# Questions tagged [recreational-mathematics]

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

214
questions

**6**

votes

**1**answer

140 views

### 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 ...

**0**

votes

**0**answers

125 views

### 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 ...

**11**

votes

**1**answer

399 views

### “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 ...

**5**

votes

**1**answer

1k views

### 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 ...

**5**

votes

**1**answer

219 views

### 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. ...

**5**

votes

**1**answer

319 views

### 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, ...

**18**

votes

**3**answers

509 views

### 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, ...

**1**

vote

**2**answers

265 views

### 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 ...

**3**

votes

**1**answer

141 views

### 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 ...

**5**

votes

**1**answer

165 views

### 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 ...

**3**

votes

**1**answer

110 views

### 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 ...

**10**

votes

**1**answer

606 views

### 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). ...

**2**

votes

**0**answers

123 views

### 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 ...

**4**

votes

**0**answers

157 views

### 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 ...

**4**

votes

**2**answers

176 views

### 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 ...

**12**

votes

**4**answers

2k views

### 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, ...

**4**

votes

**1**answer

235 views

### 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\...

**10**

votes

**2**answers

180 views

### 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 ...

**9**

votes

**0**answers

624 views

### 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 ...

**4**

votes

**1**answer

222 views

### Map $f:\mathbb{N}\to\mathbb{N}$ such that every 2-set is a neighbor exactly once

Is there a map $f:\mathbb{N}\to\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ with $a\neq b$ there is exactly one $n\in\mathbb{N}$ such that $\{a,b\} = \{f(n),f(n+1)\}$?

**3**

votes

**0**answers

361 views

### While solving the 1988 IMO problem 6, I have questions about new solutions without using Vieta Jumping [closed]

I think most of you may know the well-known problem:
"Let $x$ and $y$ be positive integers such that $xy + 1$ divides $x^{2} + y^{2}$. Show that $\frac {x^{2} + y^{2}}{xy + 1}$ is the perfect ...

**6**

votes

**3**answers

2k views

### “Gray code” for building teams

Motivation. In a team of $n$ people, we had the task to build subteams of a fixed size $k<n$ such that every day, $1$ person of the subteam is replaced by another person in the team, but not in the ...

**1**

vote

**1**answer

224 views

### Brinksmanship: how to achieve the best outcome by a single statement [closed]

This game is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which contenders practice brinksmanship to their own advantages. It goes as follows:
Anderson, Barnes, and ...

**5**

votes

**1**answer

286 views

### Game on a square grid

Not research level, comments are welcome.
Consider the following game:
The board is the vertices of an $n$ by $n$ square grid.
Two players take moves in turns.
A move is picking two vertices and ...

**8**

votes

**1**answer

686 views

### Recreational mathematical papers [closed]

Sometimes it is nice to get a less technical paper on mathematics to read and learn something different for a change. These papers often make us discover some new curiosity, to think about the process ...

**2**

votes

**1**answer

80 views

### How to turn a shuffled deck of card into bits

Suppose I fairly shuffle a deck of 52 playing cards, and I want to generate some bits. You could look at each pair of cards, see which is higher or lower, and output either a 0 or 1. That's 26 ...

**27**

votes

**1**answer

834 views

### The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...

**1**

vote

**1**answer

154 views

### Bike lock graph

Motivation. I have a bike lock with 4 dials, and I was wondering whether I can reach any combination by always turning a fixed number $k$, say $k=2$, of the dials, by $1$ position, instead of just ...

**2**

votes

**3**answers

331 views

### Generations until fixation: A nontrivial generalization of a dice convergence problem

In spite of its "recreational" aspect, this question appears to me to be research-level and (I hope) clearly formulated and tagged.
Edit 4/4/20: You can find a related question with the ...

**2**

votes

**1**answer

144 views

### Increasing the “shuffling distance” by iterating a permutation $\varphi: \omega \to \omega$

Motivation. I was wondering about the following when playing a card-shuffling game with my elder son.
If $\varphi: \omega \to \omega$ is a bijection, we define the shuffling distance of $\varphi$ by $...

**1**

vote

**1**answer

95 views

### Generating all pentominoes by cutting and pasting

Is it possible to place the twelve pentominoes around a circle in such a way that if two of the pentominoes find themselves next to each other, it is because one of the two can be obtained from the ...

**7**

votes

**0**answers

462 views

### Borderline Collatz-like problems

The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$.
We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...

**1**

vote

**1**answer

178 views

### Looking for an efficient way of maximising 'pair scores' for subset of 30 selected from 50 to 10 000 objects

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\...

