Power-law distributions in empirical data
Article by Aaron Clauset and Cosma Rohilla Shalizi and M. E. J. Newman
In collections: Drama!, Probability and statistics
Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated...
URL: arxiv.org/abs/0706.1062v2
PDF: arxiv.org/pdf/0706.1062v2

New entry!
The Strong Law of Small Numbers
Article by Richard K. Guy
In collections: Easily explained, Fun maths facts, The act of doing maths
This article is in two parts, the first of which is a do-it-yourself operation, in which I'll show you 35 examples of patterns that seem to appear when we look at several small values of n, in various problems whose answers depend on n. The question will be, in each case: do you think that...
URL: maa.org/sites/default/files/pd

Packing circles and spheres on surfaces
In collections: Basically computer science, Geometry
Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces’ incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings,...
URL: dl.acm.org/citation.cfm?id=161
PDF: geometrie.tugraz.at/wallner/pa

How do you fix an Oval Track Puzzle?
Article by David A. Nash and Sara Randall
In collections: Easily explained, Puzzles
The oval track group, $$OT_{n,k}$$, is the subgroup of the symmetric group, $$S_n$$, generated by the basic moves available in a generalized oval track puzzle with $$n$$ tiles and a turntable of size $$k$$. In this paper we completely describe the oval track group for all possible...
URL: arxiv.org/abs/1612.04476v3
PDF: arxiv.org/pdf/1612.04476v3

How to eat 4/9 of a pizza
Article by Knauer, Kolja and Micek, Piotr and Ueckerdt, Torsten
In collections: Easily explained, Protocols and strategies, Puzzles
Given two players alternately picking pieces of a pizza sliced by radial cuts, in such a way that after the first piece is taken every subsequent chosen piece is adjacent to some previously taken piece, we provide a strategy for the starting player...
URL: arxiv.org/abs/0812.2870
PDF: arxiv.org/pdf/0812.2870v4

The Fastest and Shortest Algorithm for All Well-Defined Problems
Article by Hutter, Marcus
In collections: Attention-grabbing titles, Basically computer science
An algorithm M is described that solves any well-defined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and low-order additive terms. M optimally distributes resources between the execution of...
URL: hutter1.net/ai/pfastprg.htm
PDF: arxiv.org/pdf/cs/0206022.pdf

How far can Tarzan jump?
Article by Shima, Hiroyuki
In collections: Basically physics, Easily explained
The tree-based rope swing is a popular recreation facility, often installed in outdoor areas, giving pleasure to thrill-seekers. In the setting, one drops down from a high platform, hanging from a rope, then swings at a great speed like "Tarzan", and finally jumps ahead to land on the ground. The...
URL: arxiv.org/abs/1208.4355
PDF: arxiv.org/pdf/1208.4355v1

Fractal Sequences
Web page by Clark Kimberling
In collections: Easily explained, Fun maths facts, Integerology, Puzzles
Fractal sequences have in common with the more familiar geometric fractals the property of self-containment. An example of a fractal sequence is 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, . . . If you delete the first occurrence of each positive integer,...
URL: faculty.evansville.edu/ck6/int

Two notes on notation
Article by Donald E. Knuth
In collection: Notation and conventions
The author advocates two specific mathematical notations from his popular course and joint textbook, "Concrete Mathematics". The first of these, extending an idea of Iverson, is the notation "[P]" for the function which is 1 when the Boolean condition P is true and 0 otherwise. This notation can encourage and clarify...
URL: arxiv.org/abs/math/9205211v1
PDF: arxiv.org/pdf/math/9205211v1

Tropical Arithmetic and Tropical Matrix Algebra
Article by Izhakian, Zur
In collection: Unusual arithmetic
This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for...
URL: arxiv.org/abs/math/0505458
PDF: arxiv.org/pdf/math/0505458v3

Astonishing Numbers
Article by Richard Hoshino
In collections: Attention-grabbing titles, Easily explained, Integerology
We say that an ordered pair of positive integers $$a,b$$ with $$a \lt b$$ is astonishing if the sum of the integers from $$a$$ to $$b$$, inclusive, is equal to the digits of $$a$$ followed by the digits of $$b$$. Determine all astonishing ordered pairs.
URL: cms.math.ca/publications/crux/

Article by Fisher, GL and Mellor, B.
In collections: Easily explained, Things to make and do, The groups group
URL: tandfonline.com/doi/abs/10.108

Fibonacci Jigsaw Puzzle
Article by Akio Hizume
In collections: Puzzles, Easily explained, Fibonaccinalia
URL: starcage.org/fibonacci_puzzle/

Proof of Conway's Lost Cosmological Theorem
Article by Shalosh B. Ekhad and Doron Zeilberger
In collections: Attention-grabbing titles, Basically computer science, Easily explained, The act of doing maths, About proof
John Horton Conway's Cosmological Theorem, about Audioactive sequences, for which no extant proof existed, is given a computer-generated proof, hopefully for good.
URL: arxiv.org/abs/math/9808077v1
PDF: arxiv.org/pdf/math/9808077v1

An unusual cubic representation problem
Article by Andrew Bremner and Allan Macleod
In collection: Puzzles
For a non-zero integer $$N$$, we consider the problem of finding $$3$$ integers $$(a, b, c)$$ such that $N = \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}.$ We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the...

What to do when the trisector comes
Article by Dudley, Underwood
In collections: Attention-grabbing titles, The act of doing maths
URL: web.mst.edu/~lmhall/WhatToDoWh

In retrospect: On the Six-Cornered Snowflake
Article by Ball, Philip
URL: dx.doi.org/10.1038/480455a

Overcurvature describes the buckling and folding of rings from curved origami to foldable tents
Article by Pierre-Olivier Mouthuy and Michael Coulombier and Thomas Pardoen and Jean-Pierre Raskin and Alain M. Jonas
In collections: Art, Basically physics, Easily explained, Things to make and do, Geometry, Modelling
Daily-life foldable items, such as popup tents, the curved origami sculptures exhibited...
URL: nature.com/articles/ncomms2311
PDF: nature.com/articles/ncomms2311