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

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7
votes
2answers
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 ...
1
vote
3answers
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 ...
17
votes
3answers
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 ...
8
votes
1answer
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. $$\frac{1}{7}=0.(142857)$$ $$\frac{2}{7}=0.(285714)$$ ...
1
vote
2answers
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 ...
2
votes
1answer
865 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
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 ...
3
votes
1answer
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.
-1
votes
3answers
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 ...
2
votes
1answer
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 ...
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
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 ...
3
votes
0answers
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 ...
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? ...
163
votes
2answers
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, ...
3
votes
5answers
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 ...
59
votes
1answer
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 ...
2
votes
2answers
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 ...
0
votes
1answer
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 ...
6
votes
1answer
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 ...
4
votes
0answers
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 ...
16
votes
2answers
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 ...
9
votes
4answers
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
3
votes
0answers
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, ...
5
votes
4answers
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 ...
21
votes
1answer
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 ...
0
votes
1answer
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 ...
14
votes
3answers
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, ...
195
votes
4answers
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 ...
4
votes
2answers
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 ...
2
votes
2answers
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 ...
4
votes
2answers
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
2
votes
2answers
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). ...
23
votes
2answers
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 ...
0
votes
2answers
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 ...
3
votes
4answers
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 ...
3
votes
2answers
3k views

How to construct magic squares of even order

Could someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?
5
votes
2answers
394 views

Famous Finite Sets [closed]

What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx ...
1
vote
1answer
89 views

Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...
13
votes
3answers
1k views

The Math behind rotation puzzles?

In the game Machinarium, there is the following puzzle where the goal is to get all of the green points on the green area by rotating them along any of the 3 circles engraved on the background plate. ...
19
votes
1answer
465 views

Extracting individual race results from Mario Kart final scores

In Mario Kart, one "cup" involves 4 races, and after every race each racer gets points awarded based on what place they came in (better rank means more points). After playing it enough I grew curious ...
4
votes
1answer
464 views

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?
19
votes
4answers
2k views

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff ...
7
votes
5answers
872 views

Cauchy-Schwarz inequality and three-letter identities (exercise 1.4 from “The Cauchy-Schwarz Master Class”)

Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following: For all positive $x$, $y$ and $z$, one has $$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + ...
2
votes
2answers
723 views

In Towers of Hanoi (with 3 sticks and n disks without backtracking), do all legal sequences of moves reach the solution?

Updated Question : How to show that in TH we never reach a state where there are no paths to the solution? ( without reversing moves, as if reversing is allowed this becomes trivial ) Edit : Thanks ...
4
votes
1answer
679 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
14
votes
2answers
886 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
4
votes
0answers
178 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...