# Questions tagged [recreational-mathematics]

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

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### averages - can you determine min and max [closed]

Morning, this is a question I think I know the answer to already, and that is 'no you can't'.
Say I know the total number of responses to a single question is 10,000 and the average age is said to be ...

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### Order of the "children's card shuffle"

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 ...

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### Proving a (Representing Utility) Function is Continuous [migrated]

I sincerely apologize for posting such a long question. The question involves a complicated proof of a theorem in mathematical economics. I feel it will be better for me to state my question first. I ...

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### $2$-for-$2$ asymmetric Hex

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 ...

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### Does there exists a $\sigma \in S_{2N}$ such that $\sum_{n=1}^{N} \sigma(2n-1) ^ {\sigma(2n)}$ is a perfect square?

Does there exists a $\sigma \in S_{2N}$ such that $\sum_{n=1}^{N} \sigma(2n-1) ^ {\sigma(2n)}$ is a perfect square? ($S_k$ denotes the group of permutations of $\{1, 2, 3, ..., k\}$)
To me, it seems ...

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### What are the Nash equilibria of the “aim for the middle” game?

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 $3$. (...

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### Permutation graph with insert-and-shift

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 ...

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3
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172
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### Graph on $\mathbb{N}$ where almost every vertex is shy

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 $N(v) = \big\{w\in V:\{v,w\}\in E\big\}$ ...

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### 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?

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 ...

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190
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### Proof of an unknown source Fibonacci identity with double modulo

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} \...

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107
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### Proof that a pandiagonal Latin square of order $n$ exists iff $n$ is not a multiple of $2$ or $3$?

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 ...

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### Does every integer appear in the modular sum sequence?

$\newcommand{\N}{\mathbb{N}}$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{ ...

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### Expected value of maximal cycle length in fixed-point free bijections

$\newcommand{\n}{\{1,\ldots,n\}}$
$\newcommand{\FF}{\text{FF}}$
$\newcommand{\lc}{\text{lc}}$
Motivation. A group of my son's peers decided to have a few days of Secret Santa before last year's ...

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### Reorganizational 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). ...

2
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### Finite $k$-set-respecting splitting of $\mathbb{N}$

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 ...

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### How can I evaluate the following sum?

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 ...

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### The $n$ queens problem with no three on a line

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 ...

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### Multiplicative Persistence - Highest persistence found? [closed]

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 ...

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1
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113
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### About the power of numbers primes distribution

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?

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1
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### Another generalisation of euclidean division on integers

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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### Is "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 $...

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### A matrix / zero forcing game

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;...

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### A perfect shuffle on $\mathbb{N}$

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 ...

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### Does this number exist?

Does there exist $x\in\mathbb{R}$ such that $\lfloor 10^nx\rfloor$ is a prime number for all $n\in\mathbb{N}$?

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### What are the limits to the lengths of the sequences of consecutive completed Sudoku when order 9 Latin squares are generated in lexicographic order?

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 ...

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1
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198
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### Series involving sine and cosine

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)}\...

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### Which manhole covers fall through their holes?

Apparently one of the reasons why all manhole covers are shaped like discs is because for any other shape, the manhole cover would fall through its own hole. As stated this is not necessarily a ...

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### Inspired by a card game: finding a path through $[\mathbb{N}]^n$

Motivation. Today my sons played a card game, in which a fixed number $n$ of cards was lying on the table. A move consists of adding an unused card to the cards on the table, and removing a card from ...

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### Optimal schedule for a soccer tournament

Motivation. This weekend, my children took part in a soccer tournament consisting of $n$ teams, each of which playing once against every other team. As there was only one soccer field, the schedule ...

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### Parity of 4×4 normal magic squares

I'm writing a program that given an integer n returns all normal magic squares of size n × n. Fiddling around with it a little I started to notice that if n equals 4, every square I saw followed this ...

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### What does the best die look like?

Intransitive dice have attracted a lot of attention - especially in the context of recreational math - since their introduction by Efron in the 1960s. More recently, there has been work studying ...

