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Interesting Esoterica Summation, volume 8

Mathematicians! Stop what you’re doing! I’ve almost certainly got something more interesting for you here. It’s been a good while since I last updated you on the contents of my Interesting Esoterica collection, and I have a proportionate number of mathematical curiosities to entertain and bewilder you with.

In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Some things might not be freely available, or even available for a reasonable price. Sorry.

A Line of Sages

A new variation of an old hat puzzle, where sages are standing in line one behind the other.

Tanya Khovanova’s favourite hats puzzle.

Pascal’s Pyramid Or Pascal’s Tetrahedron?

A lattice of octahedra and tetrahedra (oct-tet lattice) is a useful paradigm for understanding the structure of Pascal’s pyramid, the 3-D analog of Pascal’s triangle. Notation for levels and coordinates of elements, a standard algorithm for generating the values of various elements, and a ratio method that is not dependent on the calculation of previous levels are discussed. Figures show a bell curve in 3 dimensions, the association of elements to primes and twin primes, and the values of elements mod(x) through patterns arranged in triangular plots. It is conjectured that the largest factor of any element is less than the level index.

How do you generalise Pascal’s triangle to three dimensions? Let this page tell you, with pictures!

Missing Data: Instrument-Level Heffalumps and Item-Level Woozles

The purpose of this paper is to provide a brief overview of each of two missing data situations, and try to show the importance of considering which elusive creature a researcher might be hunting.  We find that much of the previous literature does not consider the distinction between missing data at the item level or instrument level.  Failure to make this distinction can partially muddle one’s treatment of missing data in important situations.

Smooth neighbours

We give a new algorithm that quickly finds smooth neighbours.

As well as being an excellent title, “smooth neighbours” are pairs of adjacent numbers, neither of which has a prime factor larger than a predetermined amount. The problem of finding smooth neighbours was posed by Størmer, as you will see…

On a problem of Störmer

The person who originally edited this paper misspelt Størmer’s name as ‘Störmer’. Boo! Anyway, the paper’s about the smooth neighbours problem, and uses a lovely typeface.

Half of a Coin: Negative Probabilities

Half-coins are strange objects with infinitely many sides. They are numbered with 0,1,2,… and the positive even numbers are taken with negative probabilities. Two half coins make a complete coin in the sense that if we flip two half coins then the sum of the outcomes is 0 or 1 with 1/2 probability as if we simply flipped a fair coin. In this paper we clarify the meaning and interpretation of negative probabilities and illustrate their importance in finance.

From Unicode to Typography, a Case Study: the Greek Script

The Greek script has a long history and, although the Greek language is spoken only by about 20 million people, the Greek script is one of the most widely known in the world.In this paper we will try to synthesize, from a typographical point of view, the various instances of Greek script. Globally, one can say that the Greek script is used for the following cases:

1. Greek language;
2. Mathematics;
3. Universally used symbols(such as μ for “micro,” Ω for “Ohm,” etc.) ;
4. International Phonetic Alphabet;
5. African languages.

I decided to find out where the weird round variant of $\pi$, $\varpi$, came from. It turns out it’s been around for ages! Section 1.2.3, point 5, has the whole story. The author also basically throws his hands in the air for maths, where anything and everything is used for alternately very specific and completely arbitrary reasons.

The copy I’ve linked to is on the Internet Archive. It’s the last URL for the paper that CiteSeerX knows about.

Only Problems, Not Solutions!

The development of mathematics continues in a rapid rhythm, some unsolved problems are elucidated and simultaneously new open problems to be solved appear.

Florentin Smarandache is an odd personage. His output abounds in multicoloured insanity on viXra, the backwards arXiv for backwards people, but he has several non-mental objects named after him. I can’t work it out. Anyway, this is a scan of a typewritered book about open problems. I showed it to my friend David when we recorded our first podcast together.

On $n$-dimensional Polytope Schemes

Pyramid schemes are a well-known way of taking bundles of money from suckers. This paper is not about them. Although on first inspection, this paper sounds like it is about pyramid schemes, we promise that it is not. In this work, we defi ne and analyze n-Dimensional Polytope schemes, which generalize pyramid schemes, but are not pyramid schemes. We derive several theoretical and empirical results demonstrating the great opportunities off ered by our n-Dimensional Polytope Schemes. In particular, we demonstrate substantially superior growth potential in contrast to all previously published work. In addition to being of theoretical interest, these results mean that you can stay at home and make money in your spare time!

I believe the abstract justifies itself. The site it’s hosted on,, is also worthy of your attention.

