Mathematics of the Rubik's Cube

A very important fact about myself is that I am a very maths based person. It is a language that I can go about my daily life with in total comfort. That's certainly something that very few people are able to say! Of course, my thought process lead me one day to find out about the maths that surround the Rubik's Cube.

Many people know that the Rubik's Cube has 'billions' of possible positions, and for someone unfamiliar with the puzzle it is MINDBLOWING! Fewer people know that there are actually many more permutations, about 43 quintillion (4.3x10^19), and even fewer people understand why there are that many positions.

To determine the number of positions, one needs to consider the different pieces and how they can be arranged. A standard Rubik's Cube has eight individual corner pieces, each of which has three different orientations. There are also twelve individual edge pieces, each of which has 2 different orientations. Since the positions of the corners and edges will depend o one another, the final position of each is disregarded, giving the number of positions as 8!x3^7x(12!/2)x2^11, or

43,252,003,274,489,856,000

possible permutations. Say that you had one standard 57mmx57mmx57mm cube for each configuration. You could cover the surface of the earth 275 times over. Now that's big. 

What is interesting is that by physically moving pieces into places they do not typically belong, one can change the 'orbit' that the cube is in. If you disregarded the fact that not every piece can reach its full number of positions, there are actually about 519 quintillion possible positions, twelve times as many as there are just by turning the cube. If one corner was rotated, or one edge flipped, or two edges or two corners switched, the cube becomes unsolvable because the solved position can no longer be reached by ordinary turns. Also, these number do not consider the centre pieces, since they do not normally have any markings on them. But with a supercube (future blog?) the number of permutations goes even higher. 

Until next time...

For more info, check out the following:

Numberphile on the Rubik's Cube:

http://www.youtube.com/watch?v=QV9k6dRQQe4

RedKB on Why is the Rubik's Cube so hard?

http://www.youtube.com/watch?v=QV9k6dRQQe4

The Half-truncated Cube-My First Mod!

The half-truncated cube is exactly what its name says: an ordinary cube with half of its corners chopped off. This mod turns an ordinary Rubik's cube into a puzzle even more fun and cool-looking!

I'm new to the modding scene and wanted to start with something easy, so the half-truncated cube is a great start. Another one to start with would be a hexagonal dipyramid-something I'm currently working on (and it will be on here when it's done!)

I started by peeling the stickers off of an old cube and marking where the cuts needed to be made. I tried to keep the puzzle aligned with some sturdy rubber bands on each side, which worked until I had made a couple cuts. After finishing this project, I feel that for this particular mod it may have been better to mark each individual piece where to be cut, since they are all the same (corners and edges). I figure that by doing this it would be easier to make sure each piece is almost exactly the same. Sanding away at each corner of the whole cube allowed for small movements out of place to affect for the pieces line up. It didn't end up being a big deal, but for better quality work in the future I will try this method.

After all the shaping of the puzzle, my next step was to tidy up the rough edges created by my Dremel. With some 100 grit sandpaper, I made sure all the edges were flat and ready to be filled. I filled the pieces with JBweld KwikPlastic putty, pretty expensive stuff I think for what you get, but I could not find Milliput or Apoxie Sculpt around here and wanted to get started. I plan on trying some plastic prototyping compound but I have yet to see how it sands down and reacts to the working that would have to be done to it. After being filled and allowed to set, much more sanding was needed to make all the surfaces as smooth as possible before painting.

I cleaned all the pieces with soap and water and then with a little paint thinner before painting. I used Krylon Fusion for plastic, a good quality spray paint that turned out well. I used stickers from Cubesmith, because quite frankly I was too lazy to cut them myself, and the lubricated it with Maru lube. And the end result:

I am very happy with it, and it turned out about as well as I could have expected for my first project! up next: 3x3 hexagonal dipyramid!

Until next time...

Online Puzzle Stores

The internet is a beautiful thing, isn't it? What makes it even more beautiful is that people like me, can order PUZZLES with it! Awesome!

There are a number of websites that sell twisty puzzles. Most of these companies are based out of China, and they all vary greatly. I want to discuss the main stores that I use and recommend.

HKnowstore

This store is the main one that I shop from. The sales manager, Calvin Fan, is a very professional young man who cares about his customers. I have not yet had a bad experience with them. They are located in China, however their shipping (free!) never takes very long, for me in Canada, it has never been more than two weeks. Other Chinese stores have taken much longer to ship!

HKnowstore carries a wide variety of puzzles, however they do not encourage the sale of Knock Off puzzles and do not carry them. Regardless, they are usually in stock with their items and have reasonable prices for what you're getting. They also have their own brand of puzzles (called Calvin's Puzzles) which are very high quality, and excellent cubes. Payment is also through Paypal, which should make everyone happy.

51morefun

I have bought from this store a few times, and have not yet had a bad experience. Supposedly, a year or two ago their quality of service was not very good, but recently they have became much more reputable among the community. There are several shipping options, namely UPS, DHL or Air Mail. The items I have ordered from here have always taken quite a while, almost a month, however arrived in good condition. They also carry a variety of puzzles, as well as KOs, i.e Shengshou puzzles. They have reasonable prices and a great variety of products.

 Witeden

Witeden is a another shop I have bought from a few times before.  To me, it seems very similar to Hknowstore, and even their logo looks similar...hmm. To my knowledge, Witeden also does not carry KO puzzles, however still has a large assortment available. They also make their own puzzles, and have many unique designs that vary greatly from one to the next. At first glance, they come across as having slightly cheaper prices than say HKnowstore, however Witeden charges for shipping, and after that charge the price is almost identical to that of other stores. Over all however, Witeden is another good store to buy twisty puzzles from, and their shipping times are between that of HKnowstore and 51morefun in my experience.

In a future post I will discuss some of the other stores that are out there, as well as sites that specialize in the sale of a specific product. Until then, I always update the links at the bottom of the page under 'Puzzle Stores'.

Until next time...

Megaminx Solving

The Megaminx is a special puzzle to me because it is the first one that I learned how to solve by myself in its entirety. What I mean is that I worked out a method to solve the puzzle without any tutorial, which was a big deal for me.

The only thing I really knew about Megaminx was that it can be solved in a similar way to a 3x3 Rubik's cube. With that in mind, I started to solve it by making a 'star' on one side, and then inserting the corner pieces. Next, I moved onto the 'second' layer, and then I hit my first stump. What I managed was to solve the next section as one 'layer' by alternating edge and corner pieces; doing them one at a time, one after another, ending with a corner piece. Next is just the next layer of edges, and then the last layer. The last layer gave me some problems, but with a couple days of on and off work on it I figured out how to apply 3x3 algorithms to manage the last layer. While my method may not be terribly robust, I don't have any interest in speedsolving Megaminx, and so I will take my satisfaction in solving it and be done.

I haven't stopped there, however. I decided to move onto something bigger, and I purchased a Master Kilominx. An excellent puzzle after it was lubricated, I enjoyed the challenges it offered as an even layered minx puzzle. I was lucky with my first solve and did not end with parity, but the second time, there was parity, and i resorted to looking up an algorithm for the case. However, I did enjoy solving this puzzle as well, and it's one I do the odd time, so the parity is not a big deal.

I find minx puzzles to be an interesting challenge, and I plan to purchase a Gigaminx in the future. Although I do not want to speedsolve minx puzzles, I think they look awesome and are fun to play with, and of course are a great addition to a collection.

Until next time...