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

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16
votes
2answers
293 views

Counting the number of polygons in a figure

I often come across figures like this on the net, or as contest problems, asking to find the number of a specific type of polygon in the figure (triangles, in this case). But I've never really found ...
3
votes
1answer
164 views

Solving circle's radius only knowing angle & lengths of external triangle OR solving for sides of a triangle partial side lengths

Is this possible? Given that I know the length of Y and Z and the angle of X can I figure out the radius A? If I can't without more information, I can produce another set of data X Y Z at a ...
3
votes
1answer
52 views

If a function is smooth is 1 over the function also smooth

If $f(x):\mathbb{R}\rightarrow\mathbb{C}$ is $C^\infty$-smooth. Is $1/f(x)$ also $C^\infty$-smooth? $f(x)\neq0$
3
votes
0answers
79 views

Why is this proof false? (Why is $e^i \neq 1$?) [duplicate]

I found this on MathOverflow: $$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = > 1.$$ I first saw this one many years ago, written on the wall of a bathroom stall in ...
2
votes
1answer
47 views

Area of a right angled triangle is an even integer

Actual Question is : Let $ABC$ be a triangle in the plane such that $BC$ is perpendicular to $AC$. Let $a,b,c$ be the lengths of $BC$, $AC$, and $AB$ respectively. Suppose that $a,b,c$ are integers ...
0
votes
0answers
33 views

Stuck on interpretation of diff eq.

First thanks to @Ross Millikan, and @Gerry Myerson for helping me get this far. I now understand my problem much better, but I am still kind of stuck on finishing the problem So if you permit me one ...
31
votes
1answer
329 views

Zero-avoiding integers

Let's say an integer $n>2$ is zero-avoiding if, for every $2\leq b < n$, the representation of $n$ in base $b$ has no $0$ digits. (Obviously every $n$ has a $0$ when written in base $n$ and no ...
7
votes
0answers
121 views

Math Behind the Dragon Illusion!

Dragon illusion has been one of the items presented in the 3rd "Gathering for Gardner". This video shows the illusion. What does it have to do with mathematics?
0
votes
0answers
23 views

finding $f(C')$ if $f(C)$ is given where $C'$ is reflection of $C$ w.r.to a line segment $AB$

Question is : For a Point $P=(x,y)$ in the plane, define $f(P)=ax+by$ where $a,b$ are given real numbers.let $f(A)=f(B)=10$. Let $C$ be a point not on the line joining $A$ and $B$. Let $C'$ be the ...
2
votes
0answers
49 views

Which number should come in the blank box? [closed]

The number should be in 60s (can't remember exact options) I have tried following: Transform A-->1, B-->2, C-->3, D-->4, E-->5 ...
1
vote
2answers
74 views

Is there an algebraic method to concat two numbers?

I'm searching an algebraic way to concat numbers in base $10$. Concatening two numbers is to put side by side their notations. Let $c$ a concatenating function. $c(2,2) = 22$ $c(8,9) = 89$ ...
2
votes
1answer
63 views

Can we make rectangle from this parts?

I have next problem: Can we using all parts from picture (every part exactly one time) to make rectangle? I was thinking like: we have $20$ small square, so we have three possibility: $1 \times ...
7
votes
2answers
133 views

Come up with some fun “equation Limericks”

We were discussing "Limericks" in my Calculus class. Specifically, "equation Limericks". A Limerick is a poem with five lines. The first, second, and fifth lines should have nine syllables each and ...
3
votes
0answers
203 views

Help explain why (or why not) the solution for a in $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)=0$ is 1-2$C$MRB

$C$MRB is approximately 0.1878596424620671202485179340542732. See this and this. $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)$ is formally convergent only when $a =1$. However, if you extend the ...
-1
votes
6answers
102 views

How do I construct a sequence of N numbers such that they are decreasing and they sum to 1? [closed]

How do I construct a sequence of N numbers such that they are decreasing and they sum to 1? In my case N is 22. That is, I need to create 22 values that decrease, and they sum to 1.
3
votes
1answer
101 views

Number theory: 2 numbers within a set with same difference

You have the numbers 1,2,3...,99,100. From that set you have to choose 55 different numbers. Show that: There are 2 numbers with a difference 9,10,12,13 Show that there aren't ...
2
votes
0answers
56 views

How to prove $\sum_{n=1}^\infty (-1)^n(x n^{1/n}+y n)=(c-1/2) x-1/4 y$?

