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

**18**

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**0**answers

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### Human Knot game

In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...

**28**

votes

**1**answer

1k views

### “The Two Sheriffs” puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler.
Two sheriffs in neighboring towns are on the track of a killer, in a
case involving eight ...

**6**

votes

**1**answer

149 views

### Compiling self-referential forms

Fix $1\leq d\in\mathbb{N}$ and set $D:=\{0,1,\ldots,d-1\}$.
Consider the system of equations
\begin{equation}
x_i=c_i + \sum_{j\in D}\delta_{x_j,i}
\end{equation}
with $c_i\in D$ given and $x_i\in D$ ...

**2**

votes

**2**answers

153 views

### Websites for Math Shopping [closed]

I was wondering if anyone knows about good websites or stores to buy math related products. On etsy there are normal distribution plushes and famous mathematicians in coasters. However when I search ...

**7**

votes

**2**answers

191 views

### Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...

**13**

votes

**4**answers

1k views

### Is it possible to formulate the axiom of choice as the existence of a survival strategy?

Consider the following situation:
There is an infinite set $G$ of giraffes.
A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$.
The hungry lion ...

**6**

votes

**3**answers

444 views

### sum of binary and ternary digits

A problem in group theory (indices of imprimitive groups) gives rise to the following conjectures in number theory. Suppose a positive integer $n$ has binary and ternary expansions ...

**3**

votes

**2**answers

269 views

### Magic squares with specific properties

For what $n \geq 3$ does there exist an $n \times n$ matrix such that:
All entries are in $(0, 1)$.
Each row and column sums to $1$.
Aside from the rows and columns, no other subsets of the entries ...

**22**

votes

**1**answer

608 views

### Bouncing a ball down the stairs

In a nutshell, the question is whether it can be faster to bounce a ball down an infinite flight of stairs than to bounce it down a ramp with the same slope.
To be more specific: this is a $2$ ...

**14**

votes

**0**answers

303 views

### Knight's tours in higher dimensions

I wonder if Knight's Tours have been explored in higher dimensions,
using the following definition of a knight move.
In dimension $d=2$, the knight moves left/right and forward/back
one step and two ...

**1**

vote

**0**answers

150 views

### Knight's metric: ellipse and parabola

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...

**5**

votes

**1**answer

410 views

### Lights out game

I would like to ask about the game Lights Out for a square nxn. In http://mathworld.wolfram.com/LightsOutPuzzle.html there is a list of the number of solutions to the game, and the number of solutions ...

**0**

votes

**0**answers

146 views

### Breaking a number in two different ways

I am interested in knowing if there is a name for this process:
Suppose I have positive reals $a_1,a_2,\ldots, a_k, b_1,b_2,\ldots, b_m$ such that $\sum_{i=1}^k a_i = \sum_{j=1}^m b_j.$
Then, I can ...

**3**

votes

**2**answers

237 views

### Systems similar to Erdős numbers?

As many mathematicians know, each person has an Erdős number (see: http://en.wikipedia.org/wiki/Erd%C5%91s_number). That is, Erdős himself has Erdős number zero, each person who published anything ...

**5**

votes

**0**answers

538 views

### f(2013) = 2014? [closed]

I wonder if there is some "surprising" function $f(\;)$ that, when input $2013$, produces $2014$?
What I have in mind is more in
line with the Lewis Carroll computation involving $137$ and $992$ that ...

**11**

votes

**5**answers

893 views

### Recreational mathematics: where to search?

I am not sure I can strictly define recreational mathematics. But we all feel what it is about: puzzles, problems you can ask your mathematical friends, problems that will bother them for a couple of ...

**1**

vote

**0**answers

126 views

### Multiplicative semi-magic squares

Magic squares (Wiki) and Multiplicative magic squares (Wiki) are famous.
In this question, let us suppose that we do not consider the diagonals of multiplicative magic squares. Let us call such ...

**15**

votes

**1**answer

580 views

### a game on sets of reals

A 2 player game on $\mathcal{P}(\mathbb{R})$: Players take turns playing uncountable sets of reals. Each play must be a subset of the previously played set. Player 1 wins if the intersection of all ...

**2**

votes

**1**answer

116 views

### Increasing sequence of normal magic squares

The questions below are motivated by pure curiosity. I heard of the first question from my former advisor. I have no idea how difficult they are, since I have no experience with magic squares.
By a ...

**3**

votes

**0**answers

404 views

### sum of digits in different bases

Given a natural number, What is the maximal natural number below it, whose sums of digits in base 10 and base 2 are the same? Is there a clever algorithm to do this aside from the brute force search? ...

**5**

votes

**2**answers

631 views

### easter problem - egg shapes

Inspired by an exceptionally silly article in today's newspaper I pose the following "egg parametrization problem".
Give an explicit function $ f(x,y,t) : \mathbb{R}^2\times I \to \mathbb{R}$ such ...

**15**

votes

**5**answers

1k views

### Are there results in “Digit Theory”?

Results about numbers that are related to their decimal representation are usually confined to recreational mathematics. There I have seen mainly questions about individual numbers, like finding a ...

**13**

votes

**1**answer

577 views

### Randomly switching street lights, in a square city

This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...

**5**

votes

**4**answers

1k views

### Proving a determinant = 0

The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...

**14**

votes

**3**answers

653 views

### Smallest square to wrap a cylinder

Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle)
of height $h$ and radius $r$.
Here wrap is the natural sense of covering the surface area of the cylinder ...

**19**

votes

**1**answer

2k views

### Reconstructing the argument that yields Graham's number

Graham's number achieved a kind of cult status, thanks to Martin Gardner, as the largest finite number appearing in a mathematical proof. (It may no longer hold that record, but that is not my concern ...

