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

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1
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0answers
18 views

Has a simple optimal or provably near optimal strategy been shown for backgammon bearing off?

I aplogize in advance for the somewhat long post. I've tried to split it into manageable paragraphs. So in backgammmon, in the so-called "end-game", both players have their pieces in their respective ...
3
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0answers
249 views

Evaluating $\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$

I need to evaluate $$\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x $$ I've been told that the way forward is ...
5
votes
3answers
130 views

Dwarfs over a bridge

300 dwarfs go over a bridge in the middle of the night. The bridge is rickety and manages at most two dwarfs at a time. With them is a lantern that they must provide at each transition. Dwarfs need ...
4
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2answers
89 views

How can one find the zeroes of $f(x)=ae^{bx}+cx+d$?

A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form $$f(x)=ae^{bx}+cx+d = 0$$ where $a, b, c,$ ...
2
votes
2answers
38 views

Multidimensional Riemann integration and notion of volume or Lebesgue theory and notion of measure

I have finished 9 chapters of "Introduction to Analysis" by Maxwell Rosenlicht (1968). The last chapter treats about "Multiple Integrals". I find the notation a bit complicated. Also, author ...
5
votes
1answer
160 views

Dividing numbers with dots?!

OK. This intrigues me. I recently came across this video. Which presumably tells you how to divide 133,342 with 121 only using hand drawn dots! Fair enough but I don't think this works for every ...
1
vote
3answers
35 views

How many definitions of a list of 30 would I need to know so that I could answer at least 10 from any 18?

While studying for my English exam, I noticed that the way the definitions portion of the final is set up posed an interesting problem. While I will study all the definitions, I thought trying to ...
1
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1answer
24 views

Number of way that set of point can be colinear

Assume I have $n$ points in a plane. and I want arrange them in the way that for any point at least I can find two other points that are all the three points are collinar. I want to know how many way ...
0
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1answer
69 views

How can we draw $14$ squares to obtain an $8 \times 8$ table divided into $64$ unit squares?

How can we draw $14$ squares to obtain an $8\times8$ table divided into $64$ unit squares? Notes: -The squares to be drawn can be of any size. -There will be no drawings outside the table.
4
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1answer
97 views

What is the minimum number of squares to be drawn on a paper in order to obtain an 8x8 table divided into 64 unit squares? [closed]

What is the minimum number of squares to be drawn on a paper in order to obtain an $8\times8$ table divided into $64$ unit squares. Notes: -The squares to be drawn can be of any size. -There ...
2
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0answers
60 views

Compute shooting targets for the gunmen

This is an extension of the well known "3 gunmen puzzle": N gunmen with hitting probabilities in (0,1] take turns to shoot at each other. Firing order is fixed (gunman 1 shoots first, then gunman ...
9
votes
5answers
500 views

Formulae of the Year $2016$ [closed]

Soon it's the year $2016$. Time to ponder how we can arrange the digits in 2016 to form a valid equation. Use any symbols you like (please explain the less obvious ones). Keep digits in the same order ...
3
votes
0answers
340 views

An interesting way to visualize the Mandelbrot Set. Proofs? Simplifications? Extensions?

This is a multi-part Question. Please chime in with any interesting insights in addition to Answers. I have noticed some interesting properties of Mandelbrot series that lead to a different way to ...
2
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0answers
55 views

Could you suggest books, papers or problems that could be used as good “general” motivating examples of calculus application?

I would like to stress the kind of reference I am looking for... In statistics there are lots of motivating (and sometimes unexpected) examples that are interested for everyone such as Birthday ...
1
vote
2answers
123 views

I want to write a Christmas message only with particular zeta values. It is possible?

I want to write a Christmas message to leave as a comment thanking the people who in the next 24th December will solve some of my problems: I wish you Math Christmas and a Happy New Year ... ...
5
votes
2answers
47 views

Deleting one digit yields a divisor

Let $N$ be a positive integer with $d\geq 4$ digits, none of which is zero. Suppose that erasing some digit of $N$ yields another number $M$ which happens to be a divisor of $N$. Examples : 1375 ...
3
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0answers
31 views

Hamiltonian path on a chessboard with prescribed endpoints

On an 8 x 8 chessboard consider two squares to be adjacent if and only if they share a common side. All paths below will consist of steps which join one square to an adjacent one. Under these ...
0
votes
1answer
44 views

2-player game, putting coins on a round table [duplicate]

Two players place coins of identical size (say quarters) on a round table. Each player has to place exactly one coin on the table without overlap with the coins already on the table. The first player ...
4
votes
1answer
69 views

Fun Q6: Side length of the pentagon in a five sided star?

