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

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0
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1answer
18 views

Armstrong numbers in base 90

Are there any Armstrong numbers (narcissistic numbers) in base 90? Of course, except the one-digit ones. There don't seem to be. Just curious.
3
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1answer
48 views

If the permuted set of $(1,2, \dots n ) $ is such that sum of any two adjacent numbers is a square. Find the generalized form of $n$.

$ \text{Let}$$ P(n) \text{be permutation of}$$ (1,2 \dots n)$$ \text{such that if}$$ P(n)={a_1,a_2, \dots a_n} $$ \text{then} $$(a_i+a_{i+1})=k^2$$ \text{where}$$ k\in \mathbb{N}$ and $i \in {1,2,3, ...
33
votes
6answers
1k views
+200

Connecting three houses to three utilities

When I was a child I was given this problem to send a wire from electricity, water, and internet to each of the houses, all three houses must have all three wires connected without being crossed ...
0
votes
1answer
355 views

Find Volume of a Wineglass - a Slice

A spherical wineglass, known diameter, is brim full of wine. What is the width of any vertical parallel slice with a volume of 1/4 of the wine? Assume that the height of the wineglass is known.
2
votes
0answers
46 views

area estimation with tiling

For any given shape drawn on a graph paper, a kid can calculate the area of any shape by counting the tiles with a simple formula: any edge covering 50% or more, mark the tile; total area = sum all ...
2
votes
1answer
323 views

Arithmetic progressions of perfect powers

Find the largest positive integer $n<100$, such that there exists an arithmetic progression of positive integers $a_1,a_2,...,a_n$ with the following properties. $1)$ All numbers ...
10
votes
3answers
314 views

A problem with 26 distinct positive integers

I am trying to solve the following problem. Assume that we are given 26 distinct positive integers. Show that either there exist 6 of them $x_1<x_2<x_3<x_4<x_5<x_6$, with $x_1$ ...
1
vote
0answers
54 views

Big list of fun mathematical book to “play” with classmates

I am searching for some fun maths books "have fun" (mathematically) with my classmates. To give you a better idea of what I'm looking for, I'll mention some books that I find suitable: Roger B. ...
2
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1answer
56 views

Brainteaser Switches

You have four switches that could be on or off that are configured in a 2x2 grid. You are given an initial configuration that is random and you are blindfolded. (a) Can you possibly find the ...
4
votes
3answers
86 views

How many perfect shuffles are needed to go back to initial state?

The other day, in a popular-science book, I saw: Given 32 cards ($c_i$ for $i=0..31$), cut the deck into two parts and shuffle them the American way (riffle shuffling). The deck now looks like: ...
1
vote
3answers
44 views

Why is the three utility problem important?

I came across this problem on this website, even though it was fun to answer it there was something sad I realized, and I wanted to ask this from the Math Stack Exchange community. The question: A ...
1
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3answers
74 views

Is there a shortcut for raising 2 to the power of a number (e.g. $2^{27}$)?

In networking, when dealing with subnetting, you convert the net mask to binary and count the number of ones (for the example in the question there would be $27$ $1$'s) and to figure out how many ...
5
votes
1answer
444 views

Space-filling polyhedra (or honeycomb) survey?

Is there a survey anywhere of space-filling polyhedra? MathWorld's article, space-filling polyhedron, mentions about 400 being seen in pre-1981 books and papers. Wikipedia mentions 28 convex uniform ...
4
votes
0answers
48 views

Quantifying infinitely large sums such as $\sum_{x\in\mathbb{R}^+} x$

I thought of this as a student in calculus years ago, and it may be a silly kind of question. I wondered if there were notions of different sizes of infinity a series might sum to, which then lead me ...
19
votes
1answer
374 views

Infinite staircase to a circle

Suppose you start at $(0,0)$ on the unit disc and repeat the following procedure again and again: Face east and walk half-way to the circumference. Face north and walk half-way to the circumference. ...
457
votes
139answers
28k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
1
vote
5answers
68 views

Game Theory - First move vs second move advantage?

This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math, and we weren't sure how to get the answer. I apologize in advance if my ...
1
vote
1answer
37 views

Find the number of people in the family

PRE-RMO 2014 question 14 (set-A) One morning,each member of Manjul's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. ...
1
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1answer
1k views

A recreational math problem, integers in a grid

I was thinking of the following recreational math problem: We have a $4\times 4$ square filled with integers $a_{1,1},...,a_{4,4}$. It has $30$ sub-squares $A_{i,j,k}$, corners of the form ...
23
votes
2answers
820 views

A strange little number - $6174$.

