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

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4
votes
3answers
7k views

Probabalistic proof of green-eyed dragons logic puzzle

I came across the "green-eyed dragons" puzzle (alternatively known as the "blue eyed villagers" puzzle). The typical proof uses a straightforward inductive strategy. I came up with a probabalistic ...
0
votes
1answer
146 views

Loonies and Toonies Combinatorics

How many ways can you make $n$ Canadian dollars using only loonies (Canadian \$1 coins) and toonies (Canadian \$2 coins) such that the numbers of loonies and toonies are different from one another? I ...
8
votes
1answer
229 views

How to prove that $\frac{1}{x_1}+\frac{1}{x_2}+…+\frac{1}{x_n}-\frac{1}{x_1x_2…x_n}\in \mathbb{N}\cup \{0\}$

Question: Show that for every natural number $n$ there exist $n$ natural numbers $ x_1 < x_2 < ... < x_n ,$ such that $$ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2......
6
votes
1answer
178 views

Why is $2^{16}=65536$ the only power of $2$ less than $2^{31000}$ that doesn't contain the digits $1$, $2$, $4$ or $8$ in its decimal representation?

$65536$ is the only power of $2$ less than $2^{31000}$ that does not contain the digits $1$, $2$, $4$ or $8$ in its decimal representation. http://en.wikipedia.org/wiki/65536_%28number%29
1
vote
1answer
1k views

How to find Fantasy Football Playoff Probabilities

A friend of mine came to me a few hours ago wondering what the probability of him making the playoffs were in our fantasy football league. I originally thought it wouldn't be too hard to figure out, ...
1
vote
1answer
55 views

Average rate of speed relative to a given point

For this question I am mainly concerned about points A and B on the image below and the image below hopefully helps illustrate my question. If point B is fixed and A has to move in a strait line in ...
2
votes
2answers
393 views

The water heater problem ( mathematician or plumber)?? [duplicate]

Isn't it absurd? $\textbf{Problem-}$ Suppose my water heater broke and heat in my apartment raised high. I went to a "person" to ask him to take a look at it, he came to my apartment, used a bunch of ...
1
vote
0answers
81 views

Maximum number of non-zero entries ,such that no two non-zero entries are on the same row or column.

In an M x N matrix such that all non-zero entries are covered in "a" rows and " b" columns. Then the maximum number of non-zero entries ,such that No two non-zero entries are on the same row or column ...
1
vote
0answers
48 views

Which $L \subset [0,1]$ equal the set of limits of a sequence of a sequence in $[0,1] \setminus L$?

I was glanced at this question here and it cause me to wonder the following: Question: Is there a simple description of the subsets $L \subset [0,1]$ with the property that there exists a sequence ...
1
vote
1answer
71 views

What's known about magic cubes of order 4?

An earlier question asked for a demonstration that there is no magic cube of order 4. The question was closed and deleted. I think it's worth having some information on magic cubes on m.se, so I'm ...
0
votes
2answers
136 views

3 Knights and Knaves

I have been struggling with this problem: Knights always tell the truth but knaves never tell the truth. In a group of three individuals (who we will label as N1, N2, and N3) each is either a knight ...
2
votes
4answers
251 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
0
votes
1answer
27 views

Possible Number Combos That I can not figure out [closed]

I am wondering, I have 4 QB's, 8 RB's, 12 WR's, 4 TE's, 4 K's, 4 Def, I can only play 1 QB, 2 RB's, 3 WR's, 1 TE, 1 K, 1 DEF for a total of nine players. How many different combinations do I ...
7
votes
1answer
95 views

For what numbers is $a_{b}= b_{a}$? (Reference?)

A student recently asked me about solutions to the equation $$a_{b} = b_{a},$$ where the subscript notation $a_{b}$ denotes interpreting the digits of $a$ in base $b$. It turns out there are tons of ...
1
vote
2answers
182 views

Finding where plots may cross with octave / matlab

I have several data points that are plotted below and I would like to find the frequency value when the amplitude value crosses 4. I've included an example note: this is an example and the amplitude ...
1
vote
1answer
103 views

how $2x=x$ , related to differential calculus [duplicate]

can anybody please tell me what's happening here ? $$1^2=1$$ $$2^2=2+2$$ $$3^2=3+3+3$$ $$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$ differentiating both the sides $$2x = 1 + 1 + \cdots+1 \mbox{ ...
1
vote
1answer
50 views

How do I calculate surface area given a three dimensional coordinates of a face?

