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

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13
votes
8answers
1k views

What would be a good outdoor maths puzzle for children?

I have to find an interesting activity for some 11-year-olds moving to high school this year. It is supposed to take about 30-45 minutes, and I thought of having a mathematical theme. I can make a ...
24
votes
1answer
1k views

Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so ...
5
votes
1answer
226 views

Always a prime between $x$ and $x+cf(x)$

What is the asymptotically slowest growing function $f(x)$, such that there exists constants $a$ and $b$, such that for all $x>a$, there is always a prime between $x$ and $x+bf(x)$? $f(x)=x$ ...
12
votes
2answers
7k views

Expected Ratio of Coin Flips

If you flip a coin until you decide to stop and you want to maximize the ratio of heads to total flips, what is that expected ratio? Assuming that you want to maximize the ratio, meaning ...
0
votes
2answers
256 views

Anti Magic Square

Are the two examples of $4\times 4$ anti-magic squares currently on Wikipedia actually anti-magic squares under the definition given there? The examples are: $$\left[ \begin {array}{cccc} ...
0
votes
2answers
379 views

Moving last digit to first

Is it possible to find all positive integers $n$ such that if we move its last digit to the first digit, we get $2n$? I.e $2(a_m\cdot 10^m+\ldots +a_0)=a_0\cdot 10^m+a_m\cdot 10^{m-1}\cdots+a_1$
3
votes
2answers
563 views

Rubik's cube puzzle

If we cut along the plane orthogonal to the largest diagonal of a Rubik's cube, what is the maximum number of small cubes can we cut? I thought this should be $9$, but apparently this is not the ...
2
votes
1answer
329 views

any Math game for kids?

I need to plan a "game" for kids about 10-11 years old that involves mathematics and some physical activity or game. It must be short-time and not very difficult because it's a stage of a big game. ...
2
votes
1answer
1k views

Why is the Connect Four gaming board 7x6? (or: algorithm for creating Connect $N$ board)

The Connect Four board is 7x6, as opposed to 8x8, 16x16, or even 4x4. Is there a specific, mathematical reason for this? The reason I'm asking is because I'm developing a program that will be able ...
5
votes
1answer
83 views

Is there any relationship between the bounding box and the period of an oscillator in the Conway's Game of Life?

Is there any relationship between the bounding box and the period of an oscillator in the Conway's Game of Life? In particular I am interested in this case: what is the maximum period for an ...
6
votes
1answer
182 views

Constructing Magic Squares over $\mathbb{Z}$ from Magic Squares over $\mathbb{Z}_m$

A magic square over $\mathbb{Z}$ is an n x n matrix whose entries are $\{1, \ldots, n^2\}$, with the sum of every row and column identical (in particular, my magic squares are all normal, but the sum ...
7
votes
5answers
236 views

Understanding analysis/integration properties over $[0,1]$ and $[0,\infty)$ from an algebraic perspective?

I've noticed that in analysis we often treat the unit-interval $[0,1]$ differently from $[0,\infty)$, particularly in improper-integration (but certainly not limited to). By lieu of example, ...
0
votes
1answer
579 views

Create math (addition/subtraction) algorithm for 3 x 3 grid

I'd like to populate a "tic tac toe" board (grid of 3 x 3 squares) with four appropriate entries at which time a user will attempt to solve. I'm having a hard time coming up with a mathematical ...
2
votes
2answers
117 views

Is there a formula that will take me from a “given number of digits” to “largest possible integer”?

Apologies for what may be a simple question, but I don't have a background in mathematics beyond high school. I'm trying to work out a formula, that when given a number, say ...
29
votes
1answer
1k views

Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
2
votes
1answer
200 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
161 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 ...
24
votes
2answers
9k 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 ...
26
votes
5answers
3k 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
206 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 ...
6
votes
3answers
754 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 ...
11
votes
2answers
264 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
129 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
5k 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
408 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 ...
10
votes
2answers
22k 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
335 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
422 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
284 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
916 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
755 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
237 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
507 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
189 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 ...
1
vote
2answers
183 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
915 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
181 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
193 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
575 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
268 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
291 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$ ?
16
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!
9
votes
1answer
165 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
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
460 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
830 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
160 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
367 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 ...