#MathStatMonth Day 23: Minimally Marked Rulers

Last week, Simon Gregg tweeted a couple of images of Golomb rulers. A Golomb ruler is a ruler with marks at integer positions such that distances are measured uniquely. In other words, no two pairs of marks are the same distance apart.

Today, Christopher Danielson tweeted about a different kind of minimally marked ruler. He wanted to use the minimum number of marks possible to measure all the distances from 1 to n on an n-unit ruler. So here, duplication of distances is fine, but all possible distances must be measurable.

For most n, it’s not possible to make a Golomb ruler that also measures all possible integer distances. For example, Danielson was looking at 12-unit rulers. We were able to show that five marks wasn’t enough to measure all distances 1 to 12 because five marks only result in ten pairs of marks, so one could only measure ten distances. That meant he had to use six marks, but six marks create fifteen pairs of marks and thus fifteen distances. A 12-unit ruler with six marks will then necessarily duplicate some distances.

There are some n where a Golomb ruler that measures all the distances could be possible, though. In particular, if n is a triangular number, then the number of pairs of marks would be exactly equal to the number of desired distances. But that doesn’t mean that it’s actually possible choose marks to make such a perfect Golomb ruler, though. I decided to investigate. Continue reading

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#MathStatMonth Day 22: Recommended Reading

I’ve collected some of my favorite math-y or stats-y articles or posts from the past couple of years. Hope you enjoy!

538 wrote a feature on tornadoes and statistics; it’s also about separating real patterns from pareidolia (perceiving patterns when there really aren’t any). It’s also about the place where I grew up, so it’s a question that was relevant to my life for a long time.

This post talks about how some folks tracked down the cause of weird disruptions in Singapore’s railway system. I thought the authors did a good job showing how they worked with the data, what kinds of visualizations they made, and how they confirmed their suspicions.

Christian Lawson-Perfect wrote about investigations into times tables and a way in which 13 is particularly “bad.” It’s a cool problem and really accessible to high school students, especially if they can program a little.

Quanta had an article about the last edition of Proofs from THE BOOK. It has some interesting thoughts on what makes a beautiful proof and what the role of ugly mathematics is. At one point, Michael Pershan wondered on Twitter about the role of surprise in beautiful proofs, and this addresses that a little. It’s something I want to write about at some point.

“Math’s Beautiful Monsters” mashes together real analysis, chaos, stochastics, and fluids, all things that I love. It tells a mathematical story that I had never thought of as a single story. I wouldn’t have thought about it this way, but it’s pretty compelling.

This Stanford news story talks about a PhD student who studied mathematical diagrams in various copies and translations of Euclid’s Elements. As someone who at various points has been really interested in both visual representations in math and the role of the Elements in math history, I found this cool.

This SIAM News Blog post is a good introduction to the shallow water equations (which are a decent first model for a lot of geophysical applications) and the numerical methods you can use in solving them.

“Normal America” is an interesting look (with some well-described stats) at what “normal America” actually is. I think about this post a lot.

#MathStatMonth Day 21: Texas 42

This post is a little different from the others I’ve done this month. For the past couple of years, I’ve been putting together a set of counting/discrete math questions based on the domino game 42 (or Texas 42). I’ve integrated the questions with information about the setup and rules of the game. This is the main set of questions, and I’d love to know if you have thoughts or suggestions. I’ve also been working on a few extensions related to variations on the game (existing ones, like the game 80 or bidding Nell-O, and ones that ask for more creativity, like adjusting the game to play with double-nines instead of double-sixes). Continue reading

#MathStatMonth Day 20: Pascal’s Triangle

Pascal’s triangle is a triangle of integers that looks like this:

1
1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5  10 10  5  1

It has 1s along the left and right edges, and all other entries are equal to the sum of the two entries above it. For example, the last row that I wrote out can be written as 1, 1+4, 4+6, 6+4, 4+1, 1 using the entries in the row above it.

One of the cool things about Pascal’s triangle is that its entries turn out to equal the binomial coefficients. The rth entry from the left in the nth row is equal to C(n,r), the number of ways of choosing r unordered things from a set of n things. (For this to work, the top row must be the 0th row, and the leftmost entry in each row must be the 0th entry.)

On Twitter tonight, someone asked for people’s favorite ways to show that these two methods of constructing the triangle, addition and binomial coefficients, are equivalent. I have a favorite proof of this, and it uses a method called block-walking. Continue reading

#MathStatMonth Day 19: Latin and Magic Squares

The Games for Young Minds newsletter this week was about magic squares, so I wanted to write about how we can build magic squares out of special pairs of Latin squares.

First off, what are these objects? For a number n, we take an n-by-n array, and we want to fill the cells of the array with positive integers according to some rules.

  • For such an array to be a Latin square, each number from 1 to n must appear exactly once in each row and each column.
  • To be a magic square, each number from 1 to n2 must be used exactly once in the grid, and the entries in each row, column, and main diagonal must have the same sum.

At least one Latin square exists for each value of n, and we can come up with a number of recipes that always generate Latin squares. Magic squares are trickier. They don’t exist for every n, and any recipe for building a magic square when one does exist is much more complicated. Continue reading

#MathStatMonth Day 18: 2018 Game

What numbers can you make with the digits of 2018?

That’s the idea behind the year game. Take the number of the year, and use the digits to make as many numbers as you can. Different people allow different operations, and there are ways to make the game more or less challenging. The official Math Forum description of the game, with a link to the rules for the solutions students can post to their website, is here. Continue reading

#MathStatMonth Day 16: Energy Balance Models

The temperature of the Earth is a quantity that we care a lot about in climate science. Paleoclimatologists have studied the temperatures of the Earth in the far past, and Earth’s temperature is central to our understanding of future climate. We know that the global mean temperature has risen and is rising, and a lot of people study how much we expect it to rise under various future scenarios. (This is the basis of the entire body of literature on something called equilibrium climate sensitivity.)
One of the most basic tools for understanding Earth’s temperature is a type of model called an energy balance model. The idea of these models is that, for an Earth system in equilibrium (not changing in time), the radiation emitted by the Earth and its atmosphere should perfectly balance the radiation absorbed from the Sun.

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#MathStatMonth Day 14: Dance Number Talks

I was at a Martha Graham Dance Company performance tonight, and during a couple of the pieces, I found myself paying particular attention to the geometry and grouping of the dancers onstage. (This happens to me a lot.) Not for the first time, I was reminded of the dot images that people sometimes use for number talks.

The piece that most inspired these thoughts tonight was Panorama. It’s a relatively short piece, under ten minutes long, and it relies heavily on geometry and grouping and regrouping people. Here‘s a video. I found the grouping in the first scene particularly surprising. There are three groups, but the dancers aren’t evenly distributed at all! The corresponding dot diagram would look something like this:

dot_diagram1

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