatever reason keyed in upon the idea of physical numbers. DIGITS is basically a type of "polyform" assembly puzzle, akin to the popular Soma Cube and Pentomino Pieces (of which I've been a fan since childhood), so the fact that my mind drifted in that direction is not at all surprising. I'm also a hobbyist font/letter designer and have always enjoyed creating simple letter and number shapes whether for puzzles, games, or computer graphics.

On that drive home, I considered a design of the digits 0-9 in a basic style such that each would fit inside a 5×3 rectangle (and if extruded to three dimensions by one unit: a 5×3×1 volume). I calculated the total volume of the shapes I had in mind and realized, at 107 unit volume, the total of the shapes might be able to be interlocked to fit into a 5×5×5 box. I say "might", of course, because just being a small enough volume does not guarantee that there would actually be a configuration possible. There was also the chance that the answer would be too easy. I'm pretty good at visualizing three dimensional geometry, but trying to determine a solution was not something I could do while speeding down I-85 toward Auburn.

Immediately upon getting home, however, I built a prototype out of LEGO bricks (of which I had many). My original version actually had a slightly different "7" shape (it had one less unit, the top did not hook downward) -- so its total volume was actually 106 units at this point). Despite this possibly easier-to-solve design, I still did not find a solution right away. The closest I came was by lopping off yet another unit from the top arm of the "6" piece (this was not the most aesthetically pleasing design, but it did distinguish the shape from the fully formed "9").

With these abbreviated pieces, I was able to find a solution in an hour or so, and I was pleased enough with this design to introduce the LEGO-built version at my 2006 Spring Puzzle Party. That event was held just a few weeks after the original inception of the puzzle idea.

Later that year, over the summer, I was visiting friends out in California, and spent one afternoon catching up with fellow puzzle enthusiast Wei Hwa Huang (seen here as we goofed around with puzzles on the campus of his former employer, Google). I had brought about a dozen puzzle prototypes with me, including DIGITS. He was intrigued by the puzzle and mentioned that he had recent cooked up a computer program to find solutions for certain classes of puzzles and was looking for puzzles to test the application on.

This was perfect, I said, because I still had not found time to determine if the 106-unit version (with the "6" piece fully formed) was even solvable (i.e. packable into a 5×5×5 box). He copied down the shapes of the pieces, and told me that he'd analyze it soon.

And soon he did... by the next morning, in fact. My email inbox held a message from Wei Hwa with an attachment listing over 10,000 solutions! No only that, but he had actually copied down the "7" piece incorrectly... it's his fault that the extra unit hooked onto the arm of that piece (a pleasant surprise to me, because it made the puzzle even harder, and the "7" still looked good... maybe even better). I immediately memorized one of the first solutions listed (in a fairly obvious textual notation), and to this day that is the solution I generally use to get all of the pieces back into the container. Over the next few months that list of solutions was pared down quite a bit. The original roster had duplicates due to symmetries as well as some configurations that were physically impossible to construct.

By the end of 2006 I had had some nice sets out of aluminum at a local machine shop. I started selling some of these sets to friends who got hooked on the puzzle, and in 2007 the puzzle was my entry into the Nob Yoshigahara Puzzle Design Competition.

At the start of 2010 I bought a laser cutter and started crafting half-sized acrylic versions (which were much less expensive than the \$150 metal sets). It was a couple of these sets that made their way to the puzzle and toy manufacturer Popular Playthings, and that company expressed an interest in mass producing them out of injection molded plastic (on a scale and at a price which I obviously could not match as an individual). The puzzle was produced in time for their 2011 catalog of products. Chances are that is the version of the set you have played with if you tried to solve it.

If you own one of the acrylic sets, or one of the older metal sets... consider those collector's items.