Richard Green

How many cylinders can touch each other? What is the largest number of identical infinite cylinders that can be arranged in three dimensional space so that every two of them touch? It is at least 7, and at most 10. tinyurl.com/3jvmnvjk #math #maths #apieceofthepi #substack

The golden ratio as a number base The golden ratio, φ=(1+√5)/2, can be used as a number base. Integers that have a symmetric representation in base φ turn out to have some interesting properties. tinyurl.com/mzutht67 #apieceofthepi #math #maths

Matroid bingo is a game of chance that can be used to define the mathematical concept of a matroid. The probabilities involved in the game have some surprising properties. tinyurl.com/9xhy58jt #apieceofthepi #math #maths #substack

Spiral Sudoku There is a unique way to fill this 5 by 5 grid with the digits 1 to 5 so that each digit appears once in each row and each column, and so that the digit k appears in exactly k of the circles. tinyurl.com/4muj2pej #math #maths #substack

Games in projective space The design of the game Dobble (or Spot It!) is based on the structure of a finite projective plane. This is a discrete version of two dimensional projective space, in which parallel lines do not exist. tinyurl.com/y5eyaesr #math #maths #substack

Intersections of chords of a circle Given two random chords of a circle, what is the probability that their intersection lies within distance r of the centre? What do we even mean by a “random” chord? tinyurl.com/3w9habhc #apieceofthepi #substack #maths #math #probability

Geodesics and polyhedra A geodesic on a cube is a curve on the cube that becomes a straight line when the cube is folded flat. There are three types of geodesics on a cube, and 15 types of “quasigeodesics”. tinyurl.com/ycx5v84s #apieceofthepi #substack #geometryy #maths

Venn diagrams and Winkler’s conjecture Can a simple Venn diagram on n sets always be extended to a simple Venn diagram on n+1 sets by adding a suitable curve? Apparently not! tinyurl.com/47msm279 #apieceofthepi #math #maths #substack

@mk270 @onehappyfellow That is the best description I’ve seen of mathematicians’ use of “morally”. I don’t know where it came from, and it is very odd usage. What does it say about mathematicians’ morals, I wonder?

@onehappyfellow this could be entirely @edfromo's fault :)

I’ve heard a lot of mathy people say something is “morally correct/true” for a statement which is technically false but the gist of it can be formalised (frequently with a ton of preconditions). Where does this phrase come from? Any math-adjacent etymology nerds around here? :)

Which way is the bike going? Can you deduce the direction of travel of a bike by the shape of the tracks it leaves? Sherlock Holmes thought so, but the answer is somewhat subtle. tinyurl.com/yetz489e #apieceofhtepi #substack #math #maths

Random coprime numbers The probability that two random large integers have no factors in common is 6 divided by π squared, which is about 60.8%. tinyurl.com/55ed8wky #apieceofthepi #piday #math #maths #substack

Rubik’s abstract polytopes If we make a 3×3×3 face-turning Rubik’s cube in the shape of a different Platonic solid, how many symmetries does it have? How about if we do it in higher dimensions? tinyurl.com/4x6js6ah #apieceofthepi #maths #math #substack

Counting dimer tilings How many ways can you cover a square grid with non-overlapping 2 by 1 dominoes, leaving at most one square empty? tinyurl.com/mrxtbp68 #apieceofthepi #math #maths #substack

Since the 1960s, mathematicians have been trying to figure out the biggest shape that can fit through an L-shaped hallway. A recent proof uncovered the answer without any computer assistance. It instead took a fresh approach to solving optimization problems. quantamagazine.org/the-largest-so…

The next time you move, consider mathematician Joseph Gerver’s 18-piece sofa. It’s the biggest shape that can slide down an L-shaped hallway… though maybe not the easiest to assemble. quantamagazine.org/the-largest-so…

If you’ve ever moved into a new home, then you know how difficult it can be to steer bulky furniture through narrow hallways or around awkward corners. A new proof reveals the biggest shape that can slide down an L-shaped hallway. Richard Green reports: quantamagazine.org/the-largest-so…

The moving sofa problem Jineon Baek recently announced a proof that Gerver’s sofa is the largest shape that can be slid around a right-angled hallway of unit width. @QuantaMagazine has just published an article by me about this.

Identifying bottlenecks in networks A complicated network often consists of highly connected regions that are linked to each other by bottlenecks. How can we identify bottlenecks computationally? tinyurl.com/bdhxwpr7 #substack #math #maths

The mathematics of the game Waffle Waffle is a word puzzle game along the lines of Wordle. In order to solve a game efficiently, it helps to know something about abstract algebra and the symmetric group. tinyurl.com/25va86hx #substack #maths #math tinyurl.com/25va86hx

On @substack Turning a triangle into a square There is a well-known way to cut an equilateral triangle into four polygonal pieces and slide the pieces back together to form a square. It is impossible to do this with three pieces. tinyurl.com/ye2995dy #math #substack