# Questions tagged [recreational-mathematics]

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

192
questions

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203 views

### Brinksmanship: how to achieve the best outcome by a single statement [closed]

This game is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which contenders practice brinksmanship to their own advantages. It goes as follows:
Anderson, Barnes, and ...

**6**

votes

**1**answer

255 views

### Game on a square grid

Not research level, comments are welcome.
Consider the following game:
The board is the vertices of an $n$ by $n$ square grid.
Two players take moves in turns.
A move is picking two vertices and ...

**8**

votes

**1**answer

628 views

### Recreational mathematical papers [closed]

Sometimes it is nice to get a less technical paper on mathematics to read and learn something different for a change. These papers often make us discover some new curiosity, to think about the process ...

**1**

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**0**answers

34 views

### How to turn a shuffled deck of card into bits

Suppose I fairly shuffle a deck of 52 playing cards, and I want to generate some bits. You could look at each pair of cards, see which is higher or lower, and output either a 0 or 1. That's 26 ...

**25**

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**1**answer

721 views

### The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...

**1**

vote

**1**answer

146 views

### Bike lock graph

Motivation. I have a bike lock with 4 dials, and I was wondering whether I can reach any combination by always turning a fixed number $k$, say $k=2$, of the dials, by $1$ position, instead of just ...

**2**

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**3**answers

290 views

### Generations until fixation: A nontrivial generalization of a dice convergence problem

In spite of its "recreational" aspect, this question appears to me to be research-level and (I hope) clearly formulated and tagged.
Edit 4/4/20: You can find a related question with the same ...

**2**

votes

**1**answer

137 views

### Increasing the “shuffling distance” by iterating a permutation $\varphi: \omega \to \omega$

Motivation. I was wondering about the following when playing a card-shuffling game with my elder son.
If $\varphi: \omega \to \omega$ is a bijection, we define the shuffling distance of $\varphi$ by $...

**1**

vote

**1**answer

92 views

### Generating all pentominoes by cutting and pasting

Is it possible to place the twelve pentominoes around a circle in such a way that if two of the pentominoes find themselves next to each other, it is because one of the two can be obtained from the ...

**7**

votes

**0**answers

383 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 $C^{\circ r}(n)$ ...

**0**

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**0**answers

43 views

### A ballot-casting problem

For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ denote the collection of subsets of $X$ with cardinality $\kappa$. If $n$ is a positive integer, let $[n]:=\{1,\ldots,n\}$.
Let $V$ and $K$ be ...

**1**

vote

**1**answer

82 views

### 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\...

**4**

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**2**answers

659 views

### Coin flipping game

Motivation. My elder son played the following game. He had a bunch of coins, all with heads up, arranged in a circle. He flipped one coin, so that it showed tails, then he moved $1$ position clockwise,...

**4**

votes

**1**answer

138 views

### Iterated product of digits

It is well-known that the interated
sum-of-digits function equally distributes
the numbers
from $1$ to $10^k-1$ to the digits $1,\ldots,9$.
And this holds true for any base $b$.
For example, see the ...

**10**

votes

**2**answers

270 views

### Algebraic properties of graph of chess pieces

For the purpose of this question, a chess piece is the King, Queen, Rook, Bishop or Knight of the game of chess. To a chess piece is attached a graph which represents the legal moves it can make on an ...

**6**

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**0**answers

169 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 ...

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**0**answers

113 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 ...

**0**

votes

**2**answers

105 views

### Majority-driven manipulations of integer vectors

Motivation. Recently I was watching two people play a game that involved arranging sticks in a number of heaps and moving them around in certain allowed ways that I think I was able to infer from ...

**11**

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**3**answers

1k views

### How can I simplify this sum any further?

Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able ...

**10**

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**0**answers

272 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 ...

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**2**answers

413 views

### Magic $\mathbb{Z}\times\mathbb{Z}$-square

Is there an injective map $j:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ satisfying: For every $z\in \mathbb{Z}$ we have $$\lim_{N\to \infty}\sum_{k=-N}^Nj(k,z) = 0 = \lim_{N\to \infty}\sum_{k=-N}^Nj(...

**1**

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**0**answers

379 views

### Magical $\mathbb{Z}\times\mathbb{Z}$-square [duplicate]

Now duplicate of Magic $\mathbb{Z}\times\mathbb{Z}$-square where it has an answer.
Is there an injective map $j:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ satisfying: For every $z\in \mathbb{Z}$ we ...

**3**

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**1**answer

161 views

### What is the form of the $(v_0,v_1)$-pizza curve?

Assume that there are two (competing) pizza houses situated at the points $0$ and $1$ on the complex plane. These pizza houses can deliver pizza to points of the plane with the largest velocities $v_0$...

