Tagged Questions

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

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2
votes
2answers
132 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
183 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 - ...
12
votes
4answers
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
3answers
422 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
262 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
1answer
499 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
0answers
279 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
0answers
133 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
383 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
0answers
142 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
2answers
226 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
0answers
525 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
5answers
640 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
0answers
108 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
1answer
560 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
1answer
112 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
0answers
379 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
478 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
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
votes
1answer
546 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
4answers
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
3answers
557 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 ...
18
votes
1answer
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
1answer
146 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
0answers
90 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
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
vote
1answer
133 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
3answers
677 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
1answer
292 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
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 ...
39
votes
1answer
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 ...
11
votes
3answers
914 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 ...
33
votes
34answers
13k 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
1answer
302 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
1answer
951 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
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 ...
7
votes
2answers
499 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
365 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
1answer
206 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
1answer
979 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
2answers
470 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
3answers
665 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
1answer
189 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
5answers
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
4answers
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
2answers
986 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
2answers
970 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 ...
3
votes
2answers
359 views

Truel extended to n persons

n players numbered 1~n play a shooting game. Their accuracy rates p1~pn are strictly between 0 and 1, and strictly increases from p1 to pn. This is common knowledge. Before the game starts, the ...
0
votes
1answer
281 views

Zermelo's stone game in 3 dimensional space

Well, first let me make this clear: I'm actually not sure about the background of the game, whether it was really posed (and solved) by Zermelo. But I'll state the game anyway (perhaps someone can ...
3
votes
1answer
778 views

Galois connections

I've been experiencing minor qualms about my preprint "A Galois Connection in the Social Network" (accepted by Mathematics Magazine, pending revisions), and one of them involves the way I describe the ...