Could someone help me prove this theorem where regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the ...
3
votes
3answers
79 views

Is there a $3\times 3$ magic square adding up to $7$.

I suspect that there is no magic square with natural number entries (matrix where each row, column and long diagonal add up to the same number) which would add up to $7$. There is no restriction on ...
7
votes
1answer
176 views

Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
2
votes
3answers
172 views

hours minutes and seconds hand will make equilateral triangle

When I am going through some aptitude questions I have got this problem How many times the hours minutes and seconds hand will make equilateral triangle in 12 hours of clock I can't understand how ...
2
votes
1answer
58 views

Fair Division: Making the Differences in Players' Valuations Believable

When teaching basic fair division algorithms, the students always propose some simple and (at the first glance) correct solutions for $n$ players, which unfortunately are not correct! The only way I ...
6
votes
3answers
183 views

If $x^3+\frac1{x^2}=1$, what is $x^3+\frac1{x^3}$?

$x^3 + \frac1{x^2} = 1$. Then, $x^3 + \frac1{x^3} = ~?$ $p + \frac1{p^2} = 47$. Then, $p + \frac1p = ~?$
11
votes
1answer
265 views

What am I getting for Christmas? Secret Santa and Graph theory

I live with four people, who thankfully don't spend much time on maths.se. We decided this year that we'd do a Secret Santa. We can represent the arrangement of who's buying for whom using a directed ...
3
votes
2answers
91 views

Argument over an induction proof

My friend gave me a problem. Define a sequence $<a_n>$ by the recurrence relation :$$ a_{n+2} - 6a_{n+1} + 8a_n = 0 $$ and $a_1 = 4, a_2 = 8 $. Find the general term $a_n$ in closed form. ...
1
vote
2answers
169 views

Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
0
votes
1answer
31 views

Calculating variable in math equation

I am not good with math, I have this equation (very simple to most) but I need help on how to get the value of x 10 = x - (1.29 + 4.99% of x) my question is how ...
5
votes
1answer
328 views

Proof Involving a Problem from “Good Will Hunting”

I don't know if any of you have seen the movie "Good Will Hunting" but there is a particular mathematics problem in the movie that is of interest to be. One of the problems used in the movie is "Draw ...
0
votes
1answer
81 views

12 base units instead of 10

Why do we use 10 as a whole unit when it can't be evenly divided by 3 or 6 without resulting in an infinite decimal I.e 3.3333etc. If units of 12 were used I.e 1 2 3 4 5 6 7 8 9 ? # 10. 3.3333 would ...
4
votes
1answer
119 views

Linear Algebra in curved space

We know that Euclidean geometry and Newtonian Physics are special cases that only work in a flat space-time. Got to thinking about linear algebra and matrices. Is linear-algebra a special subset of ...
4
votes
1answer
181 views

No primes in this sequence

Here's a fun little problem: Prove that the sequence $$10001, 100010001, 1000100010001, \cdots$$ contains no prime numbers.
3
votes
1answer
47 views

Guess the Polynomial

Player ONE has a finite degree polynomial $p$ with integer coefficients in mind whose domain is the reals. Player TWO gets to ask Player ONE to evaluate the polynomial at two points $x_0,x_1$ and ...
7
votes
1answer
622 views

How to prove the number of solutions to nine dots puzzle

The aim of the Nine Dots Puzzle is to draw a path connecting 9 dots arranged in a $3\times 3$ grid using 4 continuous straight lines, never lifting the pen/pencil from the piece of paper. A solution ...
3
votes
3answers
300 views

Minimizing perimeter given rectangle's area for 10-years-olds

I was recently in touch with some person from Russia how is busy with books for Russian elementary schools, in particularly I learned that now they give elementary set theory for the 2nd grade ...
3
votes
1answer
73 views

Minimal diameter of set of fractions

Let $p_n$ be a pairwise partition of $\{1,2,...,2n\}, n\in \bf N$ where $(a,b)\in p \implies a<b$, and $P_n$ the set of all such pairwise partition. $d(n) := \min_{p_n\in ...
0
votes
0answers
77 views

Did I make a “forced” interpretation?

