Interesting Esoterica

@christianp's collection of interesting and unusual maths references.
Posts one interesting reference a day.
http://read.somethingorotherwhatever.com

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Last Notes

2026-02-18 07:58:10 GMT

On sphere-filling ropes
Article by Gerlach, Henryk and von der Mosel, Heiko
In collections: Easily explained, Geometry, Fun maths facts
What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are...
URL: http://arxiv.org/abs/1005.4609
PDF: http://arxiv.org/pdf/1005.4609v1
Entry: https://read.somethingorotherwhatever.com/entry/Gerlach2010

2026-02-10 07:58:03 GMT

Music: a Mathematical Offering
Book by Dave Benson
In collection: Music
URL: https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-music.html
Entry: https://read.somethingorotherwhatever.com/entry/item46

2026-02-08 07:58:06 GMT

The top ten prime numbers
Article by Dubner, H
In collections: Easily explained, Lists and catalogues, Integerology
URL: http://primes.utm.edu/lists/top_ten/topten.pdf
Entry: https://read.somethingorotherwhatever.com/entry/Dubner2001

2026-02-06 14:14:02 GMT

New entry!
The Extraordinary Power of Division in Straight Line Programs
Article by Peter Borwein and Joe Hobart
In collections: Fun maths facts, Integerology
A lovely circle of ideas due primarily to Shub, Smale and Shamir says that if it is possible to divide quickly, then it is possible to factor quickly. Here, dividing quickly means modular division over a straight line program. In this context, quickly means actual computations...
URL: https://www.jstor.org/stable/10.4169/amer.math.monthly.119.07.584?seq=3
Entry: https://read.somethingorotherwhatever.com/entry/TheExtraordinaryPowerofDivisioninStraightLinePrograms

2026-02-05 14:30:14 GMT

New entry!
A formula for any real number, maybe
Article by James E. Hanson and Connor Watson
In collections: Animals, Attention-grabbing titles, Fun maths facts
We discuss how to write down three specific natural numbers \(A\), \(B\), \(C\) such that for any real number \(r\) you've probably ever thought of, it is consistent with \(\mathsf{ZFC}\) set theory that...
URL: https://arxiv.org/abs/2602.02384v1
PDF: https://arxiv.org/pdf/2602.02384v1
Entry: https://read.somethingorotherwhatever.com/entry/Aformulaforanyrealnumbermaybe

2026-01-28 14:19:20 GMT

New entry!
Optimal play in 'Guess Who?'
Article by David Cushing and Stuart Gipp and Ezra Levick and Em Rickinson and David I. Stewart
In collections: Games to play with friends, Protocols and strategies
We prove an optimal strategy for the children's game Guess Who? assuming the official rules are in use and that both players ask `classical' questions with a bipartite response. Applying a technique...
URL: https://arxiv.org/abs/2508.00799v3
PDF: https://arxiv.org/pdf/2508.00799v3
Entry: https://read.somethingorotherwhatever.com/entry/OptimalplayinGuessWho

2026-01-12 14:51:05 GMT

New entry!
Classical billiards can compute
Article by Eva Miranda and Isaac Ramos
In collection: Unusual computers
We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theory. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems...
URL: https://arxiv.org/abs/2512.19156v2
PDF: https://arxiv.org/pdf/2512.19156v2
Entry: https://read.somethingorotherwhatever.com/entry/Classicalbilliardscancompute

2026-01-05 07:58:04 GMT

Every natural number is the sum of forty-nine palindromes
Article by William D. Banks
In collections: Fun maths facts, Integerology
It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes.
URL: http://arxiv.org/abs/1508.04721v1
PDF: http://arxiv.org/pdf/1508.04721v1
Entry: https://read.somethingorotherwhatever.com/entry/Everynaturalnumberisthesumoffortyninepalindromes

2025-12-30 07:58:05 GMT

Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers
Article by Martin Erik Horn
In collections: Attention-grabbing titles, Fibonaccinalia
The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli Jacobsthal...
URL: http://arxiv.org/abs/0711.4030v1
PDF: http://arxiv.org/pdf/0711.4030v1
Entry: https://read.somethingorotherwhatever.com/entry/PauliPascalPyramidsPauliFibonacciNumbersandPauliJacobsthalNumbers

