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

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3
votes
3answers
103 views

What is the smallest number which begins with 7 and if you bring the 7 to the least significant position it becomes a third of the original number?

First I wrote the equation: $7\times 10^2+c_1\times10^1+c_0\times10^0 = 3(100c_1+10c_0+7)$ which becomes $679=290c_1+29c_0$ Then I try fix as many variables as possible. In this first iteration, ...
1
vote
2answers
260 views

Is my proof that the medians of a triangle are concurrent valid?

Consider any triangle ABC. Connect the midpoints of each of the three sides. The inscribed triangle is equal to the other three triangles and they are all congruent. It turns out that the medians of ...
7
votes
1answer
326 views

How did Euler solve the 4-whole-numbers-adding-up-to-a-perfect-square problem?

So I was watching a video on Leonhard Euler about how he amazingly solved so many difficult problems and one of the many problems that he solved was this: ...
8
votes
0answers
138 views

Take an m x n grid, and in each box pick two opposite corners at random to connect. What can be said about the resulting pattern?

Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior. Here's an example: (Note that ...
1
vote
1answer
153 views

Linear Algebra Recreational Problem

For each positive integer $k$, find the smallest number $n_k$ for which there exist real $n_k$ by $ n_k$ matrices $A_1; A_2; ....; A_k$ such that all of the following conditions hold: $$ \text{ 1. } ...
2
votes
4answers
380 views

Monty Hall problem vs. roulette systems - how are they different?

So I got interested in the Monty Hall problem - I understand what it's about, but somehow I can't wrap my head around the idea of the final choice not being 50/50. More precisely: we all know (or ...
22
votes
2answers
1k views

Question about a program generating palindromic prime numbers

I'm a programmer and software designer. I'm definitely not a mathematician and my maths is quite basic. One of my colleagues challenged me to generate a palindromic prime number, at least 1000 digits ...
3
votes
2answers
162 views

Men on a boat problem

There is the usual question of some men on a boat- various men have various speeds, the boat has a capacity of 2 men, and the boat takes on the speed of the slowest man in the boat at any given time. ...
0
votes
1answer
106 views

Closest Packing of Spherical Caps

Let the surface $S_n$ of the unit ball in $\mathbb{R}^n$ centered at the origin $O$ be defined as the set of points $P(x_1,x_2,…,x_n )$ such that $x_1^2+x_2^2+⋯+x_n^2=1$. Let the spherical cap $C(α)$ ...
0
votes
1answer
151 views

Recreational mathematics - Digit sum

Find the sum of all 3-digit positive numbers N that satisfy the condition that the digit sum of N is 3 times the digit sum of N+3 Can you help me with this question?
6
votes
0answers
139 views

Card passing game, maximum length

Quoting from this question: There are 25 people sitting around a table and each person has two cards. One of the numbers 1,2,..., 25 is written on each card, and each number occurs on exactly two ...
1
vote
3answers
643 views

Problems like the handshake problem

I am in college and my RA has been putting up little thought problems on his door for us to see as we pass by, but the ones he puts up aren't too interesting. I wrote up the handshake problem (invite ...
2
votes
2answers
114 views

combinations problem about apples and pears

Carlo has six apples and six pears: how many ways he can set in a row 6 fruits so that there should never be a pear between two apples? Thanks in advance to everyone who will help me resolving this ...
8
votes
1answer
209 views

Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?

Consider a modified version of Collatz sequence: $C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$ Let $F_n$ be the ...
6
votes
2answers
268 views

Vector spaces inquiry

Denote By $V$ the real vector spaces of all real polynomials in one variable, and let $P : V \rightarrow \mathbb{R}$ be a linear map. Suppose that $\forall$ $f,g \in V$ with $P(fg) = 0$ we have $P(f) ...
3
votes
2answers
323 views

A less challenging trivia problem

There are 25 people sitting around a table and each person has two cards. One of the numbers 1,2,..., 25 is written on each card, and each number occurs on exactly two cards. At a signal, each person ...
4
votes
2answers
391 views

A book of probability puzzles

I would like to train some recreational probability (Puzzles). Does any of you know a good collection? