Last Saturday, Associate Professor of Mathematics Mihai Stoiciu delivered an entertaining and informative lecture during Family Days entitled “The Rubik’s Cube and its Wonderful Mathematics.”
Stoiciu began the lecture by solving a scrambled Rubik’s Cube, explaining the general strategy for solving it as he finished the Rubik’s Cube. “Most people solve the Rubik’s cube using the layer-by-layer method: first you solve the bottom layer, then the middle one and in the end the top layer, which is the hardest,” Stoiciu said. The last layer is the hardest to solve because it must be finished without changing the solid patterns in either of the previous two layers.
While Stoiciu took only a few minutes to solve the cube, the world record for a single solve is a mere 5.55 seconds. Stoiciu showed this video and others to keep the audience entertained and demonstrate the principles he explained to the audience. In between videos, Stoiciu delved into the history of the Rubik’s Cube.
Invented by Ernö Rubik in 1974 out of an interest in space alteration, the transformation of objects in space and the movement of objects in space and time, the puzzle did not gain popularity until the late ’70s or early ’80s. During this period, the game’s popularity grew exponentially, becoming the world’s top-selling puzzle.
There are two main methods for solving the Rubik’s Cube, the Fridrich method and the Petrus method. Jessica Fridrich, a professor of electrical engineering, discovered the former. The method centers on the principle of solving the cube layer by layer. First, a cross is made on one layer, and then the rest of the first two layers are completed, which involves 41 algorithms. Then the last layer is oriented and permuted, a process that requires knowledge of 57 and 21 algorithms, respectively. Stoiciu focused his lecture on the Fridrich method, though he mentioned the existence of the Petrus method.
The next part of the lecture focused on the different types of cubes. The main difference in the cubes, as Stoiciu explained, is whether the cube is odd or even, like a 3×3 cube or a 4×4 cube. “Even cubes present an additional challenge because there is no center to each side,” Stoiciu said. Odd cubes, however, have a center, which means each side color is in a sense fixed and the cube must be solved so that each side matches its respective center.
In the final part of the lecture, Stoiciu explained the math of the Rubik’s Cube. “There are six basic turns you can do on the cube,” Stoiciu said. Once the cube is fixed in one direction, the moves involve turning one of the six faces clockwise, so for example, turning the front face clockwise. Each turn has order four because turning one face four times will produce no change, which corresponds to an identity element in algebraic group theory. In fact, the set of all cube moves on the Rubik’s Cube is a group. In a 3×3 cube, by considering all possible permutations of the corner cubes and the edge cubes, as well as their orientations, Stoiciu showed that there are 519,024,039,293,878,272,000 possible configurations of the cube. However, not all of these configurations are solvable. It turns out that only one in 12 of these configurations are solvable, making a total number of solvable configurations that is about 99 times larger than the age of the universe in seconds. “To get through all the configurations, making one turn per second, you would need to spend the entire age of the universe 99 times,” Stoiciu said.
The lecture concluded with what is known as God’s Number: Any configuration of the Rubik’s Cube can be solved in 20 or fewer turns. The fascinating algebraic structure of the Rubik’s Cube, along with its entertainment factor, is the reason why the cube has seen not only so much popular success, but also so much academic attention, two things that Stoiciu emphasized throughout the lecture.