Puzzles, curiosities, brain teasers and other mathematics done "just for fun".

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2
votes
1answer
190 views

Number of combinations of $k$ numbers using arithmetic operations

What is the maximum number of positive rational values that can be obtained by combining $k$ positive integers using only addition, subtraction, multiplication, division, and parentheses? Assume that ...
3
votes
2answers
151 views

How to mathematically color the regions bounded by a parametric curve?

Usually, if an implicit equation F(x,y) = 0 defines a curve (or curves) on the x-y plane, then we can use the inequalities F(x,y) < 0 or F(x,y) > 0 to color the regions bounded by the curve (or ...
22
votes
2answers
7k views

How to tell if a Rubik's cube is solvable

How can I determine if a certain Rubik's cube, that is in a certain state, is solveable? By "certain state" it could mean that the cube has been dismantled and put together again. And in my experience ...
25
votes
5answers
2k views

Is Mega Millions Positive Expected Value?

Given the rapid rise of the Mega Millions jackpot in the US (now advertised at \$640 million and equivalent to a "cash" prize of about \$448 million), I was wondering if there was ever a point at ...
8
votes
0answers
200 views

Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]

Possible Duplicate: Infinite tetration, convergence radius Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that ...
3
votes
2answers
546 views

$3 \times 3 $ Magic Square of Squares

On picture below is three-by-three magic square in which seven of the entries are squared integers, found by Andrew Bremner of Arizona State University (and independently by Lee Sallows of the ...
10
votes
2answers
231 views

How to estimate the number of articles on Wikipedia using the “random article” function?

There is a Wikipedia-type website of a fixed size of $S$ number of articles. You start at any article on Wikipedia. You then start to press the "random article" button and count the number of times ...
2
votes
1answer
122 views

Geometrical combinatorics

This question was inspired by Rush Hour game: You have a 6x6 grid, 12 pieces of size 2, and 4 pieces of size 3. A piece can be placed on the grid either horizontally or vertically. The pieces can't ...
1
vote
1answer
4k views

Combinations of a penny, nickle, dime, and quarter

You have one penny, one nickle, one dime, and one quarter. How many different amounts of money can you make using one or more of these coins? Please help me! I'm having trouble! Im having trouble I ...
3
votes
1answer
395 views

Mathematics From Futurama

Dear Professor Farnsworth, We at D.O.O.P are trying to mathematically model a rocket ship fueled by your employee Leela's pet Nibbler's pooped Black matter. Obviously this rocket ship is fueled by ...
5
votes
4answers
1k views

Calculating Gröbner basis for Sudoku

I'm trying to write a program that solves sudokus using a Gröbner basis. I introduced 81 variables $x_1$ to $x_{81}$, this is a linearisation of the sudoku board. The space of valid sudokus is ...
9
votes
2answers
15k views

What is the probability that a solitaire game be winnable?

By "solitaire", let us mean Klondike solitaire of the form "Draw 3 cards, Re-Deal infinite". What is the probability that a solitaire game be winnable? Or equivalently, what is the number of ...
2
votes
2answers
328 views

A sequence of nested fractions with a counter-intuitive limit

Given $a,b\in\mathbb C$, let us construct the following sequence: $$\begin{align} a+b&=a+b\\ \cfrac a{a+b}+\cfrac b{a+b}&=1\\ \cfrac a{\cfrac a{a+b}+\cfrac b{a+b}}+\cfrac b{\cfrac ...
2
votes
2answers
364 views

Opinions on foundational math materials to teach 8th grade, 9th grade kids at a Summer Camp

I have been asked to teach mathematics/physics to a few 8th grade/9th grade kids for a summer camp. I have been thinking about it and I realized that I could go about it in two ways: One of the ways ...
10
votes
1answer
278 views

The motorcyclist's challenge

n walkers ${A}_{i}$ ($i=1,2,...,n$) start out from X to Y simultaneously with constant speeds ${a}_{1}<{a}_{2}<...<{a}_{n}$. At the same time, motorcyclist M with speed $m=1$ starts out from ...
5
votes
3answers
1k views

What equity is necessary to offer the doubling cube in Backgammon (dice game)

