Questions tagged [recreational-mathematics]

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

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Simple disproof of Danzer — Grünbaum conjecture

I asked this question on the MSE, but I did not get an answer. I hope that one of the experienced participants will check the correctness of the proof or the truth of the statement (and, perhaps, will ...
grizzly's user avatar
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16 votes
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Division of a square and value of a disk

[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk I cam across this problem ...
user33624's user avatar
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A New York Times tiles-based graph theory question

The New York Times has a daily puzzle named Tiles that works as follows. Start with $m$ squares (in the official version, this is 30, in a 6x5 grid), and a set of $p>4$ possible patterns (typically ...
David Pepper's user avatar
12 votes
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Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible

Is the number of elements we need to construct $x$ equal to $\log_2(x) + O(1)$? This question is inspired by question 2 of the 2018 European Girls' Mathematical Olympiad. I previously posted it on ...
user133281's user avatar
11 votes
0 answers
764 views

Borderline Collatz-like problems

The usual Collatz map is $C:n \mapsto n/2$ if $n$ even, $(3n+1)/2$ if $n$ odd. Let $f^{\circ (r+1)}:=f \circ f^{\circ r}$. We suspect that for every fixed $n>0$, the sequence $C^{\circ r}(n)$ ...
Sebastien Palcoux's user avatar
11 votes
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538 views

Does Chu and Hough's solution to the mixing time of the 15-puzzle carry over to the Rubik's cube?

In his 1988 book Group Representations in Probability and Statistics , Diaconis considers mixing times of the 15-puzzle. He states: Here is a simplified version: Consider the blank as a $16$th block,...
Mark S's user avatar
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9 votes
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The $n$ queens problem with no three on a line

The $n$ queens problem asks if we can place $n$ queens on an $n\times n$ chessboard such that no two queens attack one another. For example, when $n=8$, here are two solutions (images taken from ...
ho boon suan's user avatar
8 votes
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151 views

Pursuit-evasion with many slow pursuers

Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$  ($r>0$ is distance from the origin). If you start at the origin ...
Dmytro Taranovsky's user avatar
6 votes
0 answers
474 views

Existence of an explosive prime

The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below). Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...
Sebastien Palcoux's user avatar
6 votes
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Is there a theory behind these puzzles? (communicating by modifying data)

Consider the following puzzles: Problem 1: Alice is given two data by Zora: a binary string $w$ of length $2^r$, and a position $p$ in the string (which we can view as an integer $0\leq p<2^r$). ...
Gro-Tsen's user avatar
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5 votes
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What does the best die look like?

Intransitive dice have attracted a lot of attention - especially in the context of recreational math - since their introduction by Efron in the 1960s. More recently, there has been work studying ...
Sam Hopkins's user avatar
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5 votes
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Particles sent into the same direction with uniformly distributed speed

Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters ...
Dominic van der Zypen's user avatar
5 votes
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158 views

The two Collatz-maps associated to characters modulo 8

Given a Dirichlet character $\chi$ modulo $8$ we consider the map $\mu(x)=x/2$ if $x$ is even and $\mu(x)=(3x+\chi(x))/2$ otherwise. (The corresponding map for $\chi$ the trivial Dirichlet character ...
Roland Bacher's user avatar
5 votes
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213 views

Is Domineering on any finite approximation of the Sierpinski Carpet always a second-player win?

The game of Domineering can be played on any board consisting of some subset of $\mathbb{Z} \times \mathbb{Z}$. In particular, consider the boards $K_n$ generated by iterating the following inductive ...
Noam Zeilberger's user avatar
5 votes
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161 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, 121, ...
Douglas S. Stones's user avatar
4 votes
0 answers
116 views

Reorganizational matching

Motivation. My friend works in an organization that is re-organizing itself in the following somewhat laborious way: There are $n$ people currently sitting on $n$ jobs in total (everyone has one job). ...
Dominic van der Zypen's user avatar
4 votes
0 answers
146 views

A matrix / zero forcing game

Two players, You (Y) and the Enemy (E), play the following game on a real $n\times n$ matrix. First, E selects one element from the first row of the matrix, two elements from its second row, and so on;...
Seva's user avatar
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4 votes
1 answer
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Novel examples, proofs or results in mathematics from arithmetic billiards

The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,…. Wikipedia has an ...
4 votes
0 answers
377 views