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### Has there been any progress on Conway's and Soifer's shortest paper?

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 ...

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### Eventual stabilization for repeatedly adding multiplayer games

This question is an outgrowth of a couple previous questions of mine. In order: 1,2,3. This should be fully self-contained, but those questions may help motivate this one.
To keep things readable, I'...

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### Can we arrange {1,...,9} in 3×3 grid so the set of products of rows equals the set of products of columns? [closed]

I find a interesting question of Prmo mock and Promys 2020
For which $n\in\mathbb{N}$ is it possible to arrange $\{1,…,n^2\}$ in an $n\times n$ grid so that the set of products of columns equals the ...

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### Monoid associated to $>2$-player Hackenbush

There is some literature on multiplayer combinatorial game theory, but as far as I can tell none of it follows the line of attack below. I'd love a pointer to a similar approach taken in the ...

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### For which $n$ does a y-formed $n$-polyomino tile a $n \times n \times n$-cube?

I got from my children as a gift a puzzle consisting of 25 y-shaped 5-polyominoes that form a $5 \times 5 \times 5$-cube (see picture).
I'm wondering for which $n$ does a y-formed $n$-polyomino tile a ...

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### Scheduling "parent talks" at school

Real life motivation. In my younger son's class, there are $18$ students. His teacher provided $18$ time slots for the parents of each child to have a 30-minute conversation of their kid's progress in ...

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### Are there journals for "fun mathematics"?

Are there peer-reviewed journals that focus on "fun mathematics"?
By this I mean fun things that do involve nontrivial mathematics and which I think other mathematicians would enjoy reading ...

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### The $9$th tetration of $-\sqrt2$

Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...

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### Transitive action on domino tilings

Fix a $n \times m$ rectangle and consider the set $S_{n,m}$ of all its dominos tilings.
Here are examples with $n=m=8$.
The set $S_{n,m}$ is empty if and only if $nm$ is odd, and for small $nm$, its ...

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### Another Goldbach variation for odd numbers?

Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime.
Dabbling in the dark art ...

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### On a combinatorial design inspired by a football (soccer) tournament

Real-world inspiration. My younger son was playing a micro football (soccer) tournament this afternoon with $3$ other friends. Let's label the $4$ kids $0,1,2,3$. They played $3$ matches:
$\{0,1\} \...

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2
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### Expected maximum number of "prank cigarettes" in an average pack

"Real-life" motivation. The German satirical magazine Der Postillon suggested a few measures for deterring smokers from their bad habit. I especially liked the idea of inserting one "...

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### Particles sent into the same direction with uniformly distributed speed

Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters ...

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### Lengths of paths through Conway’s Game of Life

This question is inspired by the following challenge from CodeGolf.SE: https://codegolf.stackexchange.com/q/251510/88765.
Given positive integer $N$, we can consider a version of Conway’s game of life ...

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### $3\times 3$ magic squares consisting of entries of a dense set $D\subseteq \mathbb{N}$

Starting point. The struggle for a magic square consisting of distinct square numbers is still ongoing, but it has produced an amusing landmark result called the Parker square. One of the issues is ...

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### The two Collatz-maps associated to characters modulo 8

Given a Dirichlet character $\chi$ modulo $8$ we consider the map $\mu(x)=x/2$ if $x$ is even and $\mu(x)=(3x+\chi(x))/2$ otherwise.
(The corresponding map for $\chi$ the trivial Dirichlet character ...

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### Page-turning number of a graph

Motivation. As I was travelling in the UK, I used a physical copy of the "A-Z Road Atlas BRITAIN" for getting around. I was impressed that whenever I wanted to go from the map segment shown ...

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### Two dice yielding uniform distribution, part 2

Since this question is on the front page again, a generalization.
Let $p$ be prime, and let $a$ and $b$ be positive integers with $a+b=p-1$. Is it possible to have two loaded dice, one with sides ...

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### Tiling a rectangle with squares

Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle:
The task was to determine the area of the rectangle tiled with $8$ squares, of which the ...