I have three dimensional coordinates of a face, how do I calculate surface area?
5
votes
2answers
193 views

Construct numbers using digits $123456789$ and the operations $+,-,×,÷$

From an old book I found the following question. Use the digits $1,2,3,4,5,6,7,8,9$ and the operations $"+,-,×,÷"$ with $( )$ for construct the result $100.$ During the computations the order of $...
1
vote
1answer
34 views

Mathematical reasons for hull design relative to sustainable angle of heel

I've recently been doing a comparative study of ancient Sumerian mythology relative to the book of Genesis. I am curious if there is a way to explain mathematically why a circular, square (cubic) or ...
1
vote
0answers
66 views

Game idea “square or not”

I have an idea of a quadrilateral / square game, and am looking for help. For the moment lets call it the "Square or Not " game. Imagine we have a big stack of cards with on each card some property ...
0
votes
1answer
78 views

Concatenating squares - is this solution unique?

This question asks about concatenated squares to make a square number. For example $[4][9]=49, [16][9]=169, [3136][441]=3136441, [64][009]=64009$ I've been doing a bit of investigating for the case $[...
2
votes
0answers
63 views

Shannon number upper and lower bounds

What are the best proved upper and lower bounds for the Shannon number, i.e. number of possible positions of chess? Is the upper bound 7728772977965919677164873487685453137329736522 given in http://...
0
votes
2answers
33 views

replacing numbers to get final anser

I found this question in a random problem solving book that I was reading and wanted to know how you would solve it. I am not sure as how to go about this. Take any positive integer $n$ with fewer ...
1
vote
0answers
70 views

Which numbers have the sum of their digits equal to the sum of the digits of their inverse?

$n$ is a number such as $n \in \mathbb{N}$ and $n >0$.(Eg. $n = 8$) $p$ is the sum of the digits of $n$ in base $10$.(Eg. $n=80$, $a = 8+0 = 8$) $q$ is the sum of the digits of $1/n$ in base $10$....
2
votes
1answer
145 views

How to conceptualize unintuitive topology?

I found Project Origami: Activities for Exploring Mathematics in my university's library the other day and quickly FUBAR'd (folded-up beyond all recognition) the couple sheets of paper I had with me ...
1
vote
0answers
54 views

Has the mathematics of 4d-tetris, or any other 4-dimensional polyforms been studied?

There are a few variations of 4d tetris games floating around the internet, but I'm more interested to know if there's been mathematical research done in the area of 4d polyforms. I assume that the 4d-...
2
votes
1answer
239 views

Pig Wheel question

A friend of mine was playing the bar game Pig Wheel recently and posed some interesting questions to me. He was playing with others as a group of four and, acting collectively, they came out about ...
3
votes
2answers
446 views

Finding out a person's age in days given their birthday dd/mm/yyyy?

It has to be somebody alive today. Assume that the day is today - September 15, 2014. This is convenient because the leap years will be regular (once every for years; the weird rule applies to $1900$ ...
3
votes
0answers
93 views

Bitcoin math problem example

Disclaimer: I'm not a mathematician, if something is complicated, please use layman's terms. Thank you. I'm wondering about this bitcoin thing. I have heard that mining is using a computer to solve ...
2
votes
2answers
138 views

What is the meaning of this (potentially humorous) mathematical equation?