Edit: This question is a lot shorter than it is. Don't get intimidated. If you know backgammon, just skip to question 2. In Backgammon, each game is played for one point (or one dollar) between two ...
11
votes
1answer
776 views

Solution to Locomotive Problem (Mosteller, Fifty Challenging Problems in Probability)

My question concerns the solution Professor Mosteller gives for the Locomotive Problem in his book, Fifty Challenging Problems in Probability. The problem is as follows: A railroad numbers its ...
2
votes
5answers
655 views

Help understanding proof of generalization of Cauchy-Schwarz Inequality

I'm having trouble with an exercise in the Cauchy Schwarz Master Class by Steele. Exercise 1.3b asks to prove or disprove this generalization of the Cauchy-Schwarz inequality: The following is the ...
5
votes
2answers
207 views

The area problem!

We have to find area of the quadrilateral formed by joining the point of intersection of the four quarter circles that are drawn from each vertex in a unit square. $\hspace{4cm}$ The challenge is ...
2
votes
2answers
477 views

Jack's birthday riddle

Anytime each of three consecutive months has exactly four Fridays, Jack's birthday will fall in one of those three months. Which month is that?
4
votes
2answers
182 views

A game of numbers: When can we have 2011?

Two friends are playing a game. In every turn, after one of them says a number $k$, the other one has to say a number in form $a\cdot b$ where $a,b\in \mathbb{N}$ such that $a+b=k$ holds. The game ...
0
votes
2answers
172 views

How to get maximum number 500 out of one variable

Im designing a website and I need to make a sum which will allow the result to be a maximum of 500. So the variable can be anything from '0' to 'a billion' and second number is 12. So just by basic ...
13
votes
8answers
850 views

Approximation of $e$ using $\pi$ and $\phi$?

$$e \approx \frac{4 \phi +3 \pi-5}{4}$$ where $~\phi~$ is a Golden ratio . Is it possible to construct better approximation of $e$ using $\pi$ , $\phi$ and integers ?
5
votes
0answers
175 views

$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
1
vote
2answers
165 views

Generate Magic Square Terms

I was looking at the wikipedia page for the "Magic Square": http://en.wikipedia.org/wiki/Magic_square and it gave this equation to generate the numbers for a given square: ...
6
votes
1answer
493 views

Soccer and Probability

MOTIVATION: I will quote Wikipedia's article on a soccer goalkeeper for the motivation: Some goalkeepers have even scored goals. This most commonly occurs where a goalkeeper has rushed up to the ...
1
vote
1answer
222 views

Cube nets hexomino tilings.

I am looking for an ~12x12 rectangle (small holes and small obtrusions are okay) made entirely of cube net hexominos. It is my understanding that perfect rectangles, in general, are not possible ...
11
votes
1answer
274 views

Is there a real-valued function $f$ such that $f(f(x)) = -x$?

Is there a function $f\colon \mathbb{R} \to\mathbb{R} $ such that $ f(f(x)) = -x$ ?
13
votes
3answers
1k views

Coffee Break Riddle [closed]

Here's a little brain teaser, for your coffe break: $$ 62-63 = 1 $$ Move only one digit to make it right! Have fun!
0
votes
0answers
115 views

a follow up question to the birthday-paradox question.

The previous question. My goal is to find a function of the difference (error) between F and G generally. F = $\Pi_{0}^{n-1}\frac{365-b}{365}$ G = $ \frac {364}{365}^{\frac {n^2-n}{2}}$ Now F ...
9
votes
1answer
147 views

A game played on graphs by “flipping” the state of a vertex and its neighbors

This is a well-known game: We are given a finite undirected graph $G=(V,E)$ whose vertices are labeled by "0". At each turn, we pick a vertex, and then it and all its neighbors flip their label (0 ...
7
votes
2answers
197 views

Density of black cells in rule 110

Is there a way to compute the limit of the ratio (number of black cells)/(number of white cells), in the rule 110 or rule 30 automaton? With initial state = 1 black cell. Simulation of first 120000 ...
1
vote
3answers
400 views

Tiling with polyominos

How to prove or disprove that if a polyomino tiles the plane, it must also be able to perfectly tile some larger polyomino, which also tiles the plane? A polyomino is finite set of unit squares ...
16
votes
3answers
789 views

Is the Collatz conjecture in $\Sigma_1 / \Pi_1$?