**4**

votes

**2**answers

681 views

### Coin flipping game

Motivation. My elder son played the following game. He had a bunch of coins, all with heads up, arranged in a circle. He flipped one coin, so that it showed tails, then he moved $1$ position clockwise,...

**4**

votes

**1**answer

150 views

### Iterated product of digits

It is well-known that the interated
sum-of-digits function equally distributes
the numbers
from $1$ to $10^k-1$ to the digits $1,\ldots,9$.
And this holds true for any base $b$.
For example, see the ...

**11**

votes

**2**answers

424 views

### Algebraic properties of graph of chess pieces

For the purpose of this question, a chess piece is the King, Queen, Rook, Bishop or Knight of the game of chess. To a chess piece is attached a graph which represents the legal moves it can make on an ...

**5**

votes

**0**answers

181 views

### Is Domineering on any finite approximation of the Sierpinski Carpet always a second-player win?

The game of Domineering can be played on any board consisting of some subset of $\mathbb{Z} \times \mathbb{Z}$.
In particular, consider the boards $K_n$ generated by iterating the following inductive ...

**1**

vote

**0**answers

140 views

### Perfect squares of the form $ab^n+c$ and a Diophantine equation

The motivation for this question comes from the following problem from an international Team selection test of 2007 from Chile:
Problem: Let $p$ be a prime number. Find all pairs of positive ...

**0**

votes

**2**answers

105 views

### Majority-driven manipulations of integer vectors

Motivation. Recently I was watching two people play a game that involved arranging sticks in a number of heaps and moving them around in certain allowed ways that I think I was able to infer from ...

**11**

votes

**3**answers

1k views

### How can I simplify this sum any further?

Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...

**11**

votes

**0**answers

410 views

### Does Chu and Hough's solution to the mixing time of the 15-puzzle carry over to the Rubik's cube?

In his 1988 book Group Representations in Probability and Statistics , Diaconis considers mixing times of the 15-puzzle. He states:
Here is a simplified version: Consider the blank as a $16$th block,...

**5**

votes

**2**answers

427 views

### Magic $\mathbb{Z}\times\mathbb{Z}$-square

Is there an injective map $j:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ satisfying: For every $z\in \mathbb{Z}$ we have $$\lim_{N\to \infty}\sum_{k=-N}^Nj(k,z) = 0 = \lim_{N\to \infty}\sum_{k=-N}^Nj(...

**1**

vote

**0**answers

380 views

### Magical $\mathbb{Z}\times\mathbb{Z}$-square [duplicate]

Now duplicate of Magic $\mathbb{Z}\times\mathbb{Z}$-square where it has an answer.
Is there an injective map $j:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ satisfying: For every $z\in \mathbb{Z}$ we ...

**7**

votes

**1**answer

280 views

### What is the form of the $(v_0,v_1)$-pizza curve?

Assume that there are two (competing) pizza houses situated at the points $0$ and $1$ on the complex plane. These pizza houses can deliver pizza to points of the plane with the largest velocities $v_0$...

**6**

votes

**1**answer

227 views

### Guessing the number of other $1$'s in a binary sequence

I have posed the following question on math.stackexchange.com but have not received an answer. So I would like to seek experts' opinion here.
Consider the set of all binary sequence of length $n+1$, $...

**-4**

votes

**1**answer

289 views

### Numbers representable as in the famous IMO question number 6 (1988)

The famous problem number 6 of the 1988 International Mathematical Olympiad is about showing that if $a,b$ are non-negative integers such that $\frac{a^2+b^2}{ab+1}$ is an integer, then it is a square ...

**4**

votes

**0**answers

231 views

### Numbers with a square sum arrangement

Informal version. For which $n>1$ can the numbers $1,\ldots, n^2$ be arranged in a square form such that the sums of the numbers in the little squares (consisting of $4$ numbers) are all equal?
...

**5**

votes

**1**answer

169 views

### Complete folds and one cut

The fold-and-cut theorem states that any shape with straight sides can be cut by a single complete straight cut if the paper is the folded flat in the right way. Here is an example from an answer on ...

**7**

votes

**1**answer

292 views

### Knight's tour problem

It is known that on an infinite board, if all squares of the form $(ki,kj)$ are removed, $k$ even, $i,j\in\mathbf{Z}$, then there is no knight's tour due to unbalanced black and white squares.
My ...

**8**

votes

**0**answers

117 views

### Pursuit-evasion with many slow pursuers

Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...