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

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5
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1answer
201 views

How many convex shapes can be made with the pieces of the Stomachion?

Tangrams are a well-known dissection of the square into seven convex polygons. One fun mathematical question is: how many convex rearrangements of the seven pieces are there? Answer: there are 12 ...
71
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21answers
17k views

Examples of math hoaxes/interesting jokes published on April Fool's day?

What are examples of math hoaxes/interesting jokes published on April Fool's day? For a start P=NP.
3
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1answer
42 views

Complexity class of matrix generalization of knapsack problem

Let $n$ be a natural number, $u_+,v_+,u_-,v_-$ be real or complex column vectors of length $n$, and $M_1,M_2,\ldots,M_k$ be a finite collection of $n\times n$ real or complex matrices. Consider the ...
3
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0answers
73 views

Graph adjacency grouping with geometric criteria

I start with a list of adjacent tetrahedra, where there are tight seals to one another along faces for two tetrahedra that are adjacent. The vertices belonging to these faces for both tetrahedra are ...
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0answers
62 views

Completely incongruent box partitions

Let $B$ be a rectangular box with corners in $\mathbb{Z}^d$ and sides parallel to the axes. A completely incongruent partition of $B$ is a partition into $d$-dimensional boxes, each of whose integer ...
21
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3answers
520 views

Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues: $$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ ...
11
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1answer
231 views

Have “sturdy squares” been studied before?

Over on PPCG I've just made a question that involves arranging the numbers 1 to 9 on a 3 by 3 grid such that every 2 by 2 subgrid has the same sum. I'm calling such 3 by 3 grids and their N by N ...
5
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1answer
156 views

Intersection of rotating regular polygons

This question has a recreational flavor, but may not be entirely uninteresting. Let $P_k$ be a unit-radius regular polygon of $k$ sides, and $P_n$ a unit-radius regular polygon of $n \ge k$ sides. ...
0
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1answer
176 views

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 ...
18
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0answers
481 views

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 ...
30
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1answer
2k 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
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1answer
173 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
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2answers
195 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
2answers
195 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 - ...
15
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4answers
2k 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
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3answers
479 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
2answers
284 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 ...
24
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1answer
851 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
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0answers
354 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 ...
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0answers
182 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
1answer
469 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 ...
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0answers
163 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
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2answers
257 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
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0answers
549 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 ...
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9answers
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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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0answers
142 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 ...
16
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1answer
611 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
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1answer
120 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 ...
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0answers
504 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
2answers
767 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 ...
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5answers
2k 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
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1answer
603 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 ...
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4answers
2k 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
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3answers
750 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 ...
20
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1answer
4k 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
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1answer
165 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 ...
5
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0answers
105 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, ...
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8answers
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
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1answer
144 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 ...
4
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3answers
752 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
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1answer
309 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 ...
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1answer
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 ...
42
votes
1answer
2k 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 ...
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3answers
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 ...
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34answers
18k 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 ...
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1answer
351 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 ...
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1answer
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 ...
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2answers
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
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2answers
533 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
1answer
375 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 ...