# Questions tagged [recreational-mathematics]

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

269
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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 ...

12
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4
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499
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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 ...

0
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0
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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 ...

3
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0
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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 ...

7
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0
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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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0
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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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1
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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 ...

5
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1
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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 ...

8
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1
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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 ...

8
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1
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458
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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 ...

7
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1
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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 ...

9
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668
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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 ...

3
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0
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153
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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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0
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146
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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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0
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93
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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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0
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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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0
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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 ...

3
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1
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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 ...

5
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0
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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 ...

15
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1
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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 ...

8
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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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0
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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 ...

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1
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### 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 ...

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2
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### Primes and chirality: a definition and question in the context of tessellations for squares

These days I'm trying to research relations between prime numbers and the notion of chirality in the $xy$-plane. Wikipedia has the article Chirality.
I don't know if this relation or the problem for ...

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2
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285
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### Hamiltonian path in bike-lock graph with $1$ known digit

Motivation. My youngest son has a bike lock with dials, and he forgot the unlocking combination completely, except that he remembered that digit $0$ appeared somewhere in the combination. So it was my ...

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1
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### "Lamp-switch set-up number" of $n$ [closed]

Motivation. The following has a real-life (!) inspiration from a discussion about how to connect lamps and switches in an efficient way.
Question. Let $n\in\mathbb{N}$ be a positive integer and let $\{...

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1
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### Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property?

This is a crosspost from MSE.
Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}...

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1
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### Can an odd number of marbles jump to infinity?

Loosely inspired by the game Abalone, I've encountered the following simple problem I cannot solve.
Suppose that we are given a finite set of marbles on an infinite chessboard.
One move consists of ...

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0
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### Can you escape from two lions in a closed arena?

You're at the center of a circular arena. A pair of lions are at the border, planning to catch you. One of them moves as fast as you, but the other moves slower than you. The three of you are confined ...

3
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0
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254
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### Math videos featuring interesting data animations

I am looking for interesting videos featuring pure data animations (not someone talking about math, but a video featuring some math phenomenon). I am interested in videos that tell a story, rather ...

1
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1
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283
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### Runtime for Terrible "Sorting Algorithm"?

Before I begin, I apologize for the bad wording. Consider the following "sorting algorithm":
Suppose there are $n$ books on the bookshelf labeled $1$-$n$, and ordered from left to right in a ...

9
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1
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### How far away can we get by multiple rounding and unit change?

This question is inspired by xkcd #2585 (Rounding):
Let $u_0,\ldots,u_n$ be positive real numbers (we can assume w.l.o.g. that $u_0=1$) or “units”.
Consider the following directed graph: its vertices ...

6
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0
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### Existence of an explosive prime

The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below).
Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...

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1
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### Wrapping Wallpapers around Surfaces

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 ...

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### Game on a square grid (part II)

Related to this question, where there the solution was unexpected for us.
Let $n,m$ be positive integers, $n \le m \le n^2/2$.
The board is $n \times n$ square grid.
Phase 1:
Two players, $A,B$ make $...

2
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0
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### Can Chang and Wang's proof of Thue’s Theorem on circular packing be extended into other dimentions?

The simplicity of Chang and Wang's proof of Thue’s Theorem (link on arxiv) on circular packing took me by surprise. Have similar ideas been found helpful in other dimensions? For example, partition ...

0
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1
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### Pathfinder Olympiad book's question [closed]

Let $$x_{n}=\sqrt[2]{2+\sqrt[3]{3+\sqrt[4]{4+\cdots+\sqrt[n]{n}}}};$$ prove that
$$x_{n+1}-x_{n}<\frac{1}{n !}, \quad n=2,3, \dotsc.$$

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### How 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."
...

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1
answer

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### Generalized random harmonic series

Let $Z_n=\sum_{k=1}^n a_k X_k$ with $(a_k)$ a strictly decreasing sequence of positive real numbers that tend to zero. The random variables $X_k$ are independent and satisfy $P(X_k=1) =p_k, P(X_k=-1)=...

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### Name of a game : Remove two chips from a vertex or one chip from both ends of an edge

Consider a finite graph $\Gamma$ with a positive number $n_v\geq 0$ of chips stacked at each vertex $v$ of $\Gamma$. Two players play in turn with moves consisting either of removing two chips from a ...

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### Can the thief escape (from a smooth, simple closed curve)?

Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite ...

4
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1
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863
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### Who wins this two player game of making squares?

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. ...

2
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1
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### Distribution of stopping time for a 2D random walk

Consider the following process on $\mathbb{C}$:
Start at the point 1.
At each step, move by adding $e^{i\theta}$, where $\theta$ is uniformly drawn from $\mathbb{S}^1$.
Stop at the first positive ...

7
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2
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### Can the theory of elliptic functions developed from purely geometric considerations?

I always had this question, but was unable to get a definitive answer to it.
There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...

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3
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### The devil's playground

On the $\mathbb{R}^2$ plane, the devil has trapped the angel in an equilateral triangle of firewalls.
The devil
starts at the apex of the triangle.
can move at speed $1$ to leave a trajectory of ...

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2
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474
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### Radio-playing sequence

Motivation. (Please skip if you are not in the mood for "chitchat".) Last night I listening to a classical radio station, and for the umpteenth time, they played Mendelssohn's Psalm 42, a ...

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1
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### Putting $\omega$ in two boxes

Motivation. My eldest son starts school tomorrow. His class is split in two groups of $10$ students each. From time to time, the groups are rearranged. I wondered how many rearrangements are needed ...

2
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1
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237
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### Expected value of attempts needed to find a "pair" of cards

We are given an integer $n \geq 1$ and $2n$ cards, labelled $0$ to $2n-1$. We pick a card with uniform probability, put it back, and continue, until for some $k\in \{0,n-1\}$ the cards
$2k$ and $2k+1$ ...

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0
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### Random walk on 2d lattice with obstacles

Consider a random work on $L=\mathbb Z^2$ endowed with obstacles (i.e each cell $(x,y)$ of $L$ may contain a obstacle, i.e the random walk halts whenever it hits such a cell). Let $P(x,y) = 1$ if cell ...