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

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0
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0answers
49 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
43 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
121 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
99 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
50 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
40 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
10 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
29 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
34 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
32 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
98 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 $...
12
votes
3answers
181 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
136 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
99 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
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0answers
27 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
42 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
41 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
188 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
40 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
67 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
19 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
22 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
107 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
82 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
202 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
257 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
167 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
50 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
749 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
46 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
55 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 ...
0
votes
1answer
35 views

Adding the intersections of circles. [closed]

Given the grid attached, how can you place the numbers $1-20$ at the intersections so that each circle adds to the same sum. I haven't been able to figure this out.
0
votes
1answer
45 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: ...
2
votes
2answers
547 views

63% chance of event happening over repeated attempts

I saw this online: If there is a $1 / x$ chance of something happening, in $x$ attempts, for large numbers over $50$ or so, the likelihood of it happening is about $63\%$. If there's a $1$ in $...
1
vote
0answers
69 views

If two parallel lines meet at infinity, then what is their angle? [duplicate]

Since lines that meet at some point have an angle. And if parallel lines meet at infinity, then that what is the angle of two parallel lines that meet at infinity?
11
votes
2answers
155 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 ...
25
votes
3answers
767 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 ...
5
votes
1answer
34 views

On the GCD of two palindromes.

I had an observation. Which I will discuss below. My question will be Is my observation correct? If so, how can one prove it? Observation: Consider the string of palindromes below: $100...01$ and $...
2
votes
0answers
43 views

What is the average maximum value of a set of random numbers? [duplicate]

Let $a_1, a_2, a_3, \ldots, a_{10}$ be ten randomly chosen real numbers in the interval [0,1]. Let $m$ be the maximal value out of these 10 numbers. What is the expected value of $m$? (i.e. If i ...
4
votes
0answers
36 views

de Bruijn sequence and sequence waiting time

this is quite a vague question, more of a puzzle than a question. I've spotted that two problems concerning word combinatorics have the same answer. I feel like there should be a connection, but I ...
9
votes
3answers
2k views

Represent $1729$ using four fours only.

I keep trying Four fours puzzle for various numbers, i.e. express a number using four fours and only four fours along with any mathematical operation. Today, I was thinking for Ramanujan number, i.e. ...
6
votes
1answer
50 views

Max size of $B \subset \{1, 2, \ldots, 3n+1\}$ for which no distinct $x, y, z \in B$ have sum in $B$

Given a set $$A=\{1,2,3,\ldots,3n,3n+1\},(n\in N^*)$$ Let $B$ be a subset of $A$, such that for any distinct $x, y, z\in B$, we have $x+y+z\not \in B$. Find the maximum number of elements $B$ ...
1
vote
1answer
23 views

Calculating the probability that the following random subsets of $\mathbb{R}^2$ are open.

For each $t \in \mathbb{R}$, select an open interval $U_t \subseteq \mathbb{R}$ containing 0. What is the probability that the set $U = \{ (t,x) | x \in U_t\}$ will be an open subset of $\mathbb{R}^2$?...
6
votes
0answers
88 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
6
votes
2answers
97 views

Mario Party 3 Mini-game Probability Question

I have a question about a mini-game in Mario Party 3. I have extracted the mathematical information from the game below. Setup: Four players $A,B,C$, and $D$ line up in some order. There are $12$ ...
-1
votes
1answer
33 views

Arc Length and Area of a Sector

A cake has a circumference of $30 \mathrm{cm}$ and a uniform height of $7\mathrm{cm}$. A slice is to be cut from the cake with two straight cuts meeting at the centre. If the slice is to contain $50\...
20
votes
5answers
542 views

Is $11^2+12^2+13^2+14^2+15^2+16^2=1111$ special?

Is this pure coincidence or is this a special case of some well-known number-theoretic result? If the latter is true, is there some notable generalization? EDIT: Thanks to the interesting answers ...