**6**

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**1**answer

222 views

### Guessing the number of other $1$'s in a binary sequence

I have posed the following question on math.stackexchange.com but have not received an answer. So I would like to seek experts' opinion here.
Consider the set of all binary sequence of length $n+1$, $...

**-4**

votes

**1**answer

256 views

### Numbers representable as in the famous IMO question number 6 (1988)

The famous problem number 6 of the 1988 International Mathematical Olympiad is about showing that if $a,b$ are non-negative integers such that $\frac{a^2+b^2}{ab+1}$ is an integer, then it is a square ...

**4**

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**0**answers

229 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?
...

**5**

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**1**answer

142 views

### Complete folds and one cut

The fold-and-cut theorem states that any shape with straight sides can be cut by a single complete straight cut if the paper is the folded flat in the right way. Here is an example from an answer on ...

**7**

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**1**answer

253 views

### Knight's tour problem

It is known that on an infinite board, if all squares of the form $(ki,kj)$ are removed, $k$ even, $i,j\in\mathbf{Z}$, then there is no knight's tour due to unbalanced black and white squares.
My ...

**6**

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**0**answers

88 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 ...

**5**

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**1**answer

526 views

### Questions about “The best card trick”

Kleber's Best card trick proceeds as follows: The mark (audience member) freely selects five playing cards from a standard deck of $52$ and passes these five to the magician's assistant. The ...

**56**

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**2**answers

2k views

### Guessing each other's coins

I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...

**9**

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**1**answer

260 views

### Is this kind of “Gerrymandering” NP-complete?

[I posted this on math.stackexchange.com about two weeks ago, but didn't get any reply, so I'm trying it here.]
Consider the following simplified form of "Gerrymandering": You have $n^2$ squares ...

**4**

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**2**answers

169 views

### Expected value of length of interval game

I have a die that produces uniformly distributed values in $\{1,\ldots, k\}$ for some integer $k\geq 2$. Now I play the following game.
I start rolling the die and produce one integer in $\{1,\ldots,...

**26**

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**2**answers

1k views

### Runner's High (Speed)

I find the following mind-boggling.
Suppose that runner $R_1$ runs distance $[0,d_1]$ with average speed $v_1$. Runner $R_2$ runs $[0,d_2]$ with $d_2>d_1$ and with average speed $v_2 > v_1$. I ...

**5**

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**0**answers

249 views

### 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 $0\leq p<2^r$). ...

**4**

votes

**3**answers

131 views

### Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...

**1**

vote

**1**answer

626 views

### Matrix tic tac toe

So we have a 3x3 matrix and two players, a player that only puts in ones and a player that only puts in zeros. A coin flip is used to decide which player goes first. The first move is always to fill ...

**13**

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**0**answers

319 views

### 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 ...

**1**

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**0**answers

94 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 ...

**13**

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**1**answer

274 views

### Number guessing game with lying oracle

You are probably already familiar with the usual number guessing game. But for concreteness I restate it.
The usual game
The Oracle chooses a positive integer $n$ between 1 and 1024 (or any power of ...

**2**

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**1**answer

134 views

### Is this cycling problem computable?

We have a group of $n$ people who must make a journey of length $d$. They are to start together, and their goal is to arrive at the destination at same time. They have a single bicycle, which they ...

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**2**answers

1k views

### Can we make 101 almost perfect banknotes from 100?

Disclaimer. The practical execution of the algorithm in question might be illegal in certain jurisdictions, and is thus strongly discouraged by the poser of the problem.
This recent post on the ...

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**2**answers

233 views

### How to describe the common boundaries between regions in a infinite Sudoku?

This relates to the answer to a question "Who wins two player sudoku?" and this awesome blog.
A Sudoku can be $N \times N$ where $\sqrt{N}$ is a natural number because $N \times N / \sqrt{N} \times \...

**1**

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**0**answers

74 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 ...

**3**

votes

**1**answer

233 views

### Klarner's theorem

Klarner's theorem (http://mathworld.wolfram.com/KlarnersTheorem.html) says in a special case that you cannot tile a $10 \times 10$-board with $1\times 4$-tiles (that can also be rotated and used as $4 ...

**10**

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**1**answer

312 views

### Odds on rolling a rhombicosidodecahedron

This is more of a curiosity to me, but I'm sure I don't have the mathematical skills to answer it. That said... I took a look at several other posts with questions that relate to this one, but I ...

**11**

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**0**answers

280 views

### 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 ...

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**1**answer

624 views

### Who wins the Rubik's cube game?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier).
Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...

**34**

votes

**2**answers

3k views

### Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...

**17**

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**5**answers

1k views

### Number of collinear ways to fill a grid

A way to fill a finite grid (one box after the other) is called collinear if every newly filled box (the first excepted) is vertically or horizontally collinear with a previously filled box. See the ...