Just a few years ago I wrote an article called The Geometry of the MRB constant. Since then I've wondered if there is a better, more natural geometric analysis of the following summation. $$ ...
1
vote
1answer
42 views

In what kind of Banach algebras is 0 the only topological zero divisor?

On page 33 of http://math.aalto.fi/opetus/harmanal/pruju/calg04.pdf it is asked in what kind of Banach algebras is 0 the only topological zero divisor. What do they mean by kind of Banach algebras. ...
5
votes
5answers
222 views

Leisure reading for an undergraduate student

I am a freshman at a local university. I never really had much passion for math, but I always did well in math exams . I attribute this lack of passion to rote learning/emphasis on methods/formulas ...
2
votes
1answer
58 views

Why all near integers

$C$MRB is a symbol for the MRB constant. Why do I get all near 0, 1 or 2 for all values of n for sin(Pi/$C$MRB*(5060936308 + 78389363/24*Floor(n)))? The results near 0 are very small and the ...
1
vote
2answers
49 views

Recursive formular and closed-form questions

Follow the question the $f(n)=4n-1$ and $F(n)=\sum_{k=0}^nf(k)$. And it ask you to write the recursive of $F(n)$. But I only know the recursive of $f(n)$ is $$f(n)=\begin{cases} -1,&\text{if ...
0
votes
1answer
56 views

Conditions for convergence of a geometric series [duplicate]

This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series. Consider the infinite geometric series ...
1
vote
2answers
72 views

Swap two integers in different ways

This is a famous rudimentary problem : how to use mathematical operations (not any other temporary variable or storage) to swap two integers A and B. The most well-known way is the following: ...
-3
votes
1answer
42 views

Computation of integral [closed]

I want to compute this integral: \begin{equation*} J=\int_{0}^{1}\ln(p)\ln(1-p)p^{2}dp \end{equation*} It will be great if you can detail the proof. I tryed to do change of variable it does not ...
1
vote
1answer
57 views

For how many seconds do I need to turn the pedal?

For how many seconds do I need to turn the pedal of a bike, so that the number of turns is equal to the value of my velocity in the given moment measured in km/h.
0
votes
1answer
57 views

Did I compute this eta formula correctly

In the question $C$ MRB proofs wanted , I gave the following excerpt from http://www.perfscipress.com/papers/UniversalTOC25.pdf . . In this question I just want to know if I accurately worked ...
0
votes
0answers
17 views

Card Game Puzzle [duplicate]

We play a game, where I shuffle a standard deck of cards and turn them over one at a time. At any time before the last card is turned up you say "Choose", and you win if the next card that I turn up ...
4
votes
0answers
58 views

Finding every $n$ such that $n\times$ ('reverse' number of $n)=m^2$ such as $1584\times 4851={2772}^2$

Let $r(n)$ be the 'reverse' number of $n$ in the decimal system. For example, $r(1234)=4321$. Then, here is my question. Question : Can we find every $n(\in\mathbb N)$, which is not a square ...
1
vote
1answer
93 views

Monotonically increasing path in a complete graph

Given a complete graph with n vertices such that all edge weights are distinct. Prove that we can find a monotonically increasing path of length n-1. I tried finding such a path by sorting the edges ...
1
vote
0answers
30 views

Converting dot producto to set of arithmetic mean differences?

Ok so I am reading a book on linear algebra ( Gilbert Strang to be specific) and I am on second problem set, challenge problem, problem 29. In solutions it appears that the author states that: ...
2
votes
1answer
93 views

roulette wheel sequence

Is the sequence of numbers around a European roulette wheel (the integers from 0 to 36 inclusive) random or is there a pattern to it? It is said to have been devised by Pascal, which might be thought ...
0
votes
1answer
114 views

such of sum of any two adjacent numbers is equal to the sum of the opposite numbers

Give the numbers 1 to 10 on the edges of the diametric chords for the image given below such that such of sum of any two adjacent numbers is equal to the sum of the opposite numbers