Take a 4 digit number such that it isn't made out the same digit $(1111, 2222, .. . $ etc$)$ Define an operation on such a four digit number by taking the largest number that can be constructed out of ...
1
vote
4answers
92 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 ...
-1
votes
1answer
63 views

Computation of integral $\int_{0}^{1}\ln(p)\ln(1-p)p^{2}\,dp$

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 tried to do change of variable it does not ...
0
votes
1answer
33 views

Calculate winner of soccer match

I am writing a program that simulates a soccer tournament between countries using their FIFA rankings. I am looking for a function that takes two country rankings and outputs a number between (about) ...
4
votes
2answers
111 views

I have used Cauchy and Jensen. It is not helping me very much. Advice on solving this problem.

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
0
votes
0answers
17 views

External tangent to the circle

http://www.artofproblemsolving.com/Wiki/index.php/2006_AMC_12A_Problems/Problem_19 I don't understand why L1 'clearly' bisects the angle formed by L2 and the x-axis. Is this a standard result ...
1
vote
0answers
53 views

measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
9
votes
4answers
1k views

Secret santa problem

We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem. We have 4 people ...
2
votes
1answer
4k 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 ...
4
votes
0answers
83 views

Can -9 to 9 be placed in 41 lines of zero?

The cubic curve $2x^3-4x^2y+2xy^2-8x+y^3-y$ can be used to get lattice points allowing the placement of the numbers $-8$ to $8$ so that all 32 triplets that sum to 0 will be a straight line of three. ...
1
vote
0answers
42 views

Are there tricks to solve seating arrangement problems?

How are seating problems solved in general? I am stuck on this one for example. There are 8 houses in a line and in each house only one boy lives with the conditions as given below: Jack is not the ...
1
vote
1answer
36 views

7th grade percentage question

In the year $2010$, John's monthly salary was $\$3500$. John's monthly salary in $2010$ was $25\%$ more than it was in $2009$. Calculate his monthly salary in $2009$. Many of us debated the answer ...
6
votes
0answers
55 views

Name of a certain set

I want to know if there is any already-standard way to refer to the set described as follows. Take the set of all primes in $\mathbb{Z}$, call it $\mathbb{P}$. Take the set of all finite products of ...
8
votes
1answer
191 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 $$ ...
0
votes
1answer
103 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 ...
5
votes
2answers
160 views

Rooks in 3D chess board

How many rooks are needed for a 3D chess board of size NxNxN so that every empty cube on the board can be reached by a rook in a single move?
1
vote
0answers
81 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
59 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
42 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 ...
1
vote
2answers
212 views

A problem for math lovers to count the digits

Today a classmate of mine asked a question which is based on counting. Question. Find a positive integer which when multiplied up to $6$ times will give numbers having the same digits but rearranged ...
0
votes
2answers
36 views

Suppose T(k) denotes the smallest number of steps needed to move from k to 100.Find y & z such that T(9)= 1+ min (T(y),T(z)).

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers i , j ...
0
votes
2answers
55 views

The water heater problem ( mathematician or plumber)??

Isn't it absurd, I mean doesn't it make probability 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 ...
1
vote
1answer
24 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 ...
1
vote
0answers
21 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
19 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 ...
0
votes
0answers
73 views

Why does $\infty !$ equal $\sqrt{2\pi}$? [duplicate]

It seems at least counterintuitive. Is this related to the way we do non-classical summations?($\displaystyle\sum_{k=1}^{\infty} k=\zeta(-1)=-\frac{1}{12}$ etc)
0
votes
1answer
42 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
0answers
513 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
3
votes
2answers
568 views

How to create number six using three zeroes?

How to create number 6 using only three 0, any arithmetic operation is allowed? I know it is possible, but I don't know how...
9
votes
2answers
176 views

Good Reference for Justifying (less well-known fields of) Math?

How do we as mathematicians justify the study of math to students? Or, indeed, how do we justify it to the general public? How do you justify your particular field? I'm particularly interested in ...
0
votes
1answer
291 views

Striking off a digit from each of the numbers written in seven rows, while preserving arithmetical operations

Problem Strike off any digit from each number in seven rows (need not be at same place) and combine the same operations with 3 digit numbers to get the same addition. After this strike off another ...