Prompted by some of the comments on this question, I'm wondering if anything is known about the place of the Collatz Conjecture in the arithmetic hierarchy. More specifically, is Collatz known to be ...
8
votes
1answer
147 views

cyclic permutations of periods of recurring fractions

In base 10, the recurring bits of the fractions $\frac{1}7,\ldots,\frac{6}7$ are cyclic permutations of each other. e.g. $$\frac{1}{7}=0.(142857)$$ $$\frac{2}{7}=0.(285714)$$ ...
1
vote
2answers
334 views

Find the angle between two lines using a compass and straight edge.

I've drawn two random, non-parallel, straight lines on a plane. They cross over, forming two angles, a and b, where (a + b + a + b) = 1 (or 360°) and a ≤ b. (Making a either the acute angle or a right ...
2
votes
1answer
658 views

Dot on forehead riddle

A riddle was posted in this mathoverflow question: http://mathoverflow.net/questions/85439/how-does-intuitionism-handle-this-riddle A riddle: You and another person are kidnapped and knocked ...
2
votes
3answers
549 views

A binet-like formula for $\small 1 , 2 , 3 , 6 , 9 , 18 , 27, 54, 81, \ldots $?

This is more a "recreational" problem. By another question I came to the question for a closed-form-formula for this sequence $\small 1 , 2 , 3 , 6 , 9 , 18 , 27, \ldots $ which is just the mixture of ...
3
votes
1answer
123 views

Number of knots possible with length L string

What is the asymptotic growth in L for the numer of topological different knots possible using a length L closed string of radius 1? In 3 dimension euclidean space.
-1
votes
3answers
141 views

From a mathematical point of view is it optimal in no limit texas hold em to play with more money than less?

I noticed the other time a friend of mine went to a casino and bought in 100 dollars for a 1-2 table. Other players had heavier buy ins. I have received two opposing arguments. One says that buying in ...
2
votes
1answer
146 views

Optimal polyomino induced coloring

Which polyominos (with orientation) of $n$ squares, requires the least number of different colors, $c(n)$, such that if this polyomino is placed anywhere on an optimally colored infinite square grid ...
22
votes
4answers
1k views

Something that I found, and would like to see if it's known.

Well I am quite sure it's known (I mean number theory exists thousands of years), warning beforehand, it may look like numerology, but I try not to go to mysticism. So I was in a bus, and from ...
6
votes
1answer
686 views

Math and Logic of Infinite Chess

Hello could you help me in this... Two players (White and Black) are playing on an infinite chess board (extending infinitely in all directions). First, White places a certain number of queens (and ...
2
votes
0answers
560 views

Four candle problem: Using candles as timers

The candles each take one hour to burn completely. Cutting off bits of the candles is forbidden, but the candles are placed on a raft of fork handles so they may be burnt at both ends (e.g. to time ...
1
vote
0answers
110 views

Boggle dice set letter distribution algorithm [duplicate]

Possible Duplicate: Boggle letter probability What algorithm have the creators of the word game Boggle used to come up with these dice sets? ...
154
votes
2answers
13k views

Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual

An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, ...
3
votes
4answers
10k views

Probability of winning a prize in a raffle

My work is having it's annual Christmas raffle today. 1600 tickets have been sold, and there are 40 prizes to win. I have bought ten tickets. What are the odds I will win a prize? While an initial ...
56
votes
1answer
2k views

What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
2
votes
2answers
607 views

expected number of guesses in game of mastermind

Sequence X consists of N pegs each randomly assigned on of M colours. One each go, the player places N coloured pegs in a line. If they exactly match sequence X, the game terminates. Otherwise, the ...
0
votes
1answer
581 views

Roulette betting options

I'm learning binomial distributions and I came across this problem: Let r.v X be winning from a bet on a split in roulette and Y be be winnings from a bet on red color. X = 17 (2/38 chance) and -1 ...