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

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10
votes
1answer
270 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
654 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 ...
5
votes
6answers
967 views

Math Jeopardy Answers and Questions (community wiki) [closed]

This question is just for fun, and should be community wiki, but I don't know how to set that here. Just as on Jeopardy, give an answer as the clue. Bonus points if the answer is in the Jeopardy ...
10
votes
1answer
607 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
543 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
185 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
423 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
181 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
161 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 ...
12
votes
8answers
808 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
162 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
139 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
406 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
198 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
251 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$ ?
8
votes
3answers
842 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
113 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 ...
8
votes
1answer
126 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
193 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
354 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
725 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
126 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
278 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
529 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
398 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
119 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.
0
votes
3answers
139 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
135 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 ...
3
votes
1answer
519 views

Does the math behind this joke work? [closed]

Is the mathematics implied by this joke accurate? As in, is the mathematical portion of the pun a correct mathematical statement? Q: Why do motorcycle gang members use their motorcycles to get to ...
6
votes
1answer
642 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 ...
1
vote
0answers
496 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? ...
138
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, ...
2
votes
4answers
6k 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 ...
52
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
501 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
473 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 ...
6
votes
1answer
177 views

How many states in the game of hex?

I am trying to calculate how many unique states are possible to be in during a game of hex. The upper bound for an $n\times n$ board is $3^{n^2}$. This is ignoring gameplay and simply considering ...
4
votes
0answers
165 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
14
votes
2answers
314 views

Who has the upper hand in a generalized game of Risk?

So, I played a game of Risk the other day for the first time since I was very little. I was frustrated to discover that I couldn't compute (at least not in my head) whether the attacker or the ...
9
votes
3answers
962 views

Secret santa problem

We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem. We have 4 people ...
0
votes
2answers
1k views

Solving an inequality, the equality is facing the wrong way?

I'm suppose to solve a problem that goes like this. The graph for the following function f given by $f(x) = 115.82 \cdot 0.94^x + 5$, with $x \geq 5$, gives the temperature of the water after ...
0
votes
1answer
66 views

Probability, why my solution doesn't work out? (P of drawing a pair)

The task is simple, the probability of drawing a pair of cards. You draw two cards from a stack, what is the chance that you get two kings or two fours. My idea was the following. There are 13 ...
3
votes
0answers
100 views

Least characters in a numerical representation of integers

I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, ...
5
votes
3answers
822 views

Two numbers, two mathematicians puzzle

This is one of the most beautiful and difficult puzzles I have encountered. I have talked to several people, but I still don't know the solution - I do however know that the solution exists. Someone ...
18
votes
1answer
516 views

How to create mazes on the hyperbolic plane?

I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the ...
0
votes
1answer
46 views

Drawing three cards of different type

I did draw a tree and found out that this can be done in 24 different ways. But is there a quicker formula? There are a total of four different types of cards, as you know. And we are to draw three of ...
13
votes
3answers
647 views

How many trees in a forest?

Some time ago I met a forester. He told that there are only larches and spruces in his forest. He also said that there are exactly $10$ spruces at the distance of exactly 1 km from each larch. Next, ...
172
votes
4answers
10k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...