Consider a regular pentagon of side length $a$. If you form a 5-sided star using the vertices of the pentagon, then you'll get a pentagon inside that star. What is the side length of that pentagon? ...
0
votes
0answers
36 views

Solutions of diophantine equation: $s^2 = (ad)^2+ (bc-ad+4ac)^2$

Given diophantine equation: $$s^2 = (ad)^2 + (bc-ad+4ac)^2$$ $s,a,b,c,d$ are all variables. They are all odd. a and b are coprime. c and d are coprime. How do you parametrize all the solutions? ...
2
votes
1answer
46 views

Count the pair of numbers that satisfy the set

I have an operation $f$ which takes two numbers $A$ and $B$ and returns a symmetric difference of digits of these two numbers. For example having $453$ and $1134$ the operation will produce a set ...
1
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1answer
18 views

Pinpointing a hidden submarine moving at constant speed

A submarine is moving along the line at a constant speed $s$, starting from position $a$. Thus its position is $a + st$. $t$ does not necessarily start from zero, it starts from some value $k$ ...
-1
votes
2answers
76 views

3 girls and 4 boys were standing in a circle . What is the probability that two girls are together but one is not with them?

Question: 3 girls and 4 boys were standing in a circle . What is the probability that two girls are together but one is not with them ?
13
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1answer
117 views

Numbers whose powers are almost integers

Some real numbers $\alpha$ have the property that their powers get ever closer to being integers -- more precisely, that $$ \lim_{n\to\infty} \alpha^n-[\alpha^n] = 0 $$ where $[\cdot]$ is the ...
0
votes
1answer
28 views

How to obtain fair competition between two teams

Consider a class with $4$ students having min goals as $\big\{ 1, 3, 4, 5 \big\} $ and max goals as $\big\{ 2, 5, 8, 6 \big\}$. Find the best way to divide the class in such a way that the match is ...
0
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0answers
37 views

Fun Q3: So many circles! Find Test's area

A circle of radius $r$ is drawn at the center $O$. Two lines are drawn. Horizontal line which intersects the circle at point $A$ line angled $0 \leqslant \theta \leqslant 180$ from the horizontal ...
3
votes
0answers
39 views

Fun Q2: Polygon inside a polygon. Find Test's area.

A triangle is drawn of side length $a$. Then a square is drawn inside the triangle such that the area of the square is maximum and the bottom side is shared. Then a regular pentagon is drawn inside ...
0
votes
1answer
39 views

Genearalisation of previous MSE question regarding password combinatorics in $n$ dimensions

This question caught my atention recently. Most (if not all) of the answers aproached the problem via a brute force attack. Surely there is a more elegant way to deal with this, given the inherent ...
0
votes
1answer
8 views

Finding maximal and minimal values under certain limitations

given a function that gets 3 numbers and outputs the maximal of them, I have to find using only this function the minimal number out of 3 given numbers. ...
0
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0answers
39 views

Rubik's Cube with a missing color

Suppose a color of the Rubik's cube(3x3) is missing, is it possible to find the missing color? Now a brute force method would be solve for all other colors, but that is quite naive.
-1
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1answer
75 views

Learning Differential Geometry [closed]

What levels of mathematics does one need to nail down before successfully studying Differential Geometry?
2
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0answers
71 views

When solving a big Rubik cube (100x100x100), do you reduce the solution to like 50x50x50, and then 25x25x25, and then like 10x10x10 and then 3x3x3?

My question is about Rubiks cube. Say you're solving a 100x100x100 cube (you can see examples in youtube by computer program - https://www.youtube.com/watch?v=0cedyW6JdsQ) When solving a big Rubik ...
1
vote
3answers
140 views

Bucket Puzzle Probability Problem

You have 2 buckets. One full of white marbles and the other full of black marbles (equal amounts). How do you allocate the marbles into two buckets in a way that maximizes your probability of picking ...
0
votes
1answer
20 views

Executions per second calculation

I just can't wrap my head around it. Thinking about getting some basic math courses cause it has been so long.. A user executes at a speed of 10 000 000 per second. I have 100 000 users over the ...
3
votes
2answers
148 views

How many 4 digit numbers are divisible by 29 such that their digit sum is also 29?

How many $4$ digit numbers are divisible by $29$ such that their digit sum is also $29$? Well, answer is $5$ but what is the working and how did they get it?
2
votes
3answers
51 views

How can one solve the tower of hanoi problem if there are discs of similar width in it?

For example a line with '1111' represents a disc with diameter of length 4. Similarly a line with '111' represents a disc with diameter of length 3. Below is the representation of a tower that has 5 ...
0
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1answer
24 views

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

My previous question was wrongly formulated, I wanted to know the value of $a$ as $x$ gets bigger, but due to my limited math knowledge I can't solve it. So, is $a$ a constant and what would its value ...
4
votes
0answers
107 views

Two formulae for $\pi$, probably known?

I stumbled upon (in the literature) two identities for $\pi$, but they were not referenced as they are probably well-known. Hoping someone could point out who found them first. Basically, the ...
3
votes
1answer
49 views

Setting up an English pool table

A friend and I were playing some English pool yesterday. When you rack the balls it has the specific set up in the picture where the 8 ball must be in that central position and the balls are laid out ...
0
votes
1answer
14 views

Distribution of the product of a Gaussian Random Variable with a scaled and mean shifted value of itself

Suppose we have a r.v. $X$ with pdf $N(0,\sigma^2)$ if I get another r.v. formed by $$ y=\alpha + \beta \times x$$ What is the PDF of $$ Z = x \times y $$ To be precise, how to get the joint ...
3
votes
3answers
123 views

Does there exist $n$ such that all numbers $n,2n,\dots,2000n$ have the same digits?

Does there exist a number $n$ such that all numbers $n, 2n, 3n, 4n, \dots, 2000n$ have the same multi-sets of digits except zeroes? (Having the same multi-sets of digits excepts zeroes means ...
0
votes
1answer
24 views

Fun Q1: Find the area of the region shaded by Test in the square of side length $a$

Test places a square of length $a$ on the coordinate axis such that the lower left corner coincides with the origin. He draws infinite lines of the form $y = \frac{x}{n}$ where $n \epsilon N$. Then he ...
0
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0answers
43 views

Is there a field size such that it makes perpetual “candy crush”

a.k.a Infinite Candy Crush Background: "Candy Crush Saga" is called a match3, but it has some "special" events that will eliminate all rows, eliminate all "candies" of a particular shape, or even ...
8
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4answers
1k views

Two players placing coins on a round table with the goal of making the last move [duplicate]

I came across this riddle during a job interview and thought it was worth sharing with the community as I thought it was clever: Suppose you are sitting at a perfectly round table with an ...
0
votes
2answers
30 views

Can we deduce anything given the equation of a curve and the fact that it has symmetry with $y=x$?

Question: The line $y=x$ is a line of symmetry to the curve with equation $$y=\frac{px+q}{rx+s}$$ where $p,q,r,s \neq 0$. Which of the following must be true? $p+s=0$ $p+q=0$ ...
4
votes
1answer
58 views

Starting for Self Study of Mathematics [closed]

I am a Mathematics enthusiast but after High School i took a job. Now i want to do self study in mathematics and to dive deep into the subject. What should i do ? What books and articles should i ...
1
vote
1answer
25 views

“Extending” the calculation of the golden ratio using square roots (not silver-ratio)

I'm looking at the following formula: $x =\frac{-n+\sqrt{n^{2}+4n}}{2}$ For $n=1$ this this gives $0.618...$ and then $\frac n x$ gives $1.618...$ which is $\phi$, the golden ratio. What ...
1
vote
1answer
69 views

Minimal prime numbers (no shorter subsequence of digits is prime)

I need help with minimal prime numbers, defined as primes such that no shorter subsequence of digits is prime. I calculated the prime numbers I need, but can not find a way to find minimal prime ...
0
votes
1answer
77 views

Two rook problems

I was thinking about chess problems and I couldn't find an answer for these. Are those open? If yes, how good upper or lower bounds are known? How many rooks one can put in a normal $8\times 8$ board ...
1
vote
1answer
28 views

Two player game about maximizing earnings subject to an interesting condition

Me and my friend had a bet. We each pick an integer between $1$ and $100$ inclusive and reveal it at the same time. Whoever picks the higher number has his number halved. Then the person whose number ...