Social media for a mathematics related idea buckets

Are there any good social media platforms that can recommended for communicating ideas related to mathematics? The reason for asking is that I am in the situation that, albeit having studied math, I ...
4 votes
0 answers
240 views

Numbers with a square sum arrangement

Informal version. For which $n>1$ can the numbers $1,\ldots, n^2$ be arranged in a square form such that the sums of the numbers in the little squares (consisting of $4$ numbers) are all equal? ...
Dominic van der Zypen's user avatar
4 votes
0 answers
71 views

Bound on number of steps needed for points to meet enclosing convex polygon

Let $P$ be the set of equidistant points on the unit circle which are then randomly shuffled. They then take discrete steps towards the midpoint between the 2 points that they were originally adjacent ...
martin's user avatar
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4 votes
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Combinatorial fairness property in division of goods

Given $n$ agents, and $m$ items where $v_i(g) \geq 0$ is the value of item $g$ for agent $i$, does there always exist a partition $A_1, ..., A_n$ of the $m$ items into $n$ sets s.t. for all $i, j \in \...
Daishisan's user avatar
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3 votes
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Which manhole covers fall through their holes?

Apparently one of the reasons why all manhole covers are shaped like discs is because for any other shape, the manhole cover would fall through its own hole. As stated this is not necessarily a ...
Stanley Yao Xiao's user avatar
3 votes
0 answers
182 views

Transitive action on domino tilings

Fix a $n \times m$ rectangle and consider the set $S_{n,m}$ of all its dominos tilings. Here are examples with $n=m=8$. The set $S_{n,m}$ is empty if and only if $nm$ is odd, and for small $nm$, its ...
Sebastien Palcoux's user avatar
3 votes
0 answers
135 views

Can you escape from two lions in a closed arena?

You're at the center of a circular arena. A pair of lions are at the border, planning to catch you. One of them moves as fast as you, but the other moves slower than you. The three of you are confined ...
Eric's user avatar
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3 votes
0 answers
277 views

Math videos featuring interesting data animations

I am looking for interesting videos featuring pure data animations (not someone talking about math, but a video featuring some math phenomenon). I am interested in videos that tell a story, rather ...
Vincent Granville's user avatar
3 votes
0 answers
92 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 ...
John's user avatar
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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? ...
Hans's user avatar
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2 votes
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102 views

Proof that a pandiagonal Latin square of order $n$ exists iff $n$ is not a multiple of $2$ or $3$?

A pandiagonal Latin square of order $n$ is an assignment of the numbers $\{0,\ldots,n-1\}$ to the cells of an $n \times n$ grid such that no row, column, or diagonal of any length contains the same ...
Milo B's user avatar
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2 votes
0 answers
90 views

Can Chang and Wang's proof of Thue’s Theorem on circular packing be extended into other dimentions?

The simplicity of Chang and Wang's proof of Thue’s Theorem (link on arxiv) on circular packing took me by surprise. Have similar ideas been found helpful in other dimensions? For example, partition ...
Ye Tian's user avatar
  • 161
2 votes
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67 views

Split $\{1,...,mn\}$ into $m$-tuples $x$ with $\sum_{i\gt 1} x_i=kx_1$

This question arose in Math.StackExchange with $k=3,m=3$ https://math.stackexchange.com/questions/4179825/for-which-n-in-bbb-n-can-we-divide-1-2-3-3n-into-n-subsets-each-wi For which $m,k$ are there ...
Empy2's user avatar
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2 votes
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514 views

Does there exist numerically balanced dice with odd numbers of faces?

This question is motivated by "Numerically Balanced Dice" by Bosch, Fathauer, and Segerman, in which they produced the most numerically balanced d20 and d120. After reading this paper, I ...
Ye Tian's user avatar
  • 161
2 votes
0 answers
161 views

Finding an optimal strategy for a combinatorial sequential game

We are given a set $\{p_1, p_2, \ldots, p_n\}$ of players and a set of $\{\ell_1, \ell_2, \ldots, \ell_n\}$ of locations, where $n\in\mathbb{N}$. Each location can be either free or occupied, and each ...
Let101's user avatar
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2 votes
0 answers
107 views

dividing a square into unique rectangles with the same perimeter

There's a solution for dividing a square into unique rectangles with the same area which is the blanche dissection. There's also a solution for dividing a square into unique rectangles with the same ...
elbert k's user avatar
1 vote
0 answers
101 views

Expected value of maximal cycle length in fixed-point free bijections

$\newcommand{\n}{\{1,\ldots,n\}}$ $\newcommand{\FF}{\text{FF}}$ $\newcommand{\lc}{\text{lc}}$ Motivation. A group of my son's peers decided to have a few days of Secret Santa before last year's ...
Dominic van der Zypen's user avatar
1 vote
0 answers
24 views

What are the limits to the lengths of the sequences of consecutive completed Sudoku when order 9 Latin squares are generated in lexicographic order?