2025-12-24 07:58:04 GMT

Pancake Flipping is Hard
Article by Bulteau, Laurent and Fertin, Guillaume and Rusu, Irena
In collections: Attention-grabbing titles, Puzzles, Basically computer science, Easily explained, Food
Pancake Flipping is the problem of sorting a stack of pancakes of different sizes (that is, a permutation), when the only allowed operation is to insert a spatula anywhere in the stack and to flip the pancakes...
URL: http://arxiv.org/abs/1111.0434
PDF: http://arxiv.org/pdf/1111.0434v2
Entry: https://read.somethingorotherwhatever.com/entry/Bulteau2011

2025-12-10 16:01:38 GMT

New entry!
The Census of Mathematical Notations
Article by Abdelshafi Bekhit and Paul Libbrecht
In collections: Lists and catalogues, Notation and conventions
This is the home of the notation census which aims at a list of mathematical notations around the world.
URL: https://notations.hoplahup.net/Census/
Entry: https://read.somethingorotherwhatever.com/entry/TheCensusofMathematicalNotations

2025-12-03 06:25:10 GMT

Any Monotone Boolean Function Can Be Realized by Interlocked Polygons
Article by Erik D. Demaine and Martin L. Demaine and Ryuhei Uehara
In collections: Basically computer science, Easily explained, Geometry, Fun maths facts
We show how to construct interlocked collections of simple polygons in the plane that fall apart upon removing certain combinations of pieces. Precisely, interior-disjoint simple...
URL: http://erikdemaine.org/papers/InterlockedPolygons_CCCG2010/
PDF: http://erikdemaine.org/papers/InterlockedPolygons_CCCG2010/paper.pdf
Entry: https://read.somethingorotherwhatever.com/entry/AnyMonotoneBooleanFunctionCanBeRealizedByInterlockedPolygons

2025-11-26 14:59:30 GMT

New entry!
Notation as a Tool of Thought
Article by Kenneth E. Iverson
In collection: Notation and conventions
The thesis of the present paper is that the advantages of executability and universality found in programming languages can be effectively combined, in a single coherent language, with the advantages offered by mathematical notation.
URL: https://www.jsoftware.com/papers/tot.htm
Entry: https://read.somethingorotherwhatever.com/entry/NotationasaToolofThought

2025-11-22 06:25:09 GMT

On a curious property of 3435
Article by Berkel, Daan Van
In collections: Easily explained, Fun maths facts, Integerology
Folklore tells us that there are no uninteresting natural numbers. But some natural numbers are more interesting then others. In this article we will explain why 3435 is one of the more interesting natural numbers around. We will show that 3435 is a Munchausen number in base 10, and...
URL: http://arxiv.org/abs/0911.3038
PDF: http://arxiv.org/pdf/0911.3038v2
Entry: https://read.somethingorotherwhatever.com/entry/Berkel2009

2025-11-11 06:25:10 GMT

Сетунь ВС (Setun Web Simulator)
Software by Trinarygroup
In collections: Basically computer science, History, Unusual computers
URL: http://trinary.ru/projects/setunws/
Entry: https://read.somethingorotherwhatever.com/entry/SetunSimulator

2025-11-06 06:25:06 GMT

There are not Exactly Five Objects
Article by Andreas Blass
In collection: Attention-grabbing titles
The purpose of this note is to present a solution to a problem posed informally by A. Wilkie at the 1977 ASL meeting in Wrocław: Formulate "the cardinality of the universe is not exactly five" as a Horn sentence. Although this solution is not new -- I found it and M. Morley found another solution during the same meeting -- continuing...
URL: https://www.jstor.org/stable/2274177?seq=1#metadata_info_tab_contents
Entry: https://read.somethingorotherwhatever.com/entry/TherearenotExactlyFiveObjects

2025-11-03 06:25:09 GMT

Quotients Homophones des Groupes Libres Homophonic Quotients of Free Groups
Article by Washington, Lawrence and Zagier, Don
In collections: Easily explained, The groups group
URL: http://people.mpim-bonn.mpg.de/zagier/files/exp-math-2/fulltext.pdf
Entry: https://read.somethingorotherwhatever.com/entry/Washington1986

