# Questions tagged [recreational-mathematics]

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

**24**

votes

**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 ...

**4**

votes

**0**answers

168 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

107 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**

votes

**1**answer

158 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 ...

**0**

votes

**0**answers

31 views

### Decoupling/Simplifying a long algebraic expression as a function of 5 parameters

I have the following variables defined by 5 parameters $(g_1, g_2, \kappa_1, \kappa_2, \Gamma)$:
$$
u_1=36g_1^{2}(2p-3\kappa_2)+\big(36g_2^{2}+(2p-3\kappa_1)(2p-3\kappa_2)\big)\big(4p-12\kappa_+\big)\...

**12**

votes

**0**answers

249 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**

vote

**0**answers

89 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**

votes

**1**answer

218 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**

votes

**1**answer

123 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 ...

**13**

votes

**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 ...

**1**

vote

**0**answers

238 views

### Mathematical expressions involving weird constants [closed]

I hope this is a question that fits here and is not duplicated. Also that is clear since it can be a little ambiguous.
I was wondering if you know deep expressions, theorems, isomorphisms or ...

**1**

vote

**2**answers

205 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**

vote

**0**answers

66 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

179 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**

votes

**1**answer

280 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**

votes

**0**answers

269 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 ...

**18**

votes

**1**answer

424 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). ...

**31**

votes

**2**answers

2k 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 ...

**14**

votes

**4**answers

812 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 ...

**3**

votes

**1**answer

226 views

### How to prove that all these sets equal $\mathbb N$?

Let us take two natural numbers, for example $a_1=2$ and $a_2=7$. Multiply to obtain $a_3=2 \cdot 7=14$. Obtain $a_4$ as $a_2 \cdot a_3=98$. Then $a_5$ as $a_3 \cdot a_4=14 \cdot 98=1372$. Repeat ...

**3**

votes

**2**answers

156 views

### Choose some natural number and take $k$-th power of its digits and add that. Repeat that. How many cycles will be there?

I have read in a book that has very much of a recreational flavour that if we take any natural number and square its digits and add that then if we repeat that long enough that only two outcomes can ...

**0**

votes

**1**answer

45 views

### The minimal value of k for making a inequality true [closed]

I was wondering...
What is the minimal value of $k$ for making the following inequality true.
$(\sum_{x=1}^k (n-x)*x) > nˆ2$.
Is there a way to know that? And how can I prove it?
I was ...

**0**

votes

**1**answer

171 views

### Could the sequence A287326 be generalized in order to receive expansion of natural power n>3? [closed]

The sequence https://oeis.org/A287326 - is Binomial distributed triangular array, that shows us necessary items to expand perfect cube $n^3$. Summation of $n$-th row of Triangle A287326 from $0$ to $n-...

**66**

votes

**23**answers

17k views

### Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...

**1**

vote

**0**answers

247 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 ...

**4**

votes

**1**answer

201 views

### Unusual matrix product associated with non-transitive dice

Not long ago, the Puzzle Corner of the magazine MIT Technology Review asked for a set of $N$ dice that are non-transitive in the sense that there is a cyclic ordering on them, in which each die beats ...

**-2**

votes

**1**answer

416 views

### Is there a finite or infinite number of these primes? [closed]

As I think much about primes and usually that ends up with just a formulation of some new/old conjectures I came to the following idea.
First, an example. Take a prime $1277$. Now increment every ...

**1**

vote

**0**answers

18 views

### Building sequences with the help of sum-of-digits function and concatenation and occurence of primes in them

I found very simple way of building strictly increasing sequences of strictly positive integers by starting with whatever strictly positive integer you want and with the help of sum-of-digits function ...

**2**

votes

**0**answers

161 views

### From prime to prime by squaring the digits [closed]

I took prime $131$, squared digits of it and wrote them in natural order as they appear, from left to right, and obtained $191$, then I obtained $1811$ by the same procedure, and then $16411$ and then ...

