Category Archives: Number Theory

The Great pi / e Debate

I had a very busy March, so I wasn’t able to do much for Pi Day this year. But, I am going to get in one more post before the end of Pi Month. That’s right, since this year is 2014, March 2014 (3/14) is Pi Month!

Here is a video from The Mathematical Association of America. It is “The Great pi/e Debate”, presented at Williams College during their First Year Family Weekend, in which two professors debate the relative merits of the numbers pi and e. Which is better? Watch the videos and decide for yourself.  Part 1 is mostly introductions, so if you prefer, just start in with part 2.

Rubik’s Cube Math

480px-Rubik's_cube

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:

 {8! \times 3^7 \times (12!/2) \times 2^{11}} = 43,252,003,274,489,856,000

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.

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.

PascalTriangleAnimated2

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.

Pascal's_Triangle_with_even_numbers_shaded.svg

 

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

Scientist have begun to realize that they can tackle big problems by harnessing the power of the internet to enlist “citizen scientists” to “crowdsource” their projects. The search for large prime numbers is one such project. Citizen mathematicians can become part of the search for prime numbers by joining the Great Internet Mersenne Prime Search (GIMPS). Mersenne primes (prime numbers of the form 2n-1) are extremely rare (only 48 are known).

All you need to participate is a computer, an internet connection, and patience. Simply download the GIMPS software from their website and start it running. The program runs in the background in the lowest priority, so it shouldn’t affect your computer performance. The patience is needed because you may need to run the program for weeks to complete a primality test. I downloaded the program last week and have been running it ever since, and I’m only up to 3%. It helps if you have a computer that is running most of the day. You can learn more about the math involved in the primality testing on the GIMPS math page.

And, if just knowing that you are helping advance the field of mathematics isn’t enough to entice you, there is also a $3,000 cash award to participants finding a prime having fewer than 100,000,000 digits and a $50,000 award to the first participant to find a prime with greater than 100,000,000 digits.

MarinMersenne

Mersenne primes are named for Marin Mersenne (1588– 1648), French theologian, philosopher, mathematician and music theorist.

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.