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October 2016
- A Farewell to Numberplay, by Claudia Clark
"A Numberplay Farewell," The New York Times, October 31, 2016 - On math and psychiatry, by Annette Emerson
"US mental-health chief: psychiatry must get serious about mathematics," Nature, 26 October 2016 - On math and taffy pulling, by Claudia Clark
"Using Math to Build the Ultimate Taffy Machine," Smithsonian Magazine, 21 October 2016 - On five math problems, by Rachel Crowell
"5 Simple Math Problems No One Can Solve," Popular Mechanics, 14 October 2016 - On envy-free cake cutting, by Rachel Crowell
"How to Cut Cake Fairly and Finally Eat It Too," Quanta Magazine, 6 October 2016 - On mathekniticians, by Claudia Clark
"Meet the mathekniticians - and their amazing woolly maths creations," The Guardian, 3 October 2016 - On math and credit scores, by Claudia Clark
"Weapons of Math Destruction," Discover, October 2016
A Farewell to Numberplay, by Claudia Clark
After seven years, Gary Antonick has published his last Numberplay column in The New York Times. In this final column, Antonick writes about how his thinking about the nature of problem solving has changed as a result of working on this column. In fact, he notes, "we're also in the middle of a revolution in the understanding of insight. The disciplines of decision-making and problem-solving are being completely transformed... Tools for perspective-shifting—for asking new questions as opposed to solving existing problems—are much more clear," he writes. "Among these tools is the ability to shift between problem-solving levels." He illustrates this by showing the ways in which a typical person, who falls into the category of the Casual Solver, the Technician, or the Mapmaker, solves the following problem: A ball and a bat cost $1.10. The bat costs $1.00 more than the ball. How much does the ball cost? At one point, Antonick says, it was believed that only Technicians were worth studying. "The Casual Solver approach was just dismissed as sloppy thinking, and the Mapmaking approach was seen as too individualistic or imprecise," he explains. "We're now in a transformation period in problem-solving, where all three personalities are being recognized as important."See the final challenge at "A Numberplay Farewell," Gary Antonick, The New York Times, October 31, 2016.
--- Claudia Clark
On math and psychiatry, by Annette Emerson
Psychiatrist Joshua Gordon is the new director of the U.S. National Institute of Mental Health (NIMH). He'll oversees the agency's $1.5 billion budget and direction. His priorities include "low-hanging clinical fruit, neural circuits and mathematics--lots of mathematics." In this interview he notes the development of "incredibly powerful tools to observe and alter activity in a subset of neurons... allowing us to get a questions of how neural circuits produce behavior." He explains some short-term applications, "medium-term pay-outs" (safe treatments for psychiatric disorders in humans), and long-term goals (including, ultimately, a change in mental-health care). "Fully characterizing each of these levels and then integrating them across scales requires a level of mathematical rigour that most of us, including myself, have not really brought to bear on the problem.... It's just a question of appropriate training in math for students.... We need to inject more math into every level of the NIMH portfolio. Math can also have a short-term impact in psychiatry for things such as predicting individual responses to drugs and improving precision medicine more generally. "< /p>
See "US mental-health chief: psychiatry must get serious about mathematics," by Alison Abbott, Nature, 26 October 2016.
--- Annette Emerson
On math and taffy pulling, by Claudia Clark
In this article, writer and former Math Digest contributor Evelyn Lamb describes the work of University of Wisconsin-Madison applied mathematician Jean-Luc Thiffeault, who uses topological dynamics and pseudo-Anosov mappings to model taffy pulling. (If such work seems less than critical, consider that "the same mathematics?has more serious applications: many industrial processes, including glassblowing and drug preparation, require viscous fluids to be mixed in ways that are more like pulling taffy than stirring cream into coffee," writes Lamb. In Thiffeault's words, "If you're trying to stir really viscous things, like pharmaceutical industry pastes, you can't just shake them. It's not like mixing paint.") While Thiffeault has long seen the connection between taffy pulling and viscous mixing, his exploration of the history of taffy-pulling machines is recent: his paper "A mathematical history of taffy pullers," which describes the results of his work studying the mathematics behind various taffy-puller patents, was published in July. In addition, "Thiffeault's research?inspired him and his undergraduate student Alex Flanagan to build their own model," notes Lamb. "The 6-rod device, which is still only a prototype, stretches taffy about twice as much as the standard pullers in each cycle."
See "Using Math to Build the Ultimate Taffy Machine," by Evelyn Lamb, Smithsonian Magazine, 21 October 2016.
--- Claudia Clark
On five math problems, by Rachel Crowell
Photo: White House (Pete Souza).
In this article, Avery Thompson discusses five simple, unsolved math problems. They are:
- Collatz Conjecture Let n be any number. If n is even, divide n by 2. If n is odd, multiply n by 3 and add 1. Take the number that results from whichever operation you have just done and run it through the same process. Repeat these steps until you end up at 1. Mathematicians have yet to find a number that, if run through this process enough times, doesn’t end up with a result of 1.
- Moving Sofa Problem This problem asks the question "How large can a sofa of any shape be and still fit around a corner that has a measure of 90 degrees?" This is a two-dimensional problem (the corridor is assumed to have a width of 1).The sofa constant represents the largest two-dimensional area that can negotiate this 90-degree corner. It is currently known that the sofa constant is between 2.2195 and 2.8284, but the exact value is unknown.
- Perfect Cuboid Problem Let A represent the height, B the width, and C the length of a box. G is a diagonal running from one top corner of the box to the opposite bottom corner. There are three other diagonals on this box, which we’ll call D, E and F. A perfect cuboid would be a box such that A2+B2+C2 = G2 and A, B, C, D, E and F are all integers. Mathematicians haven’t found such a cuboid, but they also haven’t disproven its existence.
- Inscribed Square Problem Draw any closed loop such that the beginning and end meet and the loop doesn’t cross itself. It is conjectured that you should be able to draw at least one square such that each corner of the square touches the loop.
- Happy Ending Problem It is known that if you have a piece of paper and draw five dots on it that are not arranged in any particular way, you will be able to draw at least one convex quadrilateral using four of these dots. If you have at least nine randomly arranged dots, you’ll be able to draw a convex pentagon. To ensure that you can draw a convex hexagon, you’ll need 17 dots. The open question is, "If you want to build a convex polygon with more than six sides, how many dots will you need?" If M is the number of dots and N is the number of sides to the shape you want to draw, it is thought that the number of dots you’ll need is governed by M=1+2N-2. However, all that has been proven is that you need at least as many dots as that equation states.
See "5 Simple Math Problems No One Can Solve," by Avery Thompson. Popular Mechanics, 14 October 2016.
--- Rachel Crowell
On envy-free cake cutting, by Rachel Crowell
It is possible to divide a cake among a group of people in an "envy-free" way, according to two young computer scientists who recently published their results. Simon Mackenzie (a 27-year-old postdoctoral researcher at Carnegie Mellon University) and Haris Aziz (a 35-year-old computer scientist at the University of New South Wales and Data 61, an Australian data research group) make up the team. Envy-free cake cutting isn’t just about dividing the cake into slices that are each the same size. Envy-free cake cutting incorporates the notion of dividing a cake that has different features among people who value those features differently. This concept applies to other real-world situations where a continuous object--such as a tract of farm land--needs to be divided among people who assign different values to the varied features. For example, one person might want a tract of land that has a lot of trees but the most important feature to another person could be an abundance of wildflowers.
Since Biblical times, the rule for making such a division fair has been to let one person decide where the division should be and let the other person decide which piece of the divided object they want. In the case of a cake, one person would cut the cake and the other person would choose which piece they wanted. This works for two people but until Aziz and Mackenzie’s algorithm, it was thought that it would not be possible to develop a bounded, envy-free algorithm for dividing a cake among n people. The pair changed that by building on a procedure for envy-free cake cutting among three people that was independently devised by mathematicians John Selfridge and John Conway in 1960.
See "How to Cut Cake Fairly and Finally Eat It Too," by Erica Klarreich, Quanta Magazine, 6 October 2016.
Image: by Helena Jacoba, Creative Commons 2.0, Wikimedia .
--- Rachel Crowell (Item posted 10/19/16)
On mathekniticians, by Claudia Clark
In this article, writer Alex Bellos writes about the work produced by married couple Pat Ashforth and Steve Plummer, recently retired math teachers, who "have been knitting and crocheting mathematical images and objects for more than two decades." The article is filled with pictures of their colorful two- and three-dimensional creations, including afghans, hexaflexagons, polyominoes, and a cube composed of eight smaller cubes that they have named an "octopush." Each piece demonstrates a mathematical concept or proof: one afghan uses color to represent the factors (from 1 to 10) of the numbers 1 to 100; another afghan shows 3 superimposed Sudoku patterns; and a cushion cover illustrates a geometric proof of the Pythagorean theorem. "Originally," writes Bellos, "the afghans were hung in their classrooms. 'They were an invaluable as a vehicle for talking about maths,' says Ashforth." Part of the enjoyment Ashforth and Plummer derive from their work is "working out how [an idea] can be made into an afghan in a way that would be easy enough for anyone else to recreate. It is like trying to solve a puzzle and refining it to give the best possible solution." (Their patterns are available for sale.) Another source of enjoyment is seeing "the effect we have had on children, either directly by them seeing our big colourful blankets and suddenly understanding something they had previously struggled with, or because other teacher have used our ideas…to help teach maths in an unconventional way. And influencing the lives of so many (most often women) math-phobics who would not dream of becoming involved with anything mathematical in other circumstances." Image: "Curve of Pursuit," by Pat Ashforth and Steve Plummer, Woolly Thoughts.
See "Meet the mathekniticians - and their amazing woolly maths creations," by Alex Bellos. The Guardian, 3 October 2016.
--- Claudia Clark
On math and credit scores, by Claudia Clark
In this article, adapted from her recently published book of the same name, mathematician and former Wall Street "quant" Cathy O'Neil takes a sober look at what she calls WMDs, or Weapons of Math Destruction: the models and algorithms that have unintentionally "encoded human prejudice, misunderstanding and bias into the software systems that increasingly manage our lives." Take, for example, the means by which people are determined to be creditworthy. Before Earl Isaac and Bill Fair came up with a model for assessing risk, called Fair, Issac, and Corporation (FICO), a person's eligibility for credit was determined by another human being, with his or her own conscious or unconscious biases. A person's FICO score, on the other hand, "was fed by a formula that looked only at a borrower's finances--mostly his or her debt load or bill-paying record." O'Neil notes that "these scores have lots of commendable, non-WMD attributes," including a clear feedback loop and relative transparency. Now, however, "the use of scoring has proliferated wildly" and many of the models used to predict our creditworthiness are "arbitrary, unaccountable, unregulated and often unfair." O'Neil also argues that using credit scores in making decisions about employment is unfair: "Framing debt as a moral issue is a mistake. Plenty of hardworking and trustworthy people lose jobs every day as companies fail, cut costs, or move jobs offshore." And, even with the Affordable Care Act, she points out, the loss of employment often means the loss of health insurance. "'The more data, the better' is the guiding principle of the Information Age," O'Neil concludes. "Yet in the name of fairness, some of this data should remain uncrunched."
See "Weapons of Math Destruction," by Cathy O'Neil. Discover, October 2016, pages 50-55, and also "'Weapons Of Math Destruction' Outlines Dangers Of Relying On Data Analytics," Interview of Cathy O’Neil by Kelly McEvers. NPR’s All Things Considered, 12 September 2016.
--- Claudia Clark
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