The equation in the image shown below (outlined in blue) was found on the cover of a magazine, along with several other "math equation jokes" like the "I heart pi" joke. My friends and I haven't been ...
3
votes
1answer
112 views

Is there a prime number ending with the natural number $n$

if $n$ not is divisible by 2 or 5? Example: given 813075843967837637675737563754361301, there is a prime 20813075843967837637675737563754361301 or given <...
0
votes
1answer
41 views

Integer solutions of an equation that is set to a number

How many integer solutions for $a$ and $b$ in $(ab)/(a+b)=3600$? My attempt: $(ab)/(a+b)=3600$ = $ab=3600(a+b)$ = $ab=3600a+3600b$ =$ab=3600a=3600b$ Dividing $3600b$ on both sides =$a(1-3600)/...
3
votes
3answers
154 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
3
votes
1answer
234 views

How are Sudoku puzzles created?

I recently read about the connection between solving Sudoku puzzles (and other graph coloring problems) and Groebner bases. This doesn't lead to an efficient solution technique, but it does link a ...
0
votes
0answers
293 views

Prime number distribution theory for dummies

For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis http://en.wikipedia.org/wiki/Riemann_hypothesis#...
2
votes
3answers
462 views

Fun proofs for layperson?

I'm not quite sure whether this question belongs here, because it has no definite answer. But I'll give it a shot. If any of the mods objects, then I will, of course, respectfully delete this ...
3
votes
1answer
543 views

Mathematics of paper fold-cutting

Take a square of paper... ... and fold it any number of times using consecutive straight folds... ... then cut off any number of pieces using consecutive straight cuts... ... and unfold the ...
0
votes
0answers
51 views

Limiting behaviour of a system

A friend of mine offered me the following problem. Suppose we have a rabbit and a fox in $\Bbb R^2$. The rabbit starts at time $t=0$ at the point $(0,0)$ and runs with constant speed $(1,0)$. The fox ...
7
votes
3answers
2k views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
-2
votes
1answer
42 views

Equation that outputs digit in 1's 10's 100's slot [duplicate]

I need an equation that outputs the digit in the slot of my choosing EX1: I want the 10's slot in 1837 EX2: I want the 10's slot in 123456789 EX3: I want the 1000's slot in 93037352 I also need it ...
-2
votes
1answer
45 views

Challenge - “Highscore” output equation

I need an equation capable of processing 2 inputs to make one output that is either = to input 1 or 2. This is how it works. Since it is working with scores and such, Input1 will be "Last Score", and ...
21
votes
10answers
3k views

Get $5$ by doing any operations with four $7$s

How can one combine four sevens with elementary operations to get $5$? For example $$\dfrac{(7+7)\times7}{7}$$ (though that does not equal $5$). I am not able to do this. Can you solve it or prove ...
0
votes
1answer
39 views

Form $4$ new symbols with the most common symbols

Suppose we have $6$ symbols, say $A,B,C,D,E,F$. We are asked to form $4$ new symbols using the $6$ symbols with the addition operation. For example, the $4$ new symbols can be $A+C+E, F+E+A, D+C+A,...
5
votes
0answers
148 views

Folding sheets of paper to measure distances

You have $n\in\mathbb{N}^*$ sheets of paper with dimensions $a,b\in\mathbb{R}_+^*$ that can be folded as many times as needed. What is the set of lengths in $\left]0,\sqrt{a^2+b^2}\right]$ one can ...
0
votes
2answers
333 views

Reciprocal of 81 being the sequence of all natural numbers?

According to this document: http://www.answering-christianity.com/fakir60/81.htm describing the theory of scientist Peter Plichta, the reciprocal of 81 is: the ...
1
vote
3answers
65 views

Easy exponentiation method

Is there a simple way of solving, say, $x^{3/2}$? For example, one way of solving $16^{3/2}$ is to calculate the square root of $16^3$, but I was wondering if there is a simpler mental trick for ...
11
votes
1answer
264 views

Mathematics of the Ice Bucket Challenge

I've been considering the mathematics of the now global ice bucket challenge. Simple model In the simplest incarnation, there is one original seed, who then nominates 3 others, each of which take ...
0
votes
3answers
251 views

Completing the square for a quartic expression

By completing the square, find (for real $x$) the minimum value of: $$x^4 + 2x^2 + 2.$$
1
vote
2answers
696 views

Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

Thanks for the help! I've written a more detailed proof. The hints were great.
1
vote
2answers
56 views

Proper way to express 0 in this case?

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use. $$\prod_{ ...