Mathematics

The "Circles Stuck in a Triangle" problem I posted last week had a rather surprising solution to it. So much so that I had to rewrite the question so that text and diagrams made sense! So here is a complete solution to what ended up being two problems; similar but different.

The solution has three steps: firstly, we need to find the relationship between the three radii; then there is a bit of number theory to find explicit numbers for a and b; and finally, some trigonometry to calculate the area of the large triangle.

The c=4 case

If you are in the northern parts of the US (and lots of Canada) you will soon have sap running - and that means maple syrup. You probably don't think about the physiological aspects of syrup production - nor should you, that is why you have Science 2.0.

But now you can learn about the mechanisms of sap exudation—processes that trigger pressure differences causing sap to flow— while you eat your pancakes. In a paper published in the SIAM Journal on Applied Mathematics, authors Maurizio Ceseri and John Stockie propose a mathematical model for the essential physiological processes that drive sap flow. 


The timber industry, including pulp and paper producers, are among Canada's most important industries  - but they are also one of the largest producers of wastewater and greenhouse gas emissions in wastewater is a concern. 

Until now, greenhouse gas emission estimates have been limited by the mathematical models used to predict them. Researchers have recently developed a new dynamic method to better predict the emission content of these gases.   


Richard Mankiewicz, our man in Bangkok, also known as Red Man (see his profile – no no, not because of Bangkok’s red light district - that would be Stickman, not Red Man!) has started a Math Puzzle Column on Science2.0, first entry: Circles Stuck in a Triangle.

I would like to start a new column of mathematical puzzles. Scientific American has one; New Scientist has one; so I hope Science 2.0 will be happy to host one!

Preamble over, here's your "started for 10".



The diagram shows five circles, each with integer radius, all touching the base of the large triangle. The four smaller circles all touch their two neighbouring circles, with the large circle touching all four. The two sides of the triangle each touch two of the circles.

Let the radii of the circles be a, b and c, such that a > b > c.

Thanks to No Child Left Behind, the gender gap in math skills tests disappeared for the first time in history.  But a new paper says the issue might never have been there if the format for math competitions was different - rather than one-shot events, switch to rounds.

Twenty-four local elementary schools in a 
Journal of Economic Behavior&Organization article changed the math format to go across five different rounds. Once the first round was over, girls performed as well or better than boys for the rest of the contest.


It is somewhat surreal to see the discovery of the largest prime number paraded on prime time broadcast media. Mathematicians around the world are asked to explain the significance of this discovery in layman’s terms, which is up there with physicists trying to explain what the Large Hadron Collider actually does.

Below you’ll see a pretty sparky Sky News interview by Eugenia Cheng, a senior lecturer at the University of Sheffield. Just notice at the very end how jolly pleased with themselves the newscasters seem to be.

How, when and where a pathogen is transmitted between two individuals in a population is crucial in understanding and predicting how a disease will spread and a new model seeks to lay the foundation for new zoonotic disease spread thinking

By outlining a predictive model of a spatial epidemic spread in a population of territorial animals and quantifying the instances of transmission events, the research team determined the propagation speed of a pathogen using parameters based on the knowledge of the demography of a species, the way animals wander and the degree of contagiousness of the disease.


There has recently been some discussion among mathematicians about whether or not a proof of a conjectured theorem put forward by a well-known mathematician, Shinichi Mochizuki, is complete and correct. Issues like this could be resolved immediately if it was practical to present, alongside research articles, proofs of the mathematical results written in a language for which the reasoning could be checked by computer.

A new paper outlines the many pitfalls associated with simulation methods such as Monte Carlo algorithms and other commonly used molecular dynamics approaches.