# Questions tagged [recreational-mathematics]

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

244
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### "Lamp-switch set-up number" of $n$

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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### 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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### 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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### 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 ...

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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 ...

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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 ...

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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 ...

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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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### 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 $...

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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 ...

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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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### 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 ...

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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. ...

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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 ...

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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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### 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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### 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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### 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 ...

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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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### 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 ...

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### Is there an equilibrium for this non-zero-sum game?

The game $G(N,M)$ is played:
$N$ ($N\geq 2$) is the number of players, labeled $1$~$N$. In the beginning they have a pot with some chips in it. Players move alternatively in the order from $1$ to $N$....

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### Split $\{1,...,mn\}$ into $m$-tuples $x$ with $\sum_{i\gt 1} x_i=kx_1$

This question arose in Math.StackExchange with $k=3,m=3$ https://math.stackexchange.com/questions/4179825/for-which-n-in-bbb-n-can-we-divide-1-2-3-3n-into-n-subsets-each-wi
For which $m,k$ are there ...

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### Finite subsets of $S\subseteq \mathbb{N}$ such that $S\setminus\{s\}$ can be partitioned with equal sum

For which integers $n>1$ is there a set of positive integers $S\subseteq \mathbb{N}$ with $n$ elements, and for every $s\in S$ the set $S\setminus\{s\}$ can be partitioned into two subsets with ...

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### Magic circle (instead of magic square)

Motivation. I stumbled over this riddle (unfortunately in German): the goal is to fill the numbers $1\ldots7$ (or, equivalently, $0\ldots6$) into the $7$ little circles so that the sums of all numbers ...

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### Escaping from infinitely many pursuers

The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards ...

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### Prime numbers and gaps of multiplications of triangular numbers

Triangular numbers: $T_n = \frac{n(n+1)}{2} = 1,3,6,10,15,21,28...$
From my observations of the first $10000$ primes:
For any prime $P$ greater than $3$:
Observation 1) There will always be at least ...

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### Does there exist numerically balanced dice with odd numbers of faces?

This question is motivated by "Numerically Balanced Dice" by Bosch, Fathauer, and Segerman, in which they produced the most numerically balanced d20 and d120. After reading this paper, I ...

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### 3D Edge matching puzzle generation

I have this weird idea for a puzzle/toy (or torture device, depending on how you look at it) I've been trying to make for years now.
I happen to be worse at this kind of math as I thought; and I'd be ...

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### For any $n-1$ elements of $\mathbb Z/n\mathbb Z$, we can make $0$ using $\{-,+,\times\}$ without parentheses

MSE: Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n? (no brackets allowed)
Is there any hope in proving the following? (Cross-posted here after a ...

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### "Drinking number" of a graph

Motivation. A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half ...

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### How many consecutive forced moves are possible in chess?

The question concerns chess. I call a move forced if, in a given position, is the unique move consistent with the rules of the game. I wonder what is the largest integer $n$ such that there exists a ...

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### Positioning ice-cream stands on a street

We want to position $n$ ice-cream stands on a street. Assume that the population on the street is modeled by a nonnegative integrable function $f$, and everyone goes to the nearest ice-cream stand. ...

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### A measurable set that acts as a speedometer

Definitions and some motivation:
Say a car is driving on a straight road. All we know is where it starts, and how much time it spends in certain stretches of the road. With just this much information, ...

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### Tic-tac-toe with one mark type

Parameters $a,b,c$ are given such that $c\leq\max(a,b)$. In an $a\times b$ board, two players take turns putting a mark on an empty square. Whoever gets $c$ consecutive marks horizontally, vertically, ...

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### Critical thinking rail track problem [closed]

On a strange railway line, there is just one infinitely long track, so overtaking is impossible. Any time a train catches up to the one in front of it, they link up to form a single train moving at ...

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### Memory game inspired problem

Motivation. As I was playing the pairs-matching game "Memory" (known as "Concentration" in some parts of the world) with my children, I was surprised that even thorough shuffling ...

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### Looking for a BAMS text about the group with commutation relations defined using meaningful words

What I definitely remember is that I saw a description of the following in the Bulletin of the American Mathematical Society, sometime in eighties (or maybe nineties?)
One considers the group ...

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### Is there a geometric way to construct $\pi \left(\frac{\alpha}{\pi}\right)\cdot\left(\frac{\beta}{\pi}\right)$, for angles $\alpha$ and $\beta$?

Given two angles $\alpha$ and $\beta$, is there a nice geometric way to construct $\pi \left(\frac{\alpha}{\pi}\right)\cdot\left(\frac{\beta}{\pi}\right)$? It does not necessarily need to be with ...

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### What is known in general about the liquid transfer problem?

In several puzzle books, I have seen the following kind of a problem: there are several containers that can hold up to certain amounts of liquid (these liquids are assumed to be infinitely divisible). ...

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### Finding an optimal strategy for a combinatorial sequential game

We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...

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### Social media for a mathematics related idea buckets

Are there any good social media platforms that can recommended for communicating ideas related to mathematics?
The reason for asking is that I am in the situation that, albeit having studied math, I ...

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### Is there more than one pseudo-Catalan solid?

This question was asked on MSE a year ago. Motivation for this question can be found in other MSE questions here, here or here.
Convex solids can have all sorts of symmetries:
the platonic solids are ...

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### Throwing a fair die until most recent roll is smaller than previous one

I roll a fair die with $n>1$ sides until the most recent roll is smaller than the previous one. Let $E_n$ be the expected number of rolls. Do we have $\lim_{n\to\infty} E_n < \infty$? If not, ...

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### Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$

Let us define the following functions:
\begin{equation*}
\small A(x)=\prod_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
B(x)=\prod_{\substack{p\leq x\\ p\...

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### Maximal in-degree in directed voting graph

Real-life motivation. Our team has $n$ members. For the next in-team presentation session, everyone had 1 talk prepared that he or she would be able to present. Now everyone could cast $1$ vote about ...

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### A New York Times tiles-based graph theory question

The New York Times has a daily puzzle named Tiles that works as follows. Start with $m$ squares (in the official version, this is 30, in a 6x5 grid), and a set of $p>4$ possible patterns (typically ...