Question: What are the maximum and minimum lengths of the sequences of consecutive completed Sudoku which occur when order 9 Latin squares are generated in (standard) lexicographic order? A minimum ...
John Palmer's user avatar
1 vote
0 answers
92 views

Eventual stabilization for repeatedly adding multiplayer games

This question is an outgrowth of a couple previous questions of mine. In order: 1,2,3. This should be fully self-contained, but those questions may help motivate this one. To keep things readable, I'...
Noah Schweber's user avatar
1 vote
0 answers
156 views

Another Goldbach variation for odd numbers?

Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime. Dabbling in the dark art ...
Roland Bacher's user avatar
1 vote
0 answers
101 views

On a combinatorial design inspired by a football (soccer) tournament

Real-world inspiration. My younger son was playing a micro football (soccer) tournament this afternoon with $3$ other friends. Let's label the $4$ kids $0,1,2,3$. They played $3$ matches: $\{0,1\} \...
Dominic van der Zypen's user avatar
1 vote
0 answers
121 views

Tiling a rectangle with squares

Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle: The task was to determine the area of the rectangle tiled with $8$ squares, of which the ...
Dominic van der Zypen's user avatar
1 vote
0 answers
163 views

Random walk on 2d lattice with obstacles

Consider a random work on $L=\mathbb Z^2$ endowed with obstacles (i.e each cell $(x,y)$ of $L$ may contain a obstacle, i.e the random walk halts whenever it hits such a cell). Let $P(x,y) = 1$ if cell ...
dohmatob's user avatar
  • 6,716
1 vote
0 answers
87 views

Prime numbers and gaps of multiplications of triangular numbers

Triangular numbers: $T_n = \frac{n(n+1)}{2} = 1,3,6,10,15,21,28...$ From my observations of the first $10000$ primes: For any prime $P$ greater than $3$: Observation 1) There will always be at least ...
Isaac Brenig's user avatar
1 vote
0 answers
172 views

For any $n-1$ elements of $\mathbb Z/n\mathbb Z$, we can make $0$ using $\{-,+,\times\}$ without parentheses

MSE: Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n? (no brackets allowed) Is there any hope in proving the following? (Cross-posted here after a ...
Vepir's user avatar
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1 vote
1 answer
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Looking for an efficient way of maximising 'pair scores' for subset of 30 selected from 50 to 10 000 objects

Context: I have a tiling program that uses a directed breadth first search algorithm. It is 'directed' by what I call 'pair scoring'. There are $N$ polyforms (pieces) used in the tiling. I have an $N\...
theonetruepath's user avatar
1 vote
0 answers
162 views

Perfect squares of the form $ab^n+c$ and a Diophantine equation

The motivation for this question comes from the following problem from an international Team selection test of 2007 from Chile: Problem: Let $p$ be a prime number. Find all pairs of positive ...
A123's user avatar
  • 111
1 vote
0 answers
104 views

1-concatenable primes

If we choose some prime, say $11$, we can concatenate one digit to the left and one to the right to obtain another prime, for example, $2113$, we can do the same with $2113$ and obtain $121139$, a ...
Right's user avatar
  • 225
1 vote
0 answers
261 views

Characterization of non-Zeno functions $f:\mathbb{R}\rightarrow \{0,1\}$

[Edit: I tried to integrate Nate's comments (see below).] In the context of automata over continuous time, consider Boolean-valued functions $f:\mathbb{R}\rightarrow \{0,1\}$. There are uncountably ...
Hans-Peter Stricker's user avatar
1 vote
0 answers
79 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 ...
Joseph O'Rourke's user avatar
1 vote
0 answers
386 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 ...
Felice Iandoli's user avatar
1 vote
0 answers
242 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 ...
mathlove's user avatar
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