Over the Christmas break, while on vacation with extended family, my son learned how to solve the Rubik’s Cube (a.k.a. magic cube) from his older cousin. He became somewhat obsessed and worked on it until he could solve it himself. One of the things he told me when he got his new cube was, “Mom, the colors can be arranged in over 43 quintillion ways!” This is why it is so hard to solve; there are too many different permutations. You’d never land on the solution by randomly turning the sides. The number of permutations is calculated with the equation:
The first term, 8!, comes from the fact that there are 8! (read 8 factorial) ways to arrange the 8 corner “cubies” (8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1). The next term is 37, because there are 3 colors on a corner cubie and 7 of the corners can be oriented independently. (“Oriented” differs from “arranged” in that arranged refers to moving a cubies to different locations on the cube, while oriented refers to the cubie being at the same corner but rotated to show a different color on top.) The orientation of the eighth cubie depends on the preceding seven. The next term, 12!, is the number of ways to arrange the edges. This term must be divided by 2 because it is an even permutation. Eleven edge cubies (with 2 colors each) can be flipped independently, with 12th depending on the preceding ones, giving 211.
If you are interested in a more in-depth explanation, there is a powerpoint presentation here, and paper here.
Most people solve the Rubik’s cube by memorizing sequences of moves called “algorithms”. In mathematics, the term algorithm means a step-by-step procedure for calculations. Many algorithms, especially those used later in the solution, are designed to move a cubie to a specific spot without moving parts of the cube that have already been moved into place. If you are interested in learning basic algorithms for solving the cube, there is a solving guide available at Rubiks.com. People into “speedcubing” learn algorithms that make it possible to solve the cube in fewer moves, but requires learning more algorithms used with specific patterns on the cube.
Puzzles like the Rubik’s Cube can also be investigated through mathematical group theory. Group theory is the study of the algebraic structures called groups. A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying the group axioms (these are closure, associativity, identity and invertibility). For example, the set of integers together with the addition operation; the addition of any two integers forms another integer. The Rubik’s group is the group corresponding to possible moves.
If you’re looking for a holiday gift for the math geek in you life, I’ve put together a list of some of the fun math-themed products out there.
There is a lot of cool math cooking tools and dining ware available. Here’s a few items:
i 8 Sum Pi Dish, $38 at Uncommon Goods
Math Glasses – set of 4, $38 at Uncommon Goods
Obsessive Chef Cutting Board, $25 at Uncommon Goods
Pizza Pi Cutter, $25 at Amazon
Pi Bottle Opener, $30 at Uncommon Goods
For your geeky guy…
Math Formulas Tie, $45 at Uncommon Goods
If you are looking for a tie, but not sure about this one, there are a bunch of math-themed ties at Zazzle for around $32-$36.
Math Cufflinks, on sale for $14 (regular $40) at Uncommon Goods
Geek Wrist Watch, $65-$78 at Uncommon Goods
There is a ton of cool math jewelry at Etsy…
…like this Laws of Motion Cuff Bracelet, $40,
…or these Math Earrings, $25,
…or this Fibonacci Nautilus Necklace, $30.
Pop Quiz Clock, $27 at ThinkGeek
I came across this Irrational Numbers Wall Clock in the SkyMall catalog while traveling for Thanksgiving, $35.
Of course, it is always possible to find lots of wonderfully geeky math t-shirts. I like the selection at the Neato Shop. For example…
Mathematicians do it to prove themselves, $15,
and, Brainier than your average bear, $15.
Does your Thanksgiving dinner need more math? Of course it does!
Here are 4 videos from Vi Hart to show you how to make green bean matherole, Borromean onion rings, optimal potatoes, and turduckenen-duckenen.
Benoit Mandelbrot and Mandelbrot Set
Today (November 20) is the birthday of Benoit Mandelbrot (1924-2010). Mandelbrot is known for his work in fractal geometry, specifically hIs “theory of roughness” and his discovery of the Mandelbrot set. Mandelbrot also coined the term “fractal”.
Mandelbrot was born in Poland, but his family fled to France to escape Nazi persecution. He studied mathematics at university in Paris, and then came to the U.S. to pursue a master’s degree in aeronatics at CalTech. Mandelbrot returned to Europe for a time, but he then took a job at IBM in New York, where he worked for 35 years.
Other mathematicians had studied the mathematical sets we now know as fractals, but Mandelbrot had the advantage of access to IBM computers. Fractal sets are created by feeding a value into an equation, and then taking that result and feeding it back into the equation. Up until the advent of computers, calculations had to be done by hand, limiting the number of iterations that were possible. Using computers, Mandelbrot was the first to do millions of iterations, creating his now famous Mandelbrot set.
Nova did a great show on fractals and Mandelbrot, which you may have seen in a previous post. But, if you missed it, hear it is again.
November 17 is the birthday of August Ferdinand Möbius a German mathematician and astronomer, who lived from 1790 to 1868. Möbius, of course, lends his name to the Möbius strip (although it was also independently discovered by Johann Benedict Listing around the same time). Other mathematical concepts sporting Möbius’ name include the Möbius configuration, Möbius transformations, the Möbius transform, the Möbius function, and the Möbius inversion formula.
Möbius is most known for his work in mathematics, but he also published work in astronomy concerning the occultation of planets and the motion of celestial objects.
One of Möbius’ teachers was Carl Friedrich Gauss, whom some consider one of the greatest mathematicians of all time.Mobius served as chair of astronomy and higher mechanics at the University of Leipzig.
One of the first posts on the blog was about the Möbius strip. If you missed it check it out here: Möbius Strip: One-Sided Wonder. Also, check out More Möbius.
If you still need to carve your pumpkin, Mathcraft has some great geometric ideas: