Q-BA-MAZE

10 Billion Trillion Combinations: Toy Math and Design Space

WHAT IS A DESIGN SPACE?

The huge sets of possible permutations for LEGO Bricks and Q-BA-MAZE cubes are called "design spaces" (Beinhocker, p. 193). It is up to the designer, the person playing with LEGO or Q-BA-MAZE, to discover the best designs among the zillions of possiblities. The enormity of these "design spaces" describes both the potential challenge and the level of freedom for the designer.

How many combinations are there? How big are these design spaces? Just six 2x4 studded LEGO bricks of a single color can be rearranged in 102,981,500 different configurations (Bedford, p. 19). The Rubik's Cube* can be scrambled in 43,252,003,274,489,856,000 ways (Walsh, p. 230). The LEGO "Creator" set contains 500 pieces of different shape and color which can be combined in roughly 10 to the power 120 combinations (1 followed by 120 zeroes) (Beinhocker, p 193). Given that the universe contains around 10 to the power 80 atoms (1 followed by 80 zeroes), the 500-piece LEGO set is pretty impressive!

I've always been impressed with these huge numbers, but also a little skeptical, because I like seeing proof. It seems the math behind these figures is never shown. And probably, the math cannot be shown, because the problems are so complex that a computer program needs to be written to calculate the combinations. This is the case with Q-BA-MAZE. I can't easily get the answer to the seemingly simple question, "how many ways can the Q-BA-MAZE cubes be reconfigured?" because it would take a custom computer program to calculate this.

It is possible, however, to make a rule that describes a subset of the ways Q-BA-MAZE cubes can be rearranged and to express this rule in a simple mathematical formula. The results of our calculations, just a subset of the total possibilities, yield these surprisingly huge numbers of ways 18 to 36 Q-BA-MAZE cubes can be rearranged:

Calculation 1B above of the 18 single-exit cubes in a 50-pack yields 9,656,357,112,229,430,000,000 and can be described as roughly 10 billion trillion combinations.

The helical construction of Q-BA-MAZE cubes in the photo at the top of this post is a single continuous pathway with no jumps. It is made with the single-exit cubes that come in the Cool Colors 50-pack (Q50C) and is one of the configurations included in both Calculation 1A and 1B.

HERE'S THE MATH

To satisfy my curiosity, and with the help of some mathematician friends, we devised this formula:

If you would like to see this formula in action just open up this Excel file and you can manipulate it as you like and see what happens as you change the variables. Here is a screen shot of the Excel calculations:

Five marble run pathways may converge on every Q-BA-MAZE cube in a construction. This formula, however, describes only a single pathway entering any particular cube, or even just a stacked connection. So the results of this formula are far lower than what an eventual computer program will find, but it will at least provide a minimum starting point for understanding how many ways the cubes may be reconfigured.

• N is the total number of cubes in a construction
• Nb = number of blue cubes, Ng = number of green cubes, Nc = number of clear cubes
• N = Nb + Ng + Nc
• C = number of connections
• when C = 4, there is a single continuous pathway with no jumps (the side joint on each cube is always engaged with the cube below)
• when C = 8, the path may be discontinuous and jumps are allowed (either the side joint or the bottom pegs may be used to connect to the cube below)
• the ! symbol means 'factorial' (as an example, 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720)
• the left side of this equation deals with the color combinations
• the right side of this equation deals with the cube configurations

* I include Rubik's Cube here because I found this huge number that describes it and so it makes a good example of how many ways something can be scrambled. Because Rubik's Cube is a puzzle, it is mostly thought of as having only one solution: all sides a single color. But there are interesting checkerboard and other patterns in the design space of 43 quintillion Rubik's combinations, its just that the mechanism of rotating cubes intentionally makes these difficult to find.

If you have an answer to the question "How many ways can the cubes of the 50-pack be reconfigured?" I'd be very interested in hearing from you!

