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

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4
votes
0answers
37 views

Formula on conference t-shirt: $\ell(\beta) = \sum_{i=1}^{N}y_i \sum_{k=0}^{K}x_{ik}\beta_{k}-\log\left(1+e^{\sum_{k=0}^{K}x_{ik}\beta_{k}}\right)$

I need some assistance figuring out what this formula signifies and what it does. It is from a shirt I got recently at a conference and am curious about it. Thanks! $$\ell(\beta) = \sum_{i=1}^{N}...
0
votes
0answers
6 views

Convex 4-polytopes requiring 6 or more colors

Projected into 3-D space, a convex 4-polytope looks like a collection of convex polyhedra. If any two convex cells sharing a face have different colors, how many colors are required? In the paper ...
1
vote
1answer
58 views

A weight problem

I am having a hard time solving the following puzzle. Could you please me to figure it out? A chemist has a set of five weights. She knows that it includes one 1-gram weight, and also one each 2-, 3-,...
-6
votes
0answers
35 views

Khan Academy Android Alternative [on hold]

Is there any alternative for Khan Academy on Android? I just want an app that will ask me a question for the topic/concept that I choose and then I can get paper and work it out. I can then put in the ...
2
votes
1answer
64 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
80 views

Is there a possible mathematical solution for this? [closed]

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
617 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 $...
1
vote
1answer
46 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^...
2
votes
2answers
36 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 ...
1
vote
4answers
104 views

Help in proving an inequality

Show that $$a^4 + b^4\ge\frac{1}{8}$$ if $a+b=1.$
0
votes
1answer
39 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 $a_i - a_{i+1} = \pm 1$.)
2
votes
1answer
86 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: ...
3
votes
0answers
102 views
+50

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. ...
0
votes
1answer
69 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
4
votes
1answer
70 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&...
0
votes
0answers
47 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
40 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 ...
7
votes
1answer
113 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 ...
0
votes
0answers
97 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
0answers
44 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 ...
-3
votes
1answer
38 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 ...
-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
28 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 ...
1
vote
2answers
31 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 ...
0
votes
1answer
31 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 ...
1
vote
3answers
92 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 $...
9
votes
1answer
120 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 ...
5
votes
2answers
121 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 ($...
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 ...
0
votes
1answer
75 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
98 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 ...
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 ...
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
38 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
2answers
185 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+...
0
votes
0answers
39 views

Find f(x) subject to contraints

Given $x_0, x_d, v_0 \in R^3$, and a scalar $a$, I'm looking for some $f(t): R \to R^3$ constrained by: $f(0) = x_0$ $\frac{d}{dt}f(0) = v_0$ $\exists t_d$ with $f(t_d) = x_d$ $\frac{d}{dt}f(t_d) = ...
4
votes
0answers
64 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 ...
0
votes
0answers
17 views

Ease out elastic function with equivalent start and end values?

I have an elastic ease out function: http://easings.net/#easeOutElastic formula in code: ...
0
votes
1answer
20 views

Product of all Square Roots, taken only Decimal Digits

How and where could I compute the decimal reminder of a product of square roots times ten: $$Dr\left( \prod_{x=1}^{k}x^\frac{1}{2} \right) \times 10$$ Where $k$ is a power of $10$. I would like to ...
6
votes
2answers
99 views

Is there a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(f(x))=x+1$?

Is there a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(f(x))=x+1$? If so, can you give an example?
4
votes
3answers
80 views

Given $a_1,a_{100}, a_i=a_{i-1}a_{i+1}$, what's $a_1+a_2$?

I've been given the following puzzle Let $a_1, a_{100}$ be given real numbers. Let $a_i=a_{i-1}a_{i+1}$ for $2\leq i \leq 99$. Further suppose that the product of the first $50$ is $27$, and the ...
0
votes
0answers
21 views

How to find Turing Machine for given arbitrary output

Are there general methods / algorithms for finding a Turing Machine that will output a given binary number? For example, I want the machine to write ...
7
votes
3answers
200 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 ...
10
votes
3answers
226 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, ...
8
votes
2answers
161 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 ...
0
votes
4answers
49 views

How do I write n ∈ all of the known number sets

If I want to say that n ∈ all of the known number sets do I have to write n ∈ $\mathbb{N} , \mathbb{Z} , \mathbb{Q} , \mathbb{R} , \mathbb{C}$ or should I just leave it blank?
20
votes
6answers
719 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 ...
4
votes
1answer
43 views

Density of a set of numbers.

Firstly, I introduce a notation. $\Bbb{N}$ denotes the set of natural numbers, $0$ included. For $E \subseteq \Bbb{N}$ and $n \in \Bbb{N}$, I denote by $$\pi_E(n) = |E \cap \{ 1, \dots , n\}|$$ and $$...
1
vote
2answers
54 views

Is there a name for the logical scenario where A does not necessarily imply B, but B implies A?

A real life example of this is the 'Active' status on Facebook Messenger. (For those interested see this article here, and some Quora answers here for details.) When you are actively using Facebook ...