On the Cookie Monster Problem

The Cookie Monster Problem supposes that the Cookie Monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The Cookie Monster number of a set is the minimum number of moves the Cookie Monster must use to empty all of the jars. This number depends on the initial distribution of cookies in the jars. We discuss bounds of the Cookie Monster number and explicitly find the Cookie Monster number for jars containing cookies in the Fibonacci, Tribonacci, n-nacci, and Super-n-nacci sequences. We also construct sequences of k jars such that their Cookie Monster numbers are asymptotically rk, where r is any real number between 0 and 1 inclusive.

I’m always happy to have a food-based paper in my collection.

Cookie Monster Devours Naccis

In 2002, Cookie Monster appeared in The Inquisitive Problem Solver. The hungry monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars.   The Cookie Monster number is the minimum number of moves Cookie Monster must use to empty all of the jars. This number depends on the initial distribution of cookies in the jars.   We discuss bounds of the Cookie Monster number and explicitly find the Cookie Monster number for Fibonacci, Tribonacci and other nacci sequences.

… which is why I’ve also included this one.

Perfect Matchings and the Octahedron Recurrence

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.


Mathematical Entertainments

Hey, it’s like this site, but in paper and twenty years ago and inexplicably still thirty quid for a digital copy! Anyway, if you have institutional access or don’t understand the value of money, there’s a good bit on Somos sequences. Closed access, Springer, £29.95.

An Infinite Set of Heron Triangles with Two Rational Medians

It would be good if I wrote down why I found papers when I added them to the collection. Oh well, a bit of geometry for you. Maybe the word ‘heron’ caught my eye? Ah! It involves Somos sequences, and cites David Gale’s column just above. It’s cited in the OEIS entry for the Somos 4-sequence. Closed access, JSTOR, $12.

The Laurent phenomenon

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes—quite unexpectedly—it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D.Gale and R.Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J.Propp, N.Elkies, and M.Kleber.

More Somos fun!

Sloane’s Gap. Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS

The Online Encyclopedia of Integer Sequences (OEIS) is made up of thousands of numerical sequences considered particularly interesting by some mathematicians. The graphic representation of the frequency with which a number n as a function of n appears in that database shows that the underlying function decreases fast, and that the points are distributed in a cloud, seemingly split into two by a clear zone that will be referred to here as “Sloane’s Gap”. The decrease and general form are explained by mathematics, but an explanation of the gap requires further considerations.

The Ubiquitous Thue-Morse Sequence

Slides about the Thue-Morse sequence, which is everywhere. Apparently.

How to Differentiate a Number

We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibnitz rule. The aim of the article is to consider the basic properties of this map and to show how to generalize the notion to the case of rational and arbitrary real numbers. We make some conjectures and find some connections with Goldbach’s Conjecture and the Twin Prime Conjecture. Finally, we solve the easiest associated differential equations and calculate the generating function.

The arithmetic derivative is pretty rad.

Practical numbers

A little note by Srinivasan, introducing the sequence that would later be known as A005153.

Proofs by Descent

The method of descent is a technique developed by Fermat for proving certain equations have no (or few) integral solutions. The idea is to show that if there is an integral solution to an equation then there is another integral solution which is smaller in some way. Repeating this process and comparing the sizes of the successive solutions leads to an in finitely decreasing sequence
\[ a_1 \gt a_2 \gt a_3 \gt \dots \]
of positive integers, and that is impossible.

So it has a name!

Is POPL Mathematics or Science?

  • When writing a joint paper, mathematicians order the names of the authors alphabetically. 
  • Scientists (experimental physicists, biologists, etc.) use a different criterion to order the names of the authors: The one with the grant goes first, or last (depending on the field); or the one that did the work goes first; or the student goes first, or last; but in any case alphabetical order is not used.

Starting from those two axioms, the authors look at the proceedings of POPL to decide once and for all if it’s about maths or science.

Random Structures from Lego Bricks and Analog Monte Carlo Procedures

Recently we discovered a phenomenon: When filled with many single Lego bricks, a washing machine generates random complexes. This generation process can be viewed as a parallel analog Monte Carlo procedure. It may be used for discovering new Lego structures and for interactive “generative design”. This report is preliminary and tentative.


Playing pool with π (the number π from a billiard point of view)

Counting collisions in a simple dynamical system with two billiard balls can be used to estimate π to any accuracy.

That’s a pretty compelling claim!

Swiss cheeses, rational approximation and universal plane curves

In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We develop a theory of allocation maps connected to such sets and we use this theory to modify examples previously constructed in the literature to obtain examples homeomorphic to the Sierpiński carpet. Our techniques also allow us to avoid certain technical difficulties in the literature.

More food maths. I am unapolagetic.