Sources for this post:

The Unofficial LEGO Builder's Guide, Allan Bedford, 2005, No Starch Press Inc., ISBN 1-59372-054-2

The Playmakers: Amazing Origins of Timeless Toys, Tim Walsh, 2004, Keys Publishing, ISBN 0-9646973-4-3

The Origin of Wealth: Evolution, Complexity, and the Radical Remaking of Economics, Eric D. Beinhocker, Harvard Business School Press, ISBN 13-978-1-57851-777-0

Desktop Sculpture in Stone or Polycarbonate

photo: cover of the book Worlds Within Worlds: The Richard Rosenblum Collection of Chinese Scholars' Rocks

DESKTOP SCULPTURE

Q-BA-MAZE is a new form of desktop sculpture, but the idea of desktop sculpture for contemplation goes back more than 1000 years in China -- in the form of 'Scholar's Rocks'. Chinese scholars collected 'fantastic' rocks and displayed them on wooden bases. The rocks might look like clouds, landscapes, people, leaping tigers and so forth, or they could just be abstract forms.

The World of Scholars' Rocks exhibit in 2000 at the Metropolitan Museum of Art (with about 30 stones from the Rosenblum Collection) was my formal introduction to the subject. I say 'formal', because several years earlier, I had unwittingly purchased a scholar's rock of my own while rummaging in an alleyway antique shop in Guangzhou, China (while on a weekend break from my job in the Architecture Department of the Chinese University of Hong Kong). Having collected rocks since childhood, this richly veined stone struck me as the perfect souvenir. Only with the exhibit did I come to realize that my stone from Guangzhou was likely a scholar's stone separated from its base.

In the introductory essay of, Worlds Within Worlds, Robert Mowry writes, "Chinese scholars' rocks might be characterized as favored stones that the Chinese literati and their followers displayed and appreciated indoors, in the rarefied atmosphere of their studios." (p 19). They are displayed "indoors on desk, table, or bookshelf, though an especially large example would be set directly on the floor. Regarded as 'stand-alone' items, scholars' rocks are shown individually and are characteristically presented on carved wooden stands -- like fine bronzes and porcelains -- in order to orient and support the rocks and to distinguish them from the mundane." (p. 20).

The stone on the Worlds Within Worlds cover (top) is a stone from Lake Tai (Taihu), near Suzhou, China. The stones may either be found in nature or carved to appear natural -- Richard Rosenblum considered this particular example a carved stone. Stones with good holes all the way through are particularly prized. Mowry writes, "Although the lake (Taihu) produced limestone rocks with naturally dissolved holes and sand-washed surfaces in early times, the supply of such rocks had been considerably depleted by the late Tang period. By the Northern Song, local families had begun to sculpt rocks from the abundant native limestone, after which they placed their creations in the lake for several decades of natural finishing." (p.27)

THOUGHTS ON COMPOSITION

This Taihu stone is a dynamic form -- it appears to be moving like rising steam, a blowing cloud, or vaguely like a human arm bent at the elbow. It cantilevers boldly to the left and the weight of this gives the sense that it could overturn (counterclockwise). The red line at 'B' highlights the angle at which the wooden base meets the stone. This meeting line is rotated clockwise from horizontal in a way that visually balances the counterclockwise overturning implied by the cantilever. In opposition to these dynamic elements, the underside of the cantilever is horizontal (as demonstrated by line 'A'). If this underside were tipped one way or the other it would imply a movement. But the horizonality suggests stability and rest -- like the horizontal line made by still water in a glass.

Notice what happens if this Taihu stone is slightly adjusted, so that the wooden base meets the stone with a more horizontal connection. This forces the bottom of the cantilever to hang at an angle. It makes it appear as if the entire structure is wilting, drooping under its own weight, as opposed to the original compostition which appears to be floating upward like smoke.

Rosenblum also makes this interesting final point in his essay, "I am most fascinated by the thought that unlike other art, scholars' rocks are not fixed objects. They can be, and often have been remounted. They can be set in different positions, conjuring up new images that change their character altogether -- in effect, remaking them. This was something the Chinese did, and I have begun to do as well. Interacting with the rocks, moving them about, creating mounting environments for them and out of them, has brought me back to sculpture." (p 121)

The bottom image is a Photoshop simulation of the Taihu Scholar Stone remounted and utterly transformed.

And finally, a translation of the original composition into a Q-BA-MAZE marble run sculpture: