Many pizza places offer special "deals" if you order two medium pies. You might think that two mediums deliver more cheesy goodness than one large pie, but usually you'd be mistaken. From Primer, a mathematical comparison of two 12" pizzas and one 18" pizza:

Area of two 12” pizzas:

12/2 = 6 6×6=36 36xπ = 113.1 in² x 2 = 226.2 in²

Area of one 18” pizza:

18/2=9 9×9=81 81xπ = 254.5 in²

And as we know, more pizza is always better.

image: igorovsyannykov (CC0)

(via Cliff Pickover) Read the rest

In Misère Tic-tac-toe, getting three-in-a-row means you lose. I can't wait to unleash this game on my kids at our next dinner out.

Here is a wonderfully-named app version: Notakto

Here is a scientific paper on the mathematics of the game: "The Secrets of Notakto: Winning at X-onlyTic-Tac-Toe" by Thane E. Plambeck, Greg Whitehead

And below is part 2 of the Numberphile video. Note from the video description:

Correction from Thane: The configuration with two X's, one in a corner and one in the middle, is a "b" and not a "b^2". It's at 5:24 in the video.

(via Cliff Pickover)

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Internet Archive founder Brewster Kahle created The Game of Oligarchy, which "shows that the 'free market' leads inexorably to one person getting all the money and everyone else going broke. And fast." Read the rest

A favorite kitchen chemistry (and physics) experiment of kids (and adults), Ooblek is the weird result of mixing cornstarch with water. Now, MIT engineers have developed a mathematical model that can predict and simulate how the non-Newtonian fluid switches between liquid and solid depending on the pressure applied to it. From MIT News:

Aside from predicting what the stuff might do in the hands of toddlers, the new model can be useful in predicting how oobleck and other solutions of ultrafine particles might behave for military and industrial applications. Could an oobleck-like substance fill highway potholes and temporarily harden as a car drives over it? Or perhaps the slurry could pad the lining of bulletproof vests, morphing briefly into an added shield against sudden impacts. With the team’s new oobleck model, designers and engineers can start to explore such possibilities.

“It’s a simple material to make — you go to the grocery store, buy cornstarch, then turn on your faucet,” says Ken Kamrin, associate professor of mechanical engineering at MIT. “But it turns out the rules that govern how this material flows are very nuanced...”

Kamrin’s primary work focuses on characterizing the flow of granular material such as sand. Over the years, he’s developed a mathematical model that accurately predicts the flow of dry grains under a number of different conditions and environments. When (grad student Aaron) Baumgarten joined the group, the researchers started work on a model to describe how saturated wet sand moves. It was around this time that Kamrin and Baumgarten saw a scientific talk on oobleck.

Fashion Institute of Technology photographer Jessica Wynne's "Don Not Erase" project documents the beautiful chalkboards of mathematicians, which will be collected in a book from Princeton University Press in 2020 (Christmas 2020 will be a lot simpler for me as a result). Read the rest

Are you the driver in the lot who parks in the first spot you see? Or do you circle around and around looking for a spot by the door? Physicists Paul Krapivsky of Boston University and Sidney Redner of the Santa Fe Institute explored the mathematics of parking. The research required different equations and simulations to model the benefits of the various parking approaches. From EurkeAlert!:

In their paper, Krapivsky and Redner map three simple parking strategies onto an idealized, single row parking lot. Drivers who grab the first space available follow what the authors call a "meek" strategy. They "waste no time looking for a parking spot," leaving spots near the entrance unfilled. Those who gamble on finding a space right next to the entrance are "optimistic." They drive all the way to the entrance, then backtrack to the closest vacancy. "Prudent" drivers take the middle path. They drive past the first available space, betting on the availability of at least one other space further in. When they find the closest space between cars, they take it. If no spaces exist between the furthest parked car and the entrance, prudent drivers backtrack to the space a meek driver would have claimed straightaway.

So which strategy is best? As the name suggests, the prudent strategy. Overall, it costs drivers the least amount of time, followed closely by the optimistic strategy. The meek strategy was "risibly inefficient," to quote the paper, as the many spaces it left empty created a lengthy walk to the entrance.

Stephen Wolfram's podcast features a 90-minute lecture that he delivered at the 2019 Wolfram Summer School (MP3), recapitulating the history of mathematics from prehistory to the present day. Read the rest

Magic: The Gathering is Turing complete. In a new scientific paper, researchers "present a methodology for embedding an arbitrary Turing machine into a game of Magic such that the first player is guaranteed to win the game if and only if the Turing machine halts." From Ars Technica:

Furthermore, (software engineer Alex Churchill) and his co-authors -- Stella Biderman of the Georgia Institute of Technology and Austin Herrick of the University of Pennsylvania -- have concluded that Magic might be as computationally complex as it's possible for any tabletop game to be. In other words, "This is the first result showing that there exists a real-world game [of Magic] for which determining the winning strategy is non-computable," the authors write...

A universal Turing machine is one capable of running any algorithm, while "Turing completeness" is a term "used to indicate that a system has a particular degree of complexity," said Churchill. "Any Turing-complete system is theoretically able to emulate any other." Being able to determine whether a given problem can be solved in principle is a key task in computer science. If Magic is Turing complete, then there should exist within the game a scenario where it's impossible to determine a winning strategy—equivalent to the famous "halting problem" in computer science.

One way to demonstrate that a system is Turing complete is to create a Turing machine within it, and that's just what Churchill et al. have done with their work

"It’s possible to build a Turing machine within Magic: The Gathering" (Ars Technica)

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A new definition of kilogram went into effect today. No longer is the kilogram defined by Le Grand K, a 140-year-old weight under glass in a secret location near Paris. Now it's determined by the Planck constant, based on physicist Max Planck's theory that "electromagnetic energy at a given frequency could only be emitted in discrete amounts, or quanta, whose energy is proportional to h, now known as the Planck constant." Scientists at the 26th General Conference on Weights and Measures also redefined the kelvin, the ampere, and the mole. UP next, the second! The good news is that the changes are so small that they won't matter to most of us. From Science News:

Currently, the second is defined by atomic clocks made of cesium atoms. Those atoms absorb a certain frequency of light. The wiggling of the light’s electromagnetic waves functions like the pendulum on a grandfather clock, rhythmically keeping time. One second is defined as 9,192,631,770 oscillations of the light.

But a new generation of atomic clocks, known as optical atomic clocks, outdo the cesium clocks (SN: 11/11/17, p. 8). “Their performance is a lot better than what currently defines the second,” says physicist Andrew Ludlow of the National Institute of Standards and Technology in Boulder, Colo. Because those optical atomic clocks operate at a higher frequency, their “ticks” are more closely spaced, making them about 100 times more precise than cesium clocks.

Ideally, the length of a second should be defined using the most precise timepieces available.

Ross Anderson (previously) is one of the world's top cryptographers; the British academic and practitioner was honored by having his classic, Security Engineering, inducted into The Cybersecurity Canon; however, he was not able to attend the awards gala himself because the US government sat on his visa application for months, and ultimately did not grant it in time. Read the rest

When this curious contraption is switched on, an inner circle of white balls appears to be rolling inside the outer circle, but that's actually not the case at all. Below is a video explaining this circular motion illusion. Learn more about the mathematics behind it, specifically Copernicus’ Theorem, and the ingenious hypocycloid mechanical gear design by Girolamo Cardano (1501-1576) over at The Kid Should See This.

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Patrick Ball and the Human Rights Data Analysis Group (HRDAG) (previously) use careful, rigorous statistical models to fill in the large blank spots left behind by acts of genocide, bringing their analysis to war crimes tribunals, truth and reconciliation proceedings, and other reckonings with gross human rights abuses. Read the rest

Retired Cambridge professor Peter Smith has distilled his experience in teaching philosophers and mathematicians about formal logic into a free, frequently updated (last updated: 2017) study guide to logic, constructed to be easily accessible, with quick-start guides for different kinds of learners, written on the assumption of very little education in either maths or philosophy. Read the rest

When you snap dry spaghetti before dropping it into the pot, it sometimes results in an explosion of shards. To understand the physics of the phenomenon, MIT mathematicians used computer simulation and a custom machine to break lots of sticks of spaghetti. It turned out that spaghetti that's twisted first reduces the strength of vibrations that cause more cracks. From Science News:

This strategy may not be much practical help in the kitchen; Patil and colleagues aren’t selling their spaghetti snapper for $19.95 — and even if they were, meticulously twisting and bending pieces of pasta one-by-one is hardly efficient meal prep. Still, the discovery of the bend-and-twist technique may lend new insight into controlling the breakage of all kinds of brittle rods, from pole vault sticks to nanotubes.

And from their scientific paper in PNAS:

Fracture processes are ubiquitous in nature, from earthquakes to broken trees and bones. Understanding and controlling fracture dynamics remain one of the foremost theoretical and practical challenges in material science and physics. A well-known problem with direct implications for the fracture behavior of elongated brittle objects, such as vaulting poles or long fibers, goes back to the famous physicist Richard Feynman who observed that dry spaghetti almost always breaks into three or more pieces when exposed to large bending stresses. While bending-induced fracture is fairly well understood nowadays, much less is known about the effects of twist. Our experimental and theoretical results demonstrate that twisting enables remarkable fracture control by using the different propagation speeds of twist and bending waves.

In 2017, Leila asked Stack Exchange for suggestions for counterintuitive probability riddles for a course on probability; the assembled list is a brain-aching adventure in Monty Hall problems, neighbors' daughters, Sleeping Beauty epistemology, colored lottery balls and birthday paradoxes. Read the rest

Math 4 Love founder Dan Finkel writes:

You’ve been chosen as a champion to represent your wizarding house in a deadly duel against two rival magic schools. Your opponents are a powerful sorcerer who wields a wand that can turn people into fish, and a powerful enchantress who wields a wand that turns people into statues. Can you choose a wand and devise a strategy that ensures you will win the duel?

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Berkeley's "Foundations of Data Science" boasts the fastest-growing enrollment of any course in UC Berkeley history, and now it's free on the university's Edx distance-education platform. Read the rest