2025-11-01 06:25:09 GMT

Tiling with Three Polygons is Undecidable
Article by Erik D. Demaine and Stefan Langerman
In collections: Basically computer science, Easily explained, Fun maths facts, Geometry
We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding...
URL: http://arxiv.org/abs/2409.11582v1
PDF: http://arxiv.org/pdf/2409.11582v1
Entry: https://read.somethingorotherwhatever.com/entry/TilingwithThreePolygonsisUndecidable

2025-10-27 06:25:14 GMT

Actegories for the Working Amthematician
Article by Matteo Capucci and Bruno Gavranović
In collection: Attention-grabbing titles
Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories have been increasingly employed in...
URL: http://arxiv.org/abs/2203.16351v3
PDF: http://arxiv.org/pdf/2203.16351v3
Entry: https://read.somethingorotherwhatever.com/entry/ActegoriesfortheWorkingAmthematician

2025-10-26 06:25:09 GMT

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...
URL: http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf
Entry: https://read.somethingorotherwhatever.com/entry/AnUnusualCubicRepresentationProblem

2025-10-15 05:25:09 GMT

2025-10-10 05:25:12 GMT

Look, There's More to Say about Conway's Look and Say Sequence
Article by Greg Dresden and Jacob Siehler
In collections: Easily explained, Fun maths facts
We take Conway's Look and Say Sequence into a base-3 world, and we discover that there are only 24 interesting and irreducible sequences in base 3.
URL: http://arxiv.org/abs/2405.11103v1
PDF: http://arxiv.org/pdf/2405.11103v1
Entry: https://read.somethingorotherwhatever.com/entry/LookTheresMoretoSayaboutConwaysLookandSaySequence

2025-10-04 05:25:37 GMT

Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves
Article by Douglas M. McKenna
In collections: Art, Combinatorics, Fibonaccinalia, Things to make and do
Within the recursive subdivision of the \(n \times n\) square, what characterizes a Hilbert-style space-filling curve motif of length \(n^2\) when—under iterated, self-similar, pure edge-replacement—a sequence of always...
URL: http://ecajournal.haifa.ac.il/Volume2022/ECA2022_S2A13.pdf
Entry: https://read.somethingorotherwhatever.com/entry/FibbinaryZippers

2025-09-30 05:25:06 GMT

The Bulgarian solitaire and the mathematics around it
Article by Vesselin Drensky
In collections: Easily explained, Games to play with friends, History, Puzzles
The Bulgarian solitaire is a mathematical card game played by one person. A pack of \(n\) cards is divided into several decks (or "piles"). Each move consists of the removing of one card from each deck and collecting the removed cards to form a...
URL: http://arxiv.org/abs/1503.00885v1
PDF: http://arxiv.org/pdf/1503.00885v1
Entry: https://read.somethingorotherwhatever.com/entry/TheBulgariansolitaireandthemathematicsaroundit

2025-08-20 07:55:05 GMT

New entry!
The Rubik's Cube Perfect Scramble
Article by Bryan Wolf
In collections: Fun maths facts, The groups group
I was playing with my son’s Rubik’s Cubes and tried to scramble a cube randomly so that no two squares with the same color were side by side. But I wanted a scramble that looked like a random scramble.
URL: https://www.solutionslookingforproblems.com/post/the-rubik-s-cube-perfect-scramble
Entry: https://read.somethingorotherwhatever.com/entry/TheRubiksCubePerfectScramble

2025-08-18 05:25:13 GMT

‘Knowable' As ‘Known After an Announcement'
Article by Balbiani, Philippe and Baltag, Alexandru and Ditmarsch, Hans Van and Herzig, Andreas and Hoshi, Tomohiro and De Lima, Tiago
URL: http://www.journals.cambridge.org/abstract_S1755020308080210
Entry: https://read.somethingorotherwhatever.com/entry/Balbiani2008

2025-08-17 05:25:06 GMT

The Amazing \(3^n\) Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his École Bordelaise gang]
Article by Doron Zeilberger
In collections: Attention-grabbing titles, Combinatorics, Fun maths facts, Things to make and do
The most amazing (at least to me) result in Enumerative Combinatorics is Dominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the...
URL: http://arxiv.org/abs/1208.2258v1
PDF: http://arxiv.org/pdf/1208.2258v1
Entry: https://read.somethingorotherwhatever.com/entry/TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang

2025-08-14 05:25:07 GMT

Neumbering
Article by O.G. Cassani and John H. Conway
In collections: Easily explained, Notation and conventions
The importance of starting at 0 when counting has not often been discussed, nor has the incompatibility between this way of numbering and the usual adjectives first, second, third ... In fact, if the first number is zero, then the fifth is four and the ninth is eight, which is perfectly coherent with the traditional...
URL: https://link.springer.com/article/10.1007/s00283-017-9720-3
Entry: https://read.somethingorotherwhatever.com/entry/Neumbering

2025-08-09 05:25:06 GMT

Markets are efficient if and only if P = NP
Article by Maymin, PZ
In collection: Computational complexity of games
I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse...
URL: https://arxiv.org/abs/1002.2284
PDF: https://arxiv.org/pdf/1002.2284
Entry: https://read.somethingorotherwhatever.com/entry/Maymin2011

2025-08-05 05:25:06 GMT

The paramagnetic and glass transitions in sudoku
Article by Williams, Alex and Ackland, Graeme . J.
In collection: Basically physics
We study the statistical mechanics of a model glassy system based on a familiar and popular mathematical puzzle. Sudoku puzzles provide a very rare example of a class of frustrated systems with a unique groundstate without symmetry. Here, the puzzle is recast as...
URL: http://arxiv.org/abs/1212.1649
PDF: http://arxiv.org/pdf/1212.1649v1
Entry: https://read.somethingorotherwhatever.com/entry/Williams2012

2025-07-26 05:25:08 GMT

Check Digits
Web page by Jonathan Mohr
In collections: Basically computer science, Easily explained, Fun maths facts, Unusual arithmetic
A decimal (or alphanumeric) digit added to a number for the purpose of detecting the sorts of errors humans typically make on data entry.
URL: https://web.archive.org/web/20230204135539/http://www.augustana.ualberta.ca/~mohrj/algorithms/checkdigit.html
Entry: https://read.somethingorotherwhatever.com/entry/CheckDigits

2025-07-13 05:25:07 GMT

How Not to Compute a Fourier Transform
Article by J. A. Grzesik
In collections: Attention-grabbing titles, The act of doing maths
We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate second order differential equation, one of...
URL: http://arxiv.org/abs/2001.11987v3
PDF: http://arxiv.org/pdf/2001.11987v3
Entry: https://read.somethingorotherwhatever.com/entry/HowNottoComputeaFourierTransform

2025-06-25 05:25:11 GMT

London Calling Philosophy and Engineering: WPE 2008
Article by Glen Miller
URL: http://www.ncbi.nlm.nih.gov/pubmed/19543813
Entry: https://read.somethingorotherwhatever.com/entry/item13

2025-06-18 05:25:04 GMT

Solving Differential Equations by Symmetry Groups
Article by John Starret
URL: https://www.jstor.org/stable/27642331
Entry: https://read.somethingorotherwhatever.com/entry/item36

2025-06-12 05:25:17 GMT

Mathematical proofs with cardboard and paper
Article by Arnaldo Gunzi and Ernée Kozyroff Filho
In collections: About proof, Things to make and do
In this tutorial, we'll show visual mathematical proofs using cardboard and paper. This ludic technique is especially good for children. These math proofs are like puzzles, and, best of all, you can do it yourself, at home.
URL: https://github.com/asgunzi/Math_proofs_cardboard_paper
Entry: https://read.somethingorotherwhatever.com/entry/asgunziMathproofscardboardpaperMathematicalartwithpaperandcardboard

2025-06-10 05:25:04 GMT

2025-06-09 05:25:07 GMT

Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
Article by Mestrovic, Romeo
In collections: History, Integerology
In 1862 Wolstenholme proved that for any prime \(p\ge 5\) the numerator of the fraction \(\) 1+\frac 12 +\frac 13+...+\frac{1}{p-1} \(\) written in reduced form is divisible by \(p^2\), \((2)\) and the numerator of the fraction...
URL: http://arxiv.org/abs/1111.3057
PDF: http://arxiv.org/pdf/1111.3057v2
Entry: https://read.somethingorotherwhatever.com/entry/Mestrovic2011

2025-06-01 05:25:07 GMT

Lyons Taming
Article by Wolfram Neutsch
In collections: Animals, The groups group
Based on Kantor's geometry, we give a new Highly symmetric construction of Lyons' sporadic simple group \(Ly\) via its minimal representation over \(\mathbb F_5^{111}\), thus obtaining elementary existence proofs for both the group and the representation at one stroke.
URL: http://arxiv.org/abs/2307.11399v1
PDF: http://arxiv.org/pdf/2307.11399v1
Entry: https://read.somethingorotherwhatever.com/entry/LyonsTaming

