Tagged Questions

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

199 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 ...
447 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 ...
826 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 ...
159 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. $$\frac{1}{7}=0.(142857)$$ $$\frac{2}{7}=0.(285714)$$ ...
360 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 $360^\circ$) and $a ≤ b$. (Making $a$ either the acute ...
865 views

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 ...
765 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 ...
124 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.
142 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 ...
153 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 ...
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 ...
721 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 ...
618 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 ...
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? ...
14k 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, ...
15k 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 ...
3k 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 ...
728 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 ...
683 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 ...
249 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 ...
186 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 ...
381 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 ...
2k 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 ...
5k 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 ...
70 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 ...
103 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, ...
2k 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 ...
652 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 ...
52 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 ...
707 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, ...
13k 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 ...
394 views

Counting ordered triples of non-negative integers not greater than 100

Can we find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100$? Source:Regional Mathematics Olympiad India (2003) Thank you.I ...
159 views

Alphametic-like fraction equaling 1/2; uniqueness of solutions

This problem is kind of like those alphametics puzzles. The challenge is to assign each whole number from 2 to 9 to the letters in $$\frac{10^3A+10^2B+10C+D}{10^4+10^3E+10^2F+10G+H}$$ such that the ...
489 views

Solving math word problems WITHOUT brute force

How can we solve these problems withing using brute force? http://edhelper.com/math/multiplication51.htm
193 views

Very simple poker holdem question

N holdem hands (just 2 cards from a standard 52 cards deck)  are dealt to N players How to compute the probability that: -exactly k players have a pair ( Two cards of the same value e.g.: 7, 7). ...
1k views

What is the millionth decimal digit of the $10^{10^{10^{10}}}$-th prime?

What is the millionth decimal digit of the $10^{10^{10^{10}}}$th prime? (This prime, with more than $10^{10^{10}}$ decimal digits, is far larger than the largest "known" prime.) The answer should ...
3k views

write text using an equation

Well like the batman equations and equations for heart, I once saw a site that draws equation for whatever text you type....but now I can't find it. Does anybody know such a site? Also a general ...
718 views

Is there an algorithm to recover a crossword grid based on the clues alone?

Suppose that we have access to only the clues of a crossword puzzle along with the number of letters that the answers are supposed to be. Is there an algorithm that we can use to reconstruct the ...