John Myles White » Mathematics

Twitter Math Puzzle and Solution

John Myles White — Thu, 07 Jul 2011 13:48:07 +0000

Yesterday I posted a very simple math puzzle to Twitter that I found in Jonathan Baron’s book, Thinking and Deciding. The puzzle is the following:

Show that every number of the form ABC,ABC is divisible by 13.

The puzzle comes up in Baron’s book as an example of an “insight problem” in which one goes from not knowing the answer at all to knowing the complete answering in a sudden moment of insight.

Several people replied to my tweet with solutions: I especially like Will Townes’s solution. In particular, if you’re familiar with modular arithmetic, I like the logic of Will’s answer because it gives a simple generalization. First, represent ABC,ABC as ABC * 1000 + ABC * 1 rather than as ABC * 1001. Then notice that

1 = 1 mod 13
1000 = -1 mod 13

Thus ABC,ABC = ABC * -1 + ABC * 1 = 0 mod 13. This logic can be easily extended to show that (ABC,ABC,)*ABC,ABC = 0 mod 13 no matter how many times you repeat the ABC,ABC pattern.

The Price of Calculation

John Myles White — Mon, 15 Mar 2010 15:13:20 +0000

In a world in which the price of calculation continues to decrease rapidly, but the price of theorem proving continues to hold steady or increase, elementary economics indicates that we ought to spend a larger and larger fraction of our time on calculation.¹

Over the next ten years, I hope that more and more mathematically minded hackers, empowered by open source tools like the R programming language and emboldened by the popularization of statistical analyses by people like Steve Levitt, will follow Tukey’s suggestion.

J. W. Tukey : The American Statistician : Sunset Salvo

Algebra and Education

John Myles White — Sun, 01 Feb 2009 06:00:15 +0000

Recently a friend asked me to help her learn enough math to take the GRE’s. My response was to give her the first problem that I thought she should be able to solve before we discussed anything else. It was a very simple problem from the perspective of a mathematician, but one that is not simple enough to solve only using the approaches to problem-solving that are usually taught in American high schools. Specifically, I asked my friend to find a and b such that,

This equation poses a problem for many students who’ve only taken American high school math classes, because you have to accept that the “solution” to this equation is not a single pair of numbers, but an infinite set of numbers. If you can handle that idea, you should be able to see pretty easily that you can pull the a over to the lefthand side of the equation and then factor it out to get

As long as b is not 1 — which can never occur in an solution to this equation –, you can divide out by (b – 1) to get

which is an explicit formula for all of the possible solutions to this equation. You can pick any b and the formula will give you the corresponding value of a; because this works for all b not equal to 1, you have an infinite set of solutions.

The math itself is bland: what is interesting about this example is the question, “why are we not teaching our children to think mathematically?” We are teaching them to memorize explicit formulas for solving certain algebraic equations, but they are not learning a general approach to solving equations. Why has the substance of mathematics been lost and replaced with the shell of mathematics?