# Questions tagged [recreational-mathematics]

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

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### averages - can you determine min and max [closed]

Morning, this is a question I think I know the answer to already, and that is 'no you can't'. Say I know the total number of responses to a single question is 10,000 and the average age is said to be ...
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### Order of the "children's card shuffle"

Motivation. My eldest son thinks the following procedure is a "perfect shuffle" for a deck of cards: Take the first card, put the second on top of it, put the 3rd below cards 2 and 1, put ...
51 views

### Proving a (Representing Utility) Function is Continuous [migrated]

I sincerely apologize for posting such a long question. The question involves a complicated proof of a theorem in mathematical economics. I feel it will be better for me to state my question first. I ...
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### $2$-for-$2$ asymmetric Hex

This is a crosspost from Math stackexchange as I left the question open a while and bountied it but received no answers. If the game of Hex is played on an asymmetric board (where the hexes are ...
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### Does there exists a $\sigma \in S_{2N}$ such that $\sum_{n=1}^{N} \sigma(2n-1) ^ {\sigma(2n)}$ is a perfect square?

Does there exists a $\sigma \in S_{2N}$ such that $\sum_{n=1}^{N} \sigma(2n-1) ^ {\sigma(2n)}$ is a perfect square? ($S_k$ denotes the group of permutations of $\{1, 2, 3, ..., k\}$) To me, it seems ...
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### What are the Nash equilibria of the “aim for the middle” game?

Consider the following three-player game: Alice chooses an integer congruent to $0$ mod $3$, Bob chooses an integer congruent to $1$ mod $3$, and Chris chooses an integer congruent to $2$ mod $3$. (...
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1 vote
156 views

### Permutation graph with insert-and-shift

Motivation. I am working with a database software that allows you to sort the fields of any given table in the following peculiar way. Suppose your fields are numbered $1,\ldots, 18$. Next to every ...
1 vote
172 views

### Graph on $\mathbb{N}$ where almost every vertex is shy

The following question is loosely based on the friendship paradox. Let $G=(V,E)$ be a simple, undirected graph. For $v\in V$, we let the neighborhood of $v$ be $N(v) = \big\{w\in V:\{v,w\}\in E\big\}$ ...
1k views

### Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

Does a function from $\mathbb{R}^2$ to $\mathbb{R}$ which sums to 0 on the corners of any unit square have to vanish everywhere? I think the answer is yes but I am not sure how to prove it. If we ...
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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 ...
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### Inspired by a card game: finding a path through $[\mathbb{N}]^n$

Motivation. Today my sons played a card game, in which a fixed number $n$ of cards was lying on the table. A move consists of adding an unused card to the cards on the table, and removing a card from ...
535 views

### Optimal schedule for a soccer tournament

Motivation. This weekend, my children took part in a soccer tournament consisting of $n$ teams, each of which playing once against every other team. As there was only one soccer field, the schedule ...
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### Parity of 4×4 normal magic squares

I'm writing a program that given an integer n returns all normal magic squares of size n × n. Fiddling around with it a little I started to notice that if n equals 4, every square I saw followed this ...
226 views

### 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 ...
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### Has there been any progress on Conway's and Soifer's shortest paper?

In 2005 Conway and Soifer published the famous shortest ever paper, asking whether an equilateral triangle of sidelength $n+\varepsilon$ can be covered by $n^2+1$ unit equilateral triangles and ...
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1 vote
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### 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'...
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### Can we arrange {1,...,9} in 3×3 grid so the set of products of rows equals the set of products of columns? [closed]

I find a interesting question of Prmo mock and Promys 2020 For which $n\in\mathbb{N}$ is it possible to arrange $\{1,…,n^2\}$ in an $n\times n$ grid so that the set of products of columns equals the ...
242 views

### Monoid associated to $>2$-player Hackenbush

There is some literature on multiplayer combinatorial game theory, but as far as I can tell none of it follows the line of attack below. I'd love a pointer to a similar approach taken in the ...
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### For which $n$ does a y-formed $n$-polyomino tile a $n \times n \times n$-cube?

I got from my children as a gift a puzzle consisting of 25 y-shaped 5-polyominoes that form a $5 \times 5 \times 5$-cube (see picture). I'm wondering for which $n$ does a y-formed $n$-polyomino tile a ...
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### Scheduling "parent talks" at school

Real life motivation. In my younger son's class, there are $18$ students. His teacher provided $18$ time slots for the parents of each child to have a 30-minute conversation of their kid's progress in ...
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### Are there journals for "fun mathematics"?

Are there peer-reviewed journals that focus on "fun mathematics"? By this I mean fun things that do involve nontrivial mathematics and which I think other mathematicians would enjoy reading ...
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### The $9$th tetration of $-\sqrt2$

Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and $$^{n+1}a=a^{^na}$$ for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
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### 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 ...
1 vote
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 ...