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

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7
votes
1answer
2k views

Number Game: 31 - Winning Strategy?

My Maths teacher taught us how to play a game called 31 on Friday. Not once did my Maths teacher lose. I want to know why. I'll explain the game... 31 is a game between two people. Let's say you've ...
8
votes
7answers
3k views

Rainbow Hats Puzzle

Seven prisoners are given the chance to be set free tomorrow. An executioner will put a hat on each prisoner's head. Each hat can be one of the seven colors of the rainbow and the hat colors ...
5
votes
1answer
1k views

Handshake problem

I was given the following math puzzle which (I thought) has an interesting solution. A mathematician and her husband attended a party with $n-1$ other couples. As is normal at parties, handshaking ...
2
votes
1answer
188 views

showing a convex function s subharmonic

Given a $C^2$ convex function $f$ and $u$ a harmonic function in an open subset of $\mathbb{R^2}$, how can I show that $f(u)$ is sub-harmonic?
2
votes
0answers
287 views

Does this approach to the Collatz Conjecture make any sense.

I was playing around with the Collatz Conjecture and came up with the following: Take any positive integer. If it's odd, multiply by three and add one. If it's even, divide by two. Repeat ...
6
votes
3answers
615 views

Puzzle: A coin rolls without slipping around another coin

If a coin rolls without slipping around another coin of the same or different size, how many times will it rotate while making one revolution? The proof given is like this: Cut the curve ...
5
votes
1answer
311 views

A cute geometry problem about angle trisectors.

Here is a cute geometry problem I saw some time ago. I know the solution, I just wanted to share ;-) (Please, don't be mad at me.) Consider an acute triangle $\triangle ABC$. Let $AP$, $AQ$ and ...
22
votes
5answers
951 views

a big number that is obviously prime?

I once heard it asserted that $91$ is the smallest composite number that is not obviously composite. The reasoning was that any composite number divisible by $2$, $3$, or $5$ is obviously composite, ...
0
votes
1answer
228 views

Are actions in the $3\times 3\times 3$ rubik cube a group?

Are actions in the $3\times 3\times 3$ rubik cube a group? You can see here Rubik's Cube Not a Group? that $4\times 4\times 4$ rubik cubes or higher arent groups. But what about $3\times 3\times ...
6
votes
1answer
326 views

Lightbulbs Puzzle

There are $10$ lightbulbs in a row, on or off. How many combinations of on and off lightbulbs can we have if no two turned on bulbs can be next to each other? It seems like it forms a Fibonacci ...
2
votes
1answer
680 views

Expected Value of the Difference between 2 Dice

What is the expected value of the absolute difference between 2 N faced dice? What about the difference between 2 dice one with N faces and one with M faces? While finding the expected value ...
0
votes
1answer
76 views

Mental math tip needed; moving decimal around on larger and smaller numbers?

When I do calculations I like to round off to say nanometers for wavelength. That means I need it in the form $whatever \times 10^{-9}$. The problem here is that regardless of the number I manage to ...
0
votes
2answers
232 views

Given a victory condition and a set strategy, what are the chances of winning on a given turn in a game of Magic: The Gathering?

Tl;DR: You have winning cards. To win, you must be able to play those cards, and have them in your hand. Your hand is randomly drawn. When might you win? How could find the answer to this (very ...
2
votes
2answers
2k views

Chords on a Circle

If there are N points on a circle, and you draw a chord between each of them, how many regions is the circle subdivided into? I'm not quite sure where to begin. I know that there are at least n ...
17
votes
4answers
2k views

Probability of Coins Flips

Person A has 5 fair coins and Person B has 4 fair coins. Person A wins only if he flips more heads than B does. What is the probability of A winning? When I initially thought about the ...
6
votes
1answer
1k views

Improving Von Neumann's Unfair Coin Solution

If a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following ...
4
votes
2answers
557 views

How do you calculate the average number of lands in your hand in a game of MTG?

(This is actually a question and a half; Please tell me if this should be done otherwise) I have a magic deck of 60 cards. Some of them (say 25) are lands. The first card I draw has a 25/60 chance of ...
5
votes
2answers
260 views

Number of Ones Puzzle

$f(n)$ is a function counting all the ones that show up in the sequence $1, 2, 3, ..., n$. IE $f(1)=1$, $f(10)=2$, $f(11)=4$ etc. Discounting the trivial case $f(1) = 1$, when is the first time ...
2
votes
1answer
1k views

Battleship probability matrix

Consider a 10 x 10 Battleship grid that hides a single ship of length = 3. This ship can be placed vertically or horizontally in any of the 100 cells. The problem is to get the 10 x 10 probability ...
12
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 ...
22
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
217 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 $x+bf(x)$? $f(x)=x$ ...
10
votes
1answer
4k 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
219 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
310 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
524 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
313 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
902 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
80 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
180 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
234 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
533 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
110 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 ...
28
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
191 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
153 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
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
202 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
551 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
233 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
124 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
398 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
16k 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
386 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 ...