**1**

vote

**1**answer

154 views

### 2D visualization of sum of divisors using Cantor pairing

Related to Gerhard's question about ascii plots. On the SeqFan mailing list
was suggested to plot an
integer sequence this way:
Let $F(x,y)= (x+y) (x+y+1)/2+y$ be the Cantor pairing.
To plot an ...

**4**

votes

**0**answers

92 views

### Does there exist $a,n \in \mathbb{Z}^+$, where $n \geq 2$, such that $a, an, an^2,an^3,\ldots,an^5$ are all palindromes in base 10?

Question: Does there exist $a,n \in \mathbb{Z}^+$, where $n \geq 2$, such that $$a, an, an^2,an^3,\ldots,an^5$$ are all palindromes in base 10?
We see that $a=1$ and $n=11$ give rise to $$1, 11, ...

**37**

votes

**8**answers

3k views

### A sudden smiley? :-)

This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the ...

**1**

vote

**1**answer

139 views

### Nonexistence of high dimensional perfect magic hypercubes of fixed side length

I apologize in advance if this question is not of sufficient level. Define a perfect magic hypercube of side length $k$ and dimension $n$ to be one in which the cells are filled with consecutive ...

**3**

votes

**3**answers

710 views

### Generalized tic-tac-toe

We begin with $2n+1$ cards, each with a distinct number from $-n$ to $+n$ on it, face up in between the two players of the game. The players take turns selecting a card and keeping it. The first ...

**0**

votes

**1**answer

301 views

### Bounding a signed sum of complex numbers [closed]

Let $z_i \in \mathbb{C}\:$ for $i=1,\dots, n\;$ be complex numbers, all with absolute value $|z_i|\le 1\;$.
Prove (or disprove) that there exists a choice of signs $s_i \in \{\pm 1\}$ such that
...

**8**

votes

**1**answer

1k views

### Looking for a “scientific” application of a recreational puzzle.

First of all the puzzle.
A barman's got 15 glasses which are initially somehow divided into several stacks. The barman repeats the following process a thousand times. He takes the top glass from ...

**39**

votes

**1**answer

1k views

### Exploding primes

Suppose every prime $n$ could "explode" once.
An explosion results in $\lfloor \alpha \ln n \rfloor$ particles being
uniformly distributed over the integers in a range $n \pm \lfloor \beta \ln n ...

**12**

votes

**3**answers

1k views

### Covering a Cube with a Square

Suppose you are given a single unit square, and you would like to completely cover the surface
of a cube by cutting up the square and pasting it onto the cube's surface.
Q1. What is the largest ...

**36**

votes

**34**answers

14k views

### Examples of interesting false proofs

According to Wikipedia False proof
For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...

**7**

votes

**1**answer

316 views

### Generalizing a square wheel to a body rolling on a surface

A square wheel rolling on a catenary road maintains the wheel center at a fixed
height, a well-known construction previously discussed on MO
(e.g.,
"Generalizing square wheels rolling on inverted ...

**9**

votes

**1**answer

1k views

### Groebner basis for Sudoku

I'm trying to write a program that solves sudoku's using a Groebner basis.
I introduced 81 variables $x_1$ to $x_{81}$, this is a linearisation of the sudoku board.
The space of valid sudokus is ...

**21**

votes

**2**answers

1k views

### How to get rich in a Hilberts Hotel?

Suppose you can make infinitely many copies of yourself. Each of them starts his/her life in a Hilberts Hotel, where each room is labeled by an element in the free group with two generators, and ...

**8**

votes

**2**answers

510 views

### The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$

Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...

**8**

votes

**1**answer

369 views

### Die-rolling Hamiltonian cycles

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$,
each of whose unit squares is labeled with a number
in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$.
Say that such a labeled $R$ is ...

**4**

votes

**1**answer

208 views

### Growth of knots possible with rope of length L

What is the asymptotics (in L) for the number of topologically different knots possible using a perfectly flexible, non-selfintersecting rope of length L and radius 1? (With ends glued together after ...

**9**

votes

**1**answer

995 views

### Self-tightening knot

Is there a way, for some finite L>1, to tie two pieces of rope together, such that any finite force is not enough to pull them apart?
The type of rope I have in mind is something like cylindrical ...

**7**

votes

**2**answers

485 views

### Chameleon Bodies

Let a body $B$ be a compact set in $\mathbb{R}^3$ with a piecewise smooth boundary.
Some pieces/patches of the boundary are perfect mirrors; others perfect matte, colored surfaces.
Imagine the view of ...

**10**

votes

**3**answers

689 views

### “Rolling Geodesics”: Designing a $k$-putt green

I am interested in what might be called rolling geodesics, paths
of physical particles confined to a surface in $\mathbb{R}^3$
under certain force conditions.
Here I will pose a specific (but ...

**0**

votes

**1**answer

190 views

### Special functions on the unit disk

Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...

**14**

votes

**5**answers

2k views

### Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...

**18**

votes

**4**answers

1k views

### Does the set of happy numbers have a limiting density?

A positive integer $n$ is said to be happy if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.
For example, 7 is ...

**9**

votes

**2**answers

1k views

### The duel problem

The following duel problem is due to Ben Polak (maybe there's earlier origin, which I'll be glad to be informed about). The rule is as follows:
Two players 1 and 2 start a duel $N$ steps away from ...

**7**

votes

**2**answers

974 views

### The motorcyclist's challenge

n walkers ${A}_{i}$ start out from X to Y simultaneously with speeds ${a}_{i}$, $i=1,2,...,n$. ${a}_{i}\neq {a}_{j}$ if $i\neq j$. At the same time, a motorcyclist M with speed $m=1$ starts out from Y ...