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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 $\...
Dominic van der Zypen'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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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 ...
Bernardo Subercaseaux's user avatar
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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 ...
Pranay Gorantla's user avatar
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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 $\{A\vert B\}\in\mathscr{G}$. ...
Noah Schweber'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 ...
Ahmad Jamil Ahmad Masad'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 ...
paul Monsky'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 ...
en-drix'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 ...
Dominic van der Zypen's user avatar
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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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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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1 vote
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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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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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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 $... \...
Noah Schweber's user avatar
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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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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 ...
Dominic van der Zypen's user avatar
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5 votes
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465 views

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 $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
Roland Bacher's user avatar
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I was recently watching a Sudoku Youtube channel which shows a large number of variants on the traditional Sudoku puzzle, some of them non-trivial to solve. I think there was some mention of a Sudoku ...
Hollis Williams's user avatar
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19 votes
4 answers
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Motivation. Suppose we are given $6$ boxes, arranged in the following manner: $$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$ Two of these boxes contain a ...
Dominic van der Zypen's user avatar
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-1 votes
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Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
Dominic van der Zypen's user avatar
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1 vote
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Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups: $$ \pi : G \rightarrow S_n, \quad g \mapsto \pi(g) $$ where ...
mathoverflowUser's user avatar
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53 votes
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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 ...
Quuxplusone's user avatar
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5 votes
2 answers
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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 ...
Dominic van der Zypen'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
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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 ...
Mattias'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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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 ...
Dominic van der Zypen's user avatar
  • 54k
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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8 votes
2 answers
940 views

Consider the following three-player game: Alice chooses an integer congruent to $0$ mod $3$, Bob chooses an integer congruent to $1$ mod $3$, and Chris chooses an integer congruent to $2$ mod $3$. (...
Gro-Tsen's user avatar
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1 vote
1 answer
229 views

Motivation. I am working with a database software that allows you to sort the fields of any given table in the following peculiar way. Suppose your fields are numbered $1,\ldots, 18$. Next to every ...
Dominic van der Zypen's user avatar
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1 vote
3 answers
237 views

The following question is loosely based on the friendship paradox. Let $G=(V,E)$ be a simple, undirected graph. For $v\in V$, we let the neighborhood of $v$ be $N(v) = \big\{w\in V:\{v,w\}\in E\big\}$ ...
Dominic van der Zypen's user avatar
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20 votes
1 answer
2k views

Does a function from $\mathbb{R}^2$ to $\mathbb{R}$ which sums to 0 on the corners of any unit square have to vanish everywhere? I think the answer is yes but I am not sure how to prove it. If we ...
Ivan Meir's user avatar
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3 votes
1 answer
266 views

Many years ago, I saw the following Fibonacci identity from somewhere online, without proof: Let usual $F(n)$ be Fibonacci numbers with $F(0) = 0, F(1) = 1$, then we have $$F(n) = \left(p ^ {n + 1} \...
Voile's user avatar
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3 votes
0 answers
247 views

A pandiagonal Latin square of order $n$ is an assignment of the numbers $\{0,\ldots,n-1\}$ to the cells of an $n \times n$ grid such that no row, column, or diagonal of any length contains the same ...
Milo B's user avatar
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5 votes
1 answer
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$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by: $a(0) = 0, a(1) = 1$ and $a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
Dominic van der Zypen's user avatar
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1 vote
0 answers
146 views

$\newcommand{\n}{\{1,\ldots,n\}}$ $\newcommand{\FF}{\text{FF}}$ $\newcommand{\lc}{\text{lc}}$ Motivation. A group of my son's peers decided to have a few days of Secret Santa before last year's ...
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). ...
Dominic van der Zypen's user avatar
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2 votes
1 answer
290 views

Motivation. My sons participated in a large football tournament recently; everyone wanted to be in a team with everyone else at least once. Tricky! Formulation of the question. For any positive ...
Dominic van der Zypen's user avatar
  • 54k
5 votes
1 answer
536 views

While studying sequences and series, I came across summations of geometric series. I am able to derive the sum of a geometric progression and that of arithmetico–geometric sequence. But taking a step ...
RajaKrishnappa's user avatar
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595 views

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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-4 votes
1 answer
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tried to ask on the math reddit but got deleted due to my account being new. Is the record for highest multiplicative persistence found still 11? As I may have just found a number with persistence of ...
mwt2212's user avatar
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1 vote
1 answer
140 views

Let $r>0$, $p\neq q$ two primes numbers and $A=\{(m,n)\in\mathbb N^2; |p^m-q^n|\leq r\}$. Is it true that $A$ is a finite set?
Dattier's user avatar
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0 votes
1 answer
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Let $n \in\mathbb N^*$. What are all the surjective functions $f: \mathbb N \rightarrow \{0,...,n-1\}=E $ such that there exist functions $g,h$ from $E^2$ to $E$ with: $\forall (m,k) \in\mathbb N^2,f(...
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12 votes
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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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5 votes
0 answers
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Two players, You (Y) and the Enemy (E), play the following game on a real $n\times n$ matrix. First, E selects one element from the first row of the matrix, two elements from its second row, and so on;...
Seva's user avatar
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0 answers
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Motivation. This weekend I was playing the pair-matching game Memory (also called Concentration in other parts of the world) against my youngest son, and wondered about what constitutes a "good ...
Dominic van der Zypen's user avatar
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21 votes
2 answers
2k views

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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1 vote
0 answers
48 views

Question: What are the maximum and minimum lengths of the sequences of consecutive completed Sudoku which occur when order 9 Latin squares are generated in (standard) lexicographic order? A minimum ...
John Palmer's user avatar
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0 votes
1 answer
250 views

Let $(a_n)_n$ be an increasing real sequence with $a_n=O(\sqrt n)$. Is it true that there exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $$\lim \left|\sum\limits_{k=1}^{\phi(n)}\...
Dattier's user avatar
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