**4**

votes

**1**answer

178 views

### Bases closed under multiplication

Let us say that a Hamel basis $H$ in an algebra $A$ is closed under multiplication, if $ab\in H$ whenever $a,b\in H$. It is an easy observation that if $A$ has such a basis then there it also has a ...

**3**

votes

**1**answer

235 views

### Evaluation of an interesting Integral [duplicate]

Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically.
$$f(n) = \frac{2}{\pi} \int_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \...

**3**

votes

**2**answers

97 views

### Game of Roller Blocks

The game of roller blocks is played on a rectangular board of size $m\times n$. For example, if $m=5$ and $n=4$ we have $5 \times 4=20$ different gridpoints/coordinates. Just so the conventions are ...

**0**

votes

**1**answer

117 views

### Can $\{0,1,2\}^n$ be partitioned into $3^{n-1}$ three-element sets where no two components are equal?

Inspired by the card game SET, the following question came up:
Laying out all 81 cards, can one find 27 Sets (in the sense of the game), all of which are Sets with four different features?
To be ...

**4**

votes

**0**answers

58 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 ...

**50**

votes

**2**answers

16k views

### Is there winning strategy in Tetris ? What if Young diagrams are falling?

Question 1
Is there a winning strategy (algorithm to play infinitely) in Tetris,
or is there a sequence of bricks which is impossible to pack without holes?
Consider generalized Tetris with Young ...

**1**

vote

**1**answer

108 views

### Share of fortunate people in some pie splitting setting

(This question is a follow-up on an older one.)
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for ...

**3**

votes

**3**answers

251 views

### Conics, string art, and Bezier-like curves

It is well documented that certain string-art patterns generate quadratic Bezier curves: let $x, y_1, y_2$ be three points in $\mathbb{E}^2$, consider the family of line segments joining $x + (1-t) (...

**2**

votes

**1**answer

101 views

### Limit of biggest share of the pie

A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest ...

**6**

votes

**1**answer

176 views

### Maximum $2$-D bootstrap percolation time for $n$ points on an $n\times n$ grid

I hesitate to ask this question here, but since it remained unanswered after a bounty on MSE, I ask it here with some reservation.
Is the maximum bootstrap percolation time for $n$ points on an $n\...

**1**

vote

**1**answer

65 views

### Are convex combinations of 0-1 Pareto efficient vectors efficient?

Let $Y$ be any subset of $\{0,1\}^n$ for $n\geq3$. A vector $\alpha\in$ $Y$ is Pareto efficient if there is no $\beta\in$ $Y$ such that $\beta_i$ $\geq$ $\alpha_i$ for each $i\in\{1,...,n\}$ and $\...

**1**

vote

**1**answer

105 views

### Expectation of changing the gift choice [closed]

Suppose we are given two boxes, with one of gift valued $n$ dollars and the other one valued twice as much. We can pick a box, and after open it we have the choice of switching to another box. Shall ...

**1**

vote

**1**answer

228 views

### Can a Lucas Carmichael number also be a Smith number?

I was wondering if a Smith number can be a Lucas Carmichael number.
Is there a proof that there is no such number?
If there is such a number, can you tell me it and how you got it?
I have written a ...

**5**

votes

**2**answers

387 views

### Can we surround a non-rectangular area with Lego fences?

My children have some Duplo fences, these you have to put down on two points, and at both ends they extend a little where you can connect several to surround some area.
So a fence is described by a ...

**16**

votes

**0**answers

364 views

### 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 ...

**11**

votes

**3**answers

488 views

### 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, ...

**22**

votes

**2**answers

989 views

### $x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?

This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...

**4**

votes

**0**answers

135 views

### 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 \...

**5**

votes

**2**answers

400 views

### Separating Heavier from the Lighter Balls

This Question was originally posted Here, where I'm more interested in the methods for manual solutions yielding $n$ or less moves on average.
I wanted to post it here as well, to see what the people ...

**5**

votes

**1**answer

409 views

### 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 ...

**82**

votes

**22**answers

23k views

### 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.
Added 2017-04-01 Anything new in 2017?