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Applications of mathematics for the design and analysis of games and puzzles

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Motivation. In my younger son's class, everyone has to give a (small) Christmas present to one other student. Let $n\in\mathbb{N}$ be the number of students in the class. If you pick a permutation $\...
weissguy's user avatar
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Consider the following situation: There is an infinite set $G$ of giraffes. A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$. The hungry lion tells ...
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I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any "$E=mc^2$...
Timothy Chow's user avatar
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I should begin by disambiguating between the currently existing journal called Recreational Mathematics Magazine and the little-known journal of the same name, published from 1961–1964, that was a ...
Timothy Chow's user avatar
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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 $N\...
Community's user avatar
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15 votes
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In 2005 Conway and Soifer published the famous shortest ever paper, asking whether an equilateral triangle of sidelength $n+\varepsilon$ can be covered by $n^2+1$ unit equilateral triangles and ...
Sam Hopkins's user avatar
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115 votes
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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?
2080's user avatar
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I am intrigued by my honey bottle. Its neck is neatly wrapped by (almost) hexagons. I checked and there are no such things as part-hexagons, quarter-hexagons, half-hexagons, etc. -- if you have seen ...
Salix alba's user avatar
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So here's a problem that has tormented me for years: You have $N$ rocks, and need to distribute them into $K$ piles, potentially some of them empty. After the initial distribution into piles of sizes ...
ConnFus's user avatar
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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 ...
Sam Hopkins's user avatar
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31 votes
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While playing with a LEGO duplo train set, which has the following railroad blocks, I noticed something interesting. One of the railroad blocks is a 3-way switch shown in the image below. It works as ...
Jun Koizumi's user avatar
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Motivation. This question was inspired by a team management problem that arose during a school soccer tournament. There were $22$ students at the tournament. The goal of the managers was to schedule ...
Sean Eberhard's user avatar
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[Edit: I tried to integrate Nate's comments (see below).] In the context of automata over continuous time, consider Boolean-valued functions $f:\mathbb{R}\rightarrow \{0,1\}$. There are uncountably ...
The Amplitwist's user avatar
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The $n$ queens problem asks if we can place $n$ queens on an $n\times n$ chessboard such that no two queens attack one another. For example, when $n=8$, here are two solutions (images taken from ...
ho boon suan's user avatar
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9 votes
1 answer
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The set of finite combinatorial games is the smallest set $\mathscr{G}$ such that $\{\vert\}\in\mathscr{G}$ and, whenever $A,B\subseteq\mathscr{G}$ are finite, we have $\{A\vert B\}\in\mathscr{G}$. ...
paste bee's user avatar
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This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier). Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...
Martin Sleziak's user avatar
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I posed a conjecture which is a generalization of Conjecture on palindromic numbers My question is to find a proof or a disproof of it. I have put it in the comments section of the page of the OEIS ...
Max Alekseyev's user avatar
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Everyone's familiar with the identity $a^3+b^3+c^3=d^3$ where $(a,b,c,d)=(3,4,5,6)$. Pretty as it is it suffers the defect that two of these four integers are divisible by $3$ while two are even. The ...
J. W. Tanner's user avatar
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Two players take turns coloring edges on an $n$-by-$n$ grid. Both players use the same color. Every time a player surrounds a square of the grid, they mark that square with their name and go again. ...
Community's user avatar
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I thought of the following problem. Due to its simple and natural formulation, I guess that it must had been already studied, but I could not find any reference. I'm interested in knowing what is the ...
Jack Edward Tisdell's user avatar
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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 $...
Noah Schweber's user avatar
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The game of roller blocks is played on a rectangular board of size $m\times n$. If $m=5$ and $n=4$ we have $5 \times 4=20$ different gridpoints. Write the lower-left corner as $(1,1)$ and the upper-...
Daniel Asimov's user avatar
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Martin Gardner kept voluminous correspondence with amateur and professional mathematicians worldwide throughout his career. His files are a treasure trove of information about all areas of ...
Timothy Chow's user avatar
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93 votes
28 answers
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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 ...
allthingsgo's user avatar
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0 votes
0 answers
102 views

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 - ...
Jinglestar's user avatar
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21 votes
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Does there exist $x\in\mathbb{R}$ such that $\lfloor 10^nx\rfloor$ is a prime number for all $n\in\mathbb{N}$?
Dattier's user avatar
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6 votes
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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 ...
Dominic van der Zypen's user avatar
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6 votes
1 answer
229 views

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). ...
Oleg Orlov's user avatar
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19 votes
5 answers
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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." ...
Ryan Budney's user avatar
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1 vote
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194 views

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 $b=a+d$, $d\ge 0$, ...
Christian Remling's user avatar
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4 votes
1 answer
782 views

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 ...
Community's user avatar
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12 votes
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557 views

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 $... \...
paste bee's user avatar
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80 votes
50 answers
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According to Wikipedia False proof For 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 ...
Jean Abou Samra's user avatar
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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 ...
Humberto José Bortolossi's user avatar
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0 answers
164 views

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 ...
Hollis Williams's user avatar
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1 vote
1 answer
157 views

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 ...
Fedor Petrov's user avatar
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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 $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
Roland Bacher's user avatar
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6 votes
0 answers
301 views

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 ...
LSpice's user avatar
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19 votes
4 answers
1k views

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 ...
Ilmari Karonen's user avatar
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-1 votes
1 answer
109 views

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 ...
kodlu's user avatar
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53 votes
3 answers
6k views

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 ...
N M's user avatar
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1 vote
0 answers
162 views

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 ...
mathoverflowUser's user avatar
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5 votes
2 answers
388 views

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 ...
Marcus Pivato's user avatar
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13 votes
0 answers
376 views

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 ...
Andrea Marino's user avatar
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1 vote
0 answers
62 views

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 ...
Community's user avatar
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10 votes
0 answers
273 views

This question has gone for a while without an answer on MSE (despite a bounty that came and went) so I am now cross-posting it here, on MO, in the hope that someone may have an idea about how to ...
Benjamin Dickman's user avatar
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32 votes
9 answers
11k views

I am not sure I can strictly define recreational mathematics. But we all feel what it is about: puzzles, problems you can ask your mathematical friends, problems that will bother them for a couple of ...
Community's user avatar
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3 votes
1 answer
642 views

Motivation. The Euro 2024 soccer football championship is in full swing, and the male part of my family are avid watchers. Right now the championship is in the group stage where every group member ...
Jukka Kohonen's user avatar
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11 votes
1 answer
1k views

Motivation. My eldest son thinks the following procedure is a "perfect shuffle" for a deck of cards: Take the first card, put the second on top of it, put the 3rd below cards 2 and 1, put ...
Dominic van der Zypen's user avatar
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8 votes
0 answers
123 views

This is a crosspost from Math stackexchange as I left the question open a while and bountied it but received no answers. If the game of Hex is played on an asymmetric board (where the hexes are ...
volcanrb's user avatar
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