Today we celebrate a Permutation Day! I call days like today permutation days because the digits of the day and the month can be rearranged to form the year.
Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!
My latest column for Quanta Magazine is about the search for sums of cubes. While most integers are neither cubes nor the sum of two cubes, it is conjectured that most numbers can be written as the sum of three cubes. Finding those three cubes, however, can be a challenge.
For example, it was only this year we learned that the number 33 could be written as a sum of three cubes:
33 = 8,866,128,975,287,528³ + (−8,778,405,442,862,239)³ + (−2,736,111,468,807,040)³
What’s so hard about expressing numbers as a sum of three cubes?
It’s not hard to see that it combines the limited choices of the sum-of-squares problem with the infinite search space of the sum-of-integers problem. As with the squares, not every number is a cube. We can use numbers like 1 = 1³, 8 = 2³ and 125 = 5³, but we can never use 2, 3, 4, 10, 108 or most other numbers. But unlike squares, perfect cubes can be negative — for example, (-2)³ = -8, and (-4)³ = -64 — which means we can decrease our sum if we need to. This access to negative numbers gives us unlimited options for our sum, meaning that our search space, as in the sum-of-integers problem, is unbounded.
To learn more, read the full article, which is freely available here.
Tonight I’ll be running a workshop for teachers titled “Building Bridges Through Computational Thinking.”
In the workshop we’ll explore the mathematical and pedagogical benefits in taking a computational approach to mathematics. Through a variety of computational thinking tasks spanning different branches of math, we’ll see how these tasks offer alternate pathways into mathematical ideas, genuine engagement in applied mathematics and mathematical modeling, and opportunities for rich pedagogical variety.
This work is a natural continuation of the work I’ve been doing at the intersection of mathematics and computer science education for the past several years. As always, I’m grateful to be supported by Math for America and MfA’s teacher community in developing and trying out new ideas for students and teachers.
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Today we celebrate a Permutation Day! I call days like today permutation days because the digits of the day and the month can be rearranged to form the year.
Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!
At first it bothered me, but now I appreciate the slow deformation of quadrilaterals from top left to bottom right in this coordinate system, centered at the sun.