Novel examples, proofs or results in mathematics from arithmetic billiards
The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….Wikipedia has an...
View ArticleLooking for an efficient way of maximising 'pair scores' for subset of 30...
Context: I have a tiling program that uses a directed breadth first search algorithm. It is 'directed' by what I call 'pair scoring'. There are $N$ polyforms (pieces) used in the tiling. I have an...
View ArticleWhat is a way to calculate the maximum number of remaining competitors in...
For those who don't know what musical chairs is, it is a game where players compete to find a seat amongst a slowly dwindling amount of chairs. While not having a chair at the end of a round usually...
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Kakuro puzzles and sheaf cohomology
This is a recreational, summer question and could be more well-suited for mathstackexchange. However, some of you on holiday could appreciate the topic. I recently came across Kakuro Puzzles, similar...
View ArticleHow do you generate math figures for academic papers?
Good day! I am looking for any tool that would allow me to generate a figure similar to the figures embedded in the paper by King et al. (2020) titled "Trigonometry: a brief conversation."King, C.,...
View ArticleMajority voting on $\{0,1\}^\mathbb{Z}$
Motivation. Sometimes in life, people seem to do what the majority of their friends are doing. Do we all become more similar over time? Do we split up into pockets of similarity? This post aims to...
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Truchet tiles with non-periodic tiling from finite group multiplication...
Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups:$$ \pi : G \rightarrow S_n, \quad g \mapsto \pi(g)$$where each...
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Is there mathematical significance to the LaGuardia floor tiles?
I noticed that the new terminal at LaGuardia Airport (in New York) has an intriguing design for the tiles on at least one of their floor areas. It bears a superficial resemblance to aperiodic tilings...
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Number of collinear ways to fill a grid
A way to fill a finite grid (one box after the other) is called collinear if every newly filled box (the first excepted) is vertically or horizontally collinear with a previously filled box. See the...
View ArticleTitles composed entirely of math symbols
I apologize for burdening MO with such a vapid, nonresearch question, butI have been curious ever sinceSuvrit's popular October 2010Most memorable titles MO questionif there were any "$E=mc^2$-titles,"...
View ArticleSeating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course...
View ArticleGeneralization of a mind-boggling box-opening puzzle
Motivation. Suppose we are given $6$ boxes, arranged in the following manner:$$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$Two of these boxes contain a present,...
View ArticleAre there Sudoku variants which are useful or mathematically deep?
I was recently watching a Sudoku Youtube channel which shows a large number of variants on the traditional Sudoku puzzle, some of them non-trivial to solve. I think there was some mention of a Sudoku...
View ArticleA Collatz-like map?
Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows:Set...
View ArticleImage and pre-image integer choice function
Let $\newcommand{\Nplus}{\mathbb{N}^+}\Nplus$ denote the set of positive integers. Is there a function $f:\Nplus\to\Nplus$ with the following property?For all $(a,b)\in \Nplus\times\Nplus$ there is a...
View ArticleMinesweeper constructions in combinatorics
In a related question I asked if constructions based on Sudoku puzzles could be used to obtain any deep results in combinatorics and noted that there were papers of Greenfeld and Tao where Sudoku...
View ArticleCentered 9-gonal numbers vs Concentric 9-gonal numbers.How does their visual...
The Online Encyclopedia of Integer Sequences (OEIS) contains two distinct sequences involving 9-sided polygons:A060544: Centered 9-gonal numbers (also known as nonagonal or enneagonal numbers)These...
View ArticleExamples of interesting false proofs
According to Wikipedia False proofFor example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as...
View ArticleCan every 2-player-coalition avoid losing in 5-player-nim?
I'm interested in $5$-player nim (it will become clear why $5$). Individual players - which I'll identify with the numbers from $1$ to $5$ - make moves as usual, with the move order going $......
View ArticleIs "do-almost-nothing" ever winning on large CHOMP boards?
This is a special case of a question asked but unanswered at MSE:Consider the combinatorial game CHOMP (presented as in the linked notes so that the "poison" square is bottom-left). In any $2$-by-$n$...
View ArticlePesky knots and shoelaces
I often find my shoelaces in complicated, tight knots, even though I don't remember tying them.Consider modeling the formation of these knots as a sequence of physical actions on a string, represented...
View ArticleWinning probabilities in a simple game
This is a piece of recreational (certainly not research) math and as such perhaps not suitable for MO, but I'll give it a try anyway. Alice and Bob start the game with $a\ge 1$ and $b=a+d$, $d\ge 0$,...
View ArticleReorganizational matching
Motivation. My friend works in an organization that is re-organizing itself in the following somewhat laborious way: There are $n$ people currently sitting on $n$ jobs in total (everyone has one job)....
View ArticleSchool-class assignment problem
Motivation. In my town, every student spends the school year with the same set of students; that set is referred to as a "school class". My eldest son is in 6th grade, and that grade consists of $3$...
View ArticleDoes this number exist?
Does there exist $x\in\mathbb{R}$ such that $\lfloor 10^nx\rfloor$ is a prime number for all $n\in\mathbb{N}$?
View ArticleDefining a Pre-Addition Hyperoperation
The hyperoperation sequence (addition, multiplication, exponentiation, etc.) is typically defined such that each level is the iteration of the previous one. For instance:Addition a + a iterated a - 1...
View ArticleExact team splitting
Motivation. This question was inspired by a team management problemthat arose during a school soccer tournament. Therewere $22$ students at the tournament. The goalof the managers was to schedule...
View ArticleWhich popular games have been studied mathematically?
I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with...
View ArticleExamples of math hoaxes/interesting jokes published on April Fool's day?
What are examples of math hoaxes/interesting jokes published on April Fool's day?For a start P=NP.Added 2025-04-01 Anything new in 2025?
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Is this two player tiling game fair?
I am analyzing a 2-player combinatorial game called "Triangle Wins" played on a (potentially infinite) triangular grid. I am interested in determining if the game is fair, i.e., whether the first or...
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