2025-05-31 05:25:06 GMT

Best Laid Plans of Lions and Men
Article by Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-Nilsen
In collections: Attention-grabbing titles, Protocols and strategies, Animals, Fun maths facts
We answer the following question dating back to J.E. Littlewood (1885 - 1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to...
URL: http://arxiv.org/abs/1703.03687v1
PDF: http://arxiv.org/pdf/1703.03687v1
Entry: https://read.somethingorotherwhatever.com/entry/BestLaidPlansofLionsandMen

2025-05-19 05:25:09 GMT

World's shortest explanation of Gödel's theorem
Article by Mark Dominus
In collections: About proof, Attention-grabbing titles, Easily explained, Fun maths facts
A while back I started writing up an article titled "World's shortest explanation of Gödel's theorem". But I didn't finish it, and later I encountered Raymond Smullyan's version, which is much shorter anyway. So here, shamelessly stolen from Smullyan, is the World's...
URL: https://blog.plover.com/math/Gdl-Smullyan.html
Entry: https://read.somethingorotherwhatever.com/entry/WorldsshortestexplanationofGdelstheorem

2025-05-10 05:25:08 GMT

On Some two way Classifications of Integers
Article by J. Lambek and L. Moser
In collections: Fun maths facts, Integerology
In this note we use the method of generating functions to show that there is a unique way of splitting the non-negative integers into two classes in such a way that the sums of pairs of distinct integers will be the same (with same multiplicities) for both classes. We prove a...
URL: https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/on-some-two-way-classifications-of-integers/C36087ADE965C59BE843D73B9FA11A87#
PDF: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/C36087ADE965C59BE843D73B9FA11A87/S0008439559051116a.pdf/div-class-title-on-some-two-way-classifications-of-integers-div.pdf
Entry: https://read.somethingorotherwhatever.com/entry/OnSometwowayClassificationsofIntegers

2025-05-08 05:25:07 GMT

Circular orbits on a warped spandex fabric
Article by Middleton, Chad A. and Langston, Michael
In collections: Basically physics, Things to make and do, Geometry
We present a theoretical and experimental analysis of circular-like orbits made by a marble rolling on a warped spandex fabric. We show that the mass of the fabric interior to the orbital path influences the motion of the marble in a nontrivial...
URL: http://arxiv.org/abs/1312.3893
PDF: http://arxiv.org/pdf/1312.3893v1
Entry: https://read.somethingorotherwhatever.com/entry/Middleton2013

2025-04-21 05:25:23 GMT

Mechanical Computing Systems Using Only Links and Rotary Joints
Article by Ralph C. Merkle and Robert A. Freitas Jr. and Tad Hogg and Thomas E. Moore and Matthew S. Moses and James Ryley
In collections: Basically computer science, Things to make and do, Unusual computers
A new paradigm for mechanical computing is demonstrated that requires only two basic parts, links and rotary joints. These basic parts...
URL: http://arxiv.org/abs/1801.03534v1
PDF: http://arxiv.org/pdf/1801.03534v1
Entry: https://read.somethingorotherwhatever.com/entry/MechanicalComputingSystemsUsingOnlyLinksandRotaryJoints

2024-07-18 06:25:44 GMT

Earliest Uses of Symbols of Calculus
Web page by Jeff Miller
In collections: Easily explained, History, Lists and catalogues, Notation and conventions
URL: http://jeff560.tripod.com/calculus.html
Entry: https://read.somethingorotherwhatever.com/entry/item26

2024-03-25 06:25:08 GMT

2023-09-15 06:25:09 GMT

Poe, E.: Near A Raven
Web page by Mike Keith
In collections: Animals, Art
At the time of its writing in 1995, this composition in Standard Pilish, a retelling of Edgar Allan Poe's "The Raven", was one of the longest texts ever written using the π constraint, in which the number of letters in each successive word "spells out" the digits of π (740 digits in this example). For length this poem was subsequently outdone by the...
URL: http://www.cadaeic.net/naraven.htm
Entry: https://read.somethingorotherwhatever.com/entry/item14