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Questions tagged [recreational-mathematics]

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

24
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2answers
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
0answers
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
3answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
4answers
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
1answer
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
2answers
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
1answer
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
1answer
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
23answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
2answers
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
3answers
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
2answers
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
0answers
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
2answers
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
1answer
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
22answers
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?