Pancake Flipping is Hard

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 above it (that is, to perform a prefix reversal). In the burnt variant, one side of each pancake is marked as burnt, and it is required to finish with all pancakes having the burnt side down. Computing the optimal scenario for any stack of pancakes and determining the worst-case stack for any stack size have been challenges over more than three decades. Beyond being an intriguing combinatorial problem in itself, it also yields applications, e.g. in parallel computing and computational biology. In this paper, we show that the Pancake Flipping problem, in its original (unburnt) variant, is NP-hard, thus answering the long-standing question of its computational complexity.

Does what it says on the tin pan.

Giuga Numbers and the arithmetic derivative

We characterize Giuga Numbers as solutions to the equation $n’=an+1$, with $a \in \mathbb{N}$ and $n’$ being the arithmetic derivative. Although this fact does not refute Lava’s conjecture, it brings doubts about its veracity.

That crazy Lava, conjecturing like he doesn’t care! They cite 103 Curiosità matematiche for the conjecture, but it costs money :(

Fibonacci numbers and Leonardo numbers

Just some handwritten notes by Dijkstra, NBD. Cited in in A001595.

2178 And All That

For integers $g \gt 3$, $k \gt 2$, call a number $N$ a $(g,k)$-reverse multiple if the reversal of $N$ in base $g$ is equal to $k$ times $N$. The numbers $1089$ and $2178$ are the two smallest $(10,k)$-reverse multiples, their reversals being $9801 = 9 \times 1089$ and $8712 = 4 \times 2178$. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all $(g,k)$-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of $k$ for bases $g = 2$ through $100$, and then show how to apply the transfer-matrix method to enumerate the $(g,k)$-reverse multiples with a given number of base-$g$ digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood.

A new Neil Sloane paper! He’s also given a talk about this stuff.

Wolstenholme’s theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862–2012)

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 \[ 1+\frac{1}{2^2} +\frac{1}{3^2}+…+\frac{1}{(p-1)^2} \] written in reduced form is divisible by $p$. The first of the above congruences, the so called Wolstenholme’s theorem, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme’s type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme’s theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks.
The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane’s On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme’s type congruences, but without the expositions of related congruences. The total number of citations given here is 189.

Just another subsequence of the primes, but not any survey paper! This is enormous!

The Math Encyclopedia of Smarandache Type Notions

About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. We structured this book using numbered Definitions, Theorems, Conjectures, Notes and Comments, in order to facilitate an easier reading but also to facilitate references to a specific paragraph. We divided the Bibliography in two parts, Writings by Florentin Smarandache (indexed by the name of books and articles) and Writings on Smarandache notions (indexed by the name of authors). We treated, in this book, about 130 Smarandache type sequences, about 50 Smarandache type functions and many solved or open problems of number theory. We also have, at the end of this book, a proposal for a new Smarandache type notion, id est the concept of “a set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors”, notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers. This encyclopedia is both for researchers that will have on hand a tool that will help them “navigate” in the universe of Smarandache type notions and for young math enthusiasts: many of them will be attached by this wonderful branch of mathematics, number theory, reading the works of Florentin Smarandache.

You know how I said Smarandache is a bit weird? Here’s my evidence. I can’t work out if it’s him under a pseudonym, or a genuine crazy other person. And check out that cover page! It makes a good contrast with the nicely-presented survey paper above.

The Maximum Throughput Rate for Each Hole on a Golf Course

We develop stochastic models to describe the times required for successive groups of golfers to play each hole on a conventional 18-hole golf course as well as the entire course. We develop models for par-$k$ holes for $k = 3,4,5$. These models include the realistic feature that $k – 2$ groups can be playing at the same time on a par-k hole, but with precedence constraints. We also consider par-3 holes with a “wave-up” rule, which allows two groups to be playing simultaneously. We mathematically determine the capacity of each hole, i.e., the maximum possible throughput rate, under natural independence and identical-distribution assumptions. To do so, we carefully analyze the associated fully loaded holes, in which new groups are always available to start when the opportunity arises. We characterize the stationary interval between the times successive groups clear the green on a fully loaded hole, showing how it depends on the stage playing times. The structure of that stationary interval evidently can be exploited to help manage the pace of play. The mean of that stationary interval is the reciprocal of the capacity. The bottleneck holes are identified as the ones with the least capacity. The bottleneck capacity is then the capacity of the golf course as a whole.

Let’s end with a round of golf. Fully analysed, of course.

2 Responses to “Interesting Esoterica Summation, volume 8”

    • Christian Perfect

      Well, yes! I saved that paper because Project Euclid and the original disagreed on the spelling, and I forgot which one was right when I wrote this up. Thanks for correcting me.


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