Tagged Questions

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

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How to sample a uniform random polyomino?

A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...
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Separating Heavier from the Lighter Balls

This Question was originally posted Here, and received no significant progress so far. I wanted to post it here as well, to see what the people of mathoverflow think about it. I think we are ...
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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 ...
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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.
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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 ...
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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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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 ...
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Evaluating an integral using real methods

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

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