Category Archives: Number Theory

The Great pi / e Debate



Rubik’s Cube Math


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 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.

Pascal’s Triangle Patterns

Pascal's_Triangle1Pascal’s triangle is a triangular array of the binomial coefficients. (But don’t worry, it isn’t important to understand binomial coefficients to do this activity.) Pascal’s Triangle is named after mathematician Blaise Pascal. Pascal did not invent this array, but he was the first to develop many of its uses and was the first to organize all the information about it together in a treatise.

To create your own Pascal’s Triangle, place a 1 in the first row (which contains only 1 number). In subsequent rows, each number is the sum of the two numbers above it, treating space outside the triangle as zero. Therefore, the next row, contains 0+1 or 1 and 1+0 or 1. The next row is 0+1, 1+1, and 1+0, as shown in the animation below.


The cool thing about Pascal’s Triangle is that there are a ton of patterns you can find. Here are just a few:

  1. The first diagonal is all ones.
  2. The second diagonal is the counting numbers.
  3. The third diagonal is the triangular numbers.
  4. The horizontal sums (adding across a row) are the powers of 2 (20=1, 21=2, 22=4, 23=8, 24=16, etc.).
  5. Each row is a power of 11 (110=1, 111=11, 112=121, 113=1331, etc.).
  6. And my favorite of all: if you color the odd numbers one color and the even numbers another color, you get a pattern that looks like the Sierpinski Triangle fractal.



Can you find any other patterns?

For more patterns, check out this cool page at Math is Fun. If you would like to challenge your kids or students to fill in Pascal’s Triangle and find some of the patterns, I have a worksheet here.

You Can Help Search for Large Prime Numbers






Happy Tau Day

As a math lover, I love that I share my birthday with τ (tau) Day. What is τ you ask? It’s 2π of course (τ = 2 x 3.14 = 6.28). Why give 2π a name and a day? Well, watch these 2 videos, one from Vi Hart and the other from Numberphile, to see what the fuss is all about.