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

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### Definition of “Expected/Unexpected Event”

Background of my question is Martin Gardner's "unexpected hanging" paradoxon, which has once again be the subject of an article in a popular-scientific magazin (this time because this year it has been ...

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### Human Knot game [duplicate]

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 ...

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**1**answer

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### “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 ...

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**1**answer

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### 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 $x_i\in D$ ...

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**2**answers

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### 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 ...

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**2**answers

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### 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 - ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 $2$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 ...

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### 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? ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 ...

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**0**answers

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 ...

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### 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{ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### “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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...