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

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10
votes
2answers
181 views

about a ninth-grade geometry problem

My brother asked me this problem, and he is studying ninth-grade. I can't solve it using primitive tools of pure geometry. Hope someone can give me a hint to solve it. Thanks. Given a circle $(O, ...
0
votes
0answers
26 views

Real life illustration of the fact that rationals have measure zero

I wonder if there's any real world phenomenon that reflects the mathematical fact that $\Bbb Q^k$ has Lebesgue measure zero in $\Bbb R^k$, or put another way, the likelihood that we get a rational ...
1
vote
3answers
74 views

Is there a possible mathematical solution for this? [on hold]

I have what might be considered an odd question. I want to see if I can find a formula/equation to help me with the following. I'm working in a software package that we are using to calculate fees. ...
16
votes
2answers
589 views

Minimal generating set of Rubik's Cube group

The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $...
29
votes
2answers
2k 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 ...
1
vote
1answer
44 views

What is the largest prime $p$ such that the decimal expansion of $1/p$ repeats with period 2017?

By this discussion on John Baez's Google+ feed, the primes $p$ such that the decimal expansion of $1/p$ repeats with period 2017 are exactly those primes which occur in the prime decomposition of $10^...
1
vote
1answer
27 views

Choosing a Non-Confederate Volunteer

A magician is performing in front of a large crowd (around a 100 people, say) and wants a volunteer for a trick. The magician knows that he has no confederates in the crowd, but the crowd doesn't. How ...
0
votes
1answer
37 views

counting walks on a finite interval

Find the number of random walks on the integers $\{1, 2, ... m\}$ of length $n$. (For the purposes of this question a "random walk" is a sequence $\{a_i\}$ such that $a_i - a_{i+1} = \pm 1$.)
1
vote
0answers
70 views

Maximal unit lengths in 3D with $n$ points.

Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. ...
1
vote
4answers
96 views
2
votes
1answer
78 views

Given a population of fish with exponential growth, what is the optimal strategy for fishing?

Suppose we have a population of fish, say $10000$, with an exponential growth each year of $30\%$. If we want to collect as many fish as possible in, say 10 years, a natural question to ask is: ...
0
votes
0answers
96 views

Evaluating integral $\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx$ [duplicate]

How can I evaluate the following integral $$I=\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx$$ where $f(x)$ is a probability density function and $\lim_{x\to 0}$ xf(x) = $\lim_{x\...
0
votes
1answer
64 views

Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b} $$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
6
votes
1answer
2k views

Concrete Mathematics - Towers of Hanoi Recurrence Relation

I've decided to dive in Concrete Mathematics despite only doing a couple of years of undergraduate maths many years ago. I'm looking to work through all the material whilst plugging gaps in my ...
3
votes
1answer
1k views

Concrete Mathematics - The Josephus Problem

I've been going through Concrete Mathematics and have a question on the inductive proof for the Josephus problem. Recurrent relation: $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$...
4
votes
1answer
69 views

Ways to squeeze $e$ by hand

Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where $b-a&...
7
votes
1answer
110 views

The grey area is equal to the white area

Problem. Show that the sum of the areas of the white regions is equal to the sum of the areas of the grey regions. All the angles between consecutive chords are $45^\circ$. A solution (not totally ...
1
vote
2answers
652 views

Determining Formula (Game Mechanics)

WARNING I believe that the data below has errors in the defense strength, so is therefore not solvable. I will update it when I have more information. Thank you. I play a game (Empire: Four ...
0
votes
0answers
44 views

Strategy of ball math game

Found math game: http://www.emathhelp.net/math-games-and-logic-puzzles/rgbw/ What is a strategy for it? I can make 15 white balls max. Any thoughts?
2
votes
0answers
37 views

Trying to understand the properties of a combinatorial game

Consider the following game for $n \geqslant 3$, which I will demonstrate with $n=4$: draw an $n$-gon and place the value 0 at each of the vertices, except one vertex which we circle and place the ...
1
vote
2answers
399 views

Arithmetic progressions of perfect powers

Find the largest positive integer $n<100$, such that there exists an arithmetic progression of positive integers $a_1,a_2,...,a_n$ with the following properties. $1)$ All numbers $a_2,a_3,...,a_{...
-3
votes
1answer
36 views

Given $ x^m=y^m ; so \; x=y \;or\; -y$ when m is even. Now $ 2^0 = 3^0 \; but \; 2 \neq 3 $ . How to reason mathematically

The question may sound silly but is there a simple logic to counter the paradox.I will be glad to know if there is. Thank You. Edit: x,y $\in R\;\; x,y \neq 0$, m is a integer. Now x = y when m is ...
0
votes
0answers
42 views

Edge-matching icosahedron puzzle

Color the edges of an icosahedron with 4 colors so that all 20 triangles have a distinct set of colors. Color the edges of an icosahedron with 6 colors so that all 20 triangles have a distinct set ...
147
votes
19answers
11k views

Mental Calculations

This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky. The problem presented on a blackboard requires computing the ...
25
votes
3answers
759 views

Sudoku with special properties

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of ...
-1
votes
0answers
25 views

Translate thinking into a equation. Finding the lesser value

First of all, bear with me if I name things wrong as I'm null at math. For the same reason please add explanation for dummies in common language if you add an answer with math notation :D I'm trying ...
0
votes
0answers
8 views

Simple Composite of Relations

My lecturer has given this simple composite of relations question; R = {(1,2), (3,4), (2,1)} S = {(2, 1), (5, 3)} R o S = {(1, 1)} is the answer i acquire. he acquires the answer R o S = {(2,2), (5, ...
0
votes
0answers
27 views

How to Solve a Function Given Some of its Solutions

Suppose you have a function that defines a series. And suppose you know Some (not all) of the elements of that series. For example, you know your function is n/J, where n is for all positive integers ...
0
votes
1answer
30 views

Which vertex-transitive planar graphs represent non-self-intersecting polyhedra?

Consider an infinite planar graph with the following properties. Its vertices all have valence $3$. The faces all have $5$ edges. Now put it in cartesian space and require that the faces are all ...
8
votes
2answers
158 views

The Hardest Sudoku Puzzle

I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers ...
5
votes
2answers
113 views

Conjugates and commutators for twisty puzzles — so what?

This question isn't just rhetorical. I want to know what I'm missing. Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators ($...
20
votes
6answers
707 views

Non-trivial “I know what number you're thinking of”

Consider the following 'trick' (WARNING: very lame) Think of a number. Multiply this number by two. Add four. Divide the number by two. Subtract the number you were originally thinking of. I guess ...
1
vote
2answers
30 views

Probability advantage on order dependency puzzle

I stumbled across this problem on the NSA website, and I am having trouble grappling with the solution. I would expect that the probabilities for each would be equal, as each square would have an ...
1
vote
3answers
90 views

find the the greatest value of $m$ such that $lcm(1,2,3,..,n)=lcm(m,m+1,..,n).$

I am stuck and unable to proceed. Value of n can be very large. For eg:if $n=6,lcm(1,2,...,6)=60$, so answer will be $4$ in this case. Since $lcm(2,3,4,5,6)=60,lcm(3,4,5,6)=60,lcm(4,5,6)=60$ and $...
7
votes
3answers
199 views

Bicycle Route Optimization Puzzle

I tried to make some expressions about where each person stops the bike, but I couldn't solve it :( There are three people who would like to cross the road. It takes $a$ minutes for the first person ...
9
votes
1answer
118 views

Intuitive ways to get formula of binomial-like sum

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
0
votes
1answer
40 views

Making ease-out-bounce formula have a linear start

I'm using a bounce ease out formula, the code for it: https://github.com/jesusgollonet/processing-penner-easing/blob/master/src/Bounce.java#L9. The function is copied here: ...
0
votes
1answer
21 views

Create a formula to compare different exchange rates (one with a fee)

While looking at exchange rates for an upcoming vacation, I decided to brush up on some old math but wanted to make sure I was thinking about it correctly. $B_1$ charges a rate for USD to EUR ...
6
votes
3answers
2k views

Are there an infinite set of sets that only have one element in common with each other?

In a card game called Dobble, there are 55 cards, each containing 8 symbols. For each group of two cards, there is only one symbol in common. (The goal of the game being to spot it faster than the ...
10
votes
1answer
152 views

Recreational problems in set theory?

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, ...
3
votes
2answers
184 views

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various 'three-number means' with as many different elements in it as possible. So far I've got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+...
11
votes
2answers
149 views

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
0
votes
1answer
73 views

Large, small but a useful number. [closed]

Today we were discussing in our class about usefulness of a number no problem how large,small may be it's value. As per my knowledge (till grade 11) Avogadro number $N_A=6.022\times 10^{23}$ is a ...
4
votes
3answers
97 views

Quarter circle train tracks 2

While drawing little railroads based on the rules given in the problem here, a question occured to me: Is it possible to ever get stuck in the construction of such a railroad, i.e. to have no legal ...
16
votes
4answers
2k views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
0
votes
0answers
26 views

Probabalistic modeling of graph topology / network structure

I'll just let you know right now that I will be using very informal language here, so if you have other questions about technicalities that need to be specified please let me know. Let's say we have ...
14
votes
0answers
519 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
2
votes
1answer
41 views

vector of eigenvalues is an eigenvector

When is it the case that the vector $\begin{bmatrix} \lambda_1 \\ \lambda_2 \\ ... \end{bmatrix}$ of eigenvalues of a matrix is in fact an eigenvector of that matrix?
0
votes
0answers
35 views

Sticky boots and modular arithmetic: Find the formula!

Suppose a trek begins and on this trek the road is paved by squares with labels on them. The warning sign next to the beginning of the first square, labeled $1$, states: Beware that due to natural ...
3
votes
0answers
62 views

I'm walking towards my car - when should I try the remote, in an optimal sense?

I'm interested to learn about how discrete/'event' based elements are incorporated into optimisation problems. Hopefully this is an interesting problem in its own regard, it's inspired by a daily ...