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

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0answers
52 views

sum of digits = sum of factors

Assume that some set $A$ is defined as $A=\{x|x\in Z \ni S(10,x)=\sigma_1(x)\}$ where $S(10,x)$ returns the sum of all of x's digits in base 10, and $\sigma_1(x)$ returns the som of all prime factors ...
4
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2answers
65 views

Simple question about dividing by zero, $y=\frac{x}{x}$ when $x=0$

Is there a rule that says you have to simplify equations before evaluating them? Would $y=\frac{x}{x}$ at $x=0$ be $1$ or undefined, since without reducing it, you'd divide by $0$. I know the equation ...
2
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1answer
25 views

Find all reals x, y such that 1<=x<=a , 1<=y <=b and (x^(1/3) + y^(1/3))^3 is integer.

The question was asked in a Twitter interview. For given integers $a$ and $b$, find all reals $x$, $y$ such that $1\leq x\leq a$ , $1\leq y\leq b$ and $(x^{1/3} + y^{1/3})^3$ is an integer.
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1answer
29 views

Curiosity: Divide by two by inserting number

I noticed that, starting with 0.5 you can divide by 2 twice by inserting a number in the previous result after the decimal point. Specifically, to go from 0.5 to 0.25 a number 2 in inserted after the ...
3
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0answers
39 views

minimum number of times to change tyres.

I saw this brain teaser. Suppose, we travel 1000 miles on a tricycle and we have 5 tyres, then how many times do we need to stop to change tyres so that each of the tyres travelled the same ...
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1answer
25 views

Totem Pole sequences

Worth a shot: Find sequences of consecutive integers with consecutive bit counts. I'm not very good at this sort of thing, so I managed 8,9.
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1answer
19 views

Expected number of person not to get shot - reloaded

Inspired from this question I came up with a seemingly simpler problem that I could not solve either. There are $n$ people sit on a round table. At noon, each person shots and kills one of their ...
2
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0answers
61 views

Start with a circular cake…

This is a random math question I saw and it piqued my interest. I like cake. Start with a circular cake and cut it with five straight slices. What is the largest number of pieces that you can create ...
8
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1answer
288 views

Best strategy to find a parking lot

New Bounty Edit (2 days remaining on the Bounty): To point out that the only answer given at this time cannot be considered an answer, because it simply gives a hint on how to formally model the ...
0
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1answer
81 views

Proving that a regular polygon with infinite sides is a circle by using limits on the formula $\frac{\pi}{n}(n-2)$

In childhood, when we were taught circles for the first time, our teacher always told us that a circle is like a polygon which has infinite sides. But how to prove it? A regular polygon's interior ...
2
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4answers
158 views

Solving large multiplications in my head

What would be the best approach to solve 73 x 42 in my head? I started with 70 x 40 and then 3 x 40 and combined, but at this point I forgot what I had done and ended up getting lost and not figuring ...
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1answer
63 views

Finding parameter paths for beautiful fractal animations

So I just got renewed interest in fractals and especially animations with fractals. To make an image or a frame, we usually need to evaluate a fractal for a subset of it's parameters. However for many ...
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2answers
962 views

Convert Equilateral triangle to Isosceles triangle

Let an equilateral triangle have the length of each side an integer $N$. I need to find if it is possible to transform the triangle keeping two sides fixed and alter the third side such that it still ...
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1answer
41 views

Using geometric construction, is there a way to construct a circle that spins another circle at 3:2 ratio?

I have been tinkering with GeoGebra to construct a point on a circle that I can move around the circumference which then in turn moves another point on another circle at a slower rotation rate ...
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1answer
69 views

Interesting Original Probability Question

I have 100 balls, which are all initially yellow. Every minute, I randomly choose a ball and paint it red. How many balls are expected to be red after 100 minutes? Note: I could pick up a ball that's ...
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0answers
41 views

Where the center of gravity be for an umbilic torus?

Due to its unusual shape I am unsure if the COG would be right in the center of closer to the side with (what looks like) more mass on the outside?
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1answer
31 views

Solve the maximal value point of B-spline basis function

Description Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denotes a non-decreasing sequence of real numbers, i.e, $u_i\leq u_{i+1} \quad i=0,1,2\ldots m-1$. and the $i$-th B-spline basis function of ...
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1answer
86 views

Previous year like 2016

How can I find the previous year $y$ which was like 2016 in sense that January first is Friday, it has a leap year and date of Easters are the same on both years?
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1answer
55 views

Roulette System that I know can't work but don't understand Why?

So I know the house always win and someone has already figured this out. But I can't see the catch for my self I was watching this video Best System for Roulette Basically progressive betting on a ...
2
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1answer
96 views

Shortest path between two points on graph?

A graph has 100 vertices. Each vertex is connected to exactly 10 other vertices such that the whole graph is connected. If I choose two random vertices of the graph, what is the length of the shortest ...
0
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0answers
41 views

Number of meetings needed for everyone to know everyone

There are 100 people on an island. Each person has an unlimited number of name-tags with his name on it. (Everyone has a different name) When two people meet, each gives the other an unlimited number ...
0
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2answers
50 views

Question about the average number of cycles?

I have 100 cards labelled from 1 to 100. I place this in a row with all the cards face down. On the first turn I flip the first card. Now this card has a number $n$ on it. I then put this card face-up ...
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3answers
1k views

Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...
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3answers
72 views

What is the expected value of this random number?

Suppose $a$ is a random number between 0 and 1 and suppose $b$ is a random number $\in (0,1)$ also. What is the expected value of $a^b$? (i.e. If I performed this operation $n$ times, what would be ...
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1answer
58 views

Random but very interesting probability question! [duplicate]

There are 100 types of cards, each type with a number from 1 to 100 on it. Each minute I am given a random card. Random means that the number of my card has an equal chance of being 1,2,3,...,100. I ...
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0answers
64 views

The Earth-Moon Map Problem

The following is the Sulanke Earth-Moon Map. A planet and moon have each been divided into eleven contiguous regions. In both maps, the regions 1-3, 3-5, 5-2, 2-4, and 4-1 do not touch, while all ...
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2answers
42 views

How to prove $\binom{n+1}{m+1}=\binom{0}{m}+\binom{1}{m}+\dots+\binom{n}{m}$ combinatorially

How can we prove combinatorially $$\binom{n+1}{m+1}=\binom{0}{m}+\binom{1}{m}+\dots+\binom{n}{m}$$ I can get LHS by asking: How many ways can we form an $m+1$ person committee from a group of $n+1$ ...
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1answer
54 views

Mathematical Symbols

I'm working on a constructed language that borrows concepts from existing languages. Does anyone know if there is a consolidated set of universal symbols out there? I looked at Wikipedia and I noticed ...
3
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1answer
207 views

A prime-finding constant

Consider $\lfloor 10^n \times 0.731926765612646213686753345587262244668218433356357832021 \rfloor$. For each $n$, reverse the digits. If the number isn't already prime, find the next prime. For the ...
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1answer
46 views

Choosing a $k$ person committee with chairperson from a group of $n$ people confusion

The following is from: http://www.math.sjsu.edu/~bremer/Teaching/Math163/Homework/HomeworkFiles/Solution03.pdf I am having trouble understanding these identities and the solutions. I am confused as ...
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0answers
26 views

Help with understanding Rating Systems

Hey guys so I'm trying to come with a rating system from 1-10 that will be determined based on how close a user is to an average value that I have. I don't have any data(other than the average value) ...
4
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2answers
126 views

Penrose Triangle and Umbilic Torus

Here are two images - one of each. It seems to me that they are the same object from the topological perspective, that one is just a smoothed-out version of the other. I think this because it is clear ...
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0answers
20 views

Geometry IMO Problem: Reflections of a line tangent to a triangle's circumcircle

Let $\triangle ABC$ be an acute triangle with circumcircle $\bigcirc T$. Let $\ell$ be a tangent line to $\bigcirc T$, and let $\ell_a$ , $\ell_b$, and $\ell_c$ be the lines obtained by reflecting ...
2
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1answer
67 views

Recreational Maths IMO

I saw this problem on a maths challenge book. Given any set $A=\{a_1 ,a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1+a_2+a_3+a_4$ by $S\{A\}$. Let $n_A$ denote the number ...
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0answers
47 views

Maths Challenge IMO

I saw this problem on a maths challenge book. Given any set A={a_1 ,a_2, a_3, a_4} of four distinct positive integers, we denote the sum a_1+a_2+a_3+a_4 by sA. Let nA denote the number of pairs ...
1
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1answer
61 views

Vertical visibility graphs — canonical icosahedral graph

Here's a vertical visibility graph for the icosahedral graph. Also called a scheduling graph. And maybe other names. Each open segment corresponds to a vertex. If there is unblocked vertical ...
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0answers
56 views

"Problems worthy of attack prove their worth by fighting back.”

That is quote has been attributed to Piet Hein, inventor of the Soma cube, which is how I know of him. Q. Is the attribution correct? I wonder because the quote has a nice ring in English that ...
3
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3answers
108 views

Evaluate limit as x approaches infinity of $\lim_{x\to\infty}\cfrac{\sqrt{x^3 +7x}}{\sqrt{4x^3+5}}$

I am having trouble figuring out how to answer this question by determining the degree of the numerator and/or denominator: $$\lim_{x\to\infty}\frac{\sqrt{x^3 +7x}}{\sqrt{4x^3+5}}$$ I have tried ...
1
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2answers
53 views

Is the fuction $f(x) = \frac{7}{e^x - 2}$ continuous on the interval $ [-1,1]$

Hi I am having trouble trying to figure this problem out. I have tried to separate the equation into a piece-wise function but was no help in making things clear. I am not sure if I use the limit as ...
0
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0answers
36 views

Are there ways of finding different properties of a circle using infinite sums?

I have read many articles that have differing opinions on how many "sides" a circle has. Some say infinite, others one, zero, or undefined. If, for the sake of my question, we use the infinite example ...
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2answers
42 views

Finding an $a$, such that $\forall x(x^x-a\cdot x!=0).$

I've been finding myself wondering about this equation for a long time, however due to my limited math knowledge, I can't solve or even determine if there is a solution to that equation. So I ask: is ...
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5answers
4k views

Which week day(s) cannot be the first day of a century?

I think the question says everything. What I want is, a very short approach. What I did: Lets call the day which is not the part of a whole week, a free day. So in a normal year, there is $1$ free ...
5
votes
3answers
344 views

figuring out an integer function

$f(1) = 1\\ f(2) = 2\\ f(3) = 6\\ f(4) = 20\\ f(5) = 70\\ f(6) = 252\\ f(7) = 924\\ f(8) = 3432\\ f(9) = 12870$ Then what is $f(n)$ (where $n > 0$)? I though about many many possibilities but ...
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2answers
113 views

Intuitive/heuristic explanation of Polya's urn

Suppose we have an urn with one red ball and one blue ball. At each step, we take out a single ball from the urn and note its color; we then put that ball back into the urn, along with an additional ...
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1answer
73 views

Mental calculation of the day of birth [closed]

I have a friend who can tell the birth day of anyone by only knowing the date and year. He is able to do this almost instantaneously. How does he do it? Please enlighten me on this subject. Thanks a ...
5
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1answer
60 views

Smallest pandigital number perfect square

What is the smallest $9$-digit number that has all the digits from $1$ to $9$ exactly once and is also a perfect square? Please give me a method that doesn't involve programming.
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0answers
43 views

All polygons satisfy the “normal” property.

A fancy explanation is below, but here's an edited simpler explanation because I think the jargon makes the problem seem inaccessible. In reality this problem is super accessible and I'm sure the ...
3
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2answers
51 views

Prime number where all (ordered) combinations of digits are prime

Are there any prime numbers with more than two digits such that all combinations of its digits (preserving order) are prime? For example, if the number abc is ...
2
votes
1answer
38 views

Diophantine-like equations

So I was solving a problem and encountered a specific system of equations that I don't know if a solutions exists for it or not. $$\begin{align} 4ny&=d^2-a^2\\ -4nx+4ny&=d^2-b^2\\ ...
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2answers
41 views

Composite numbers which divide the concatenation of their prime factors in chaos order

Since numbers divide the concatenation of their prime factors in neat ascending/descending order are have been listed at OEIS (See [$A248915$ and $A259047$ in Sloane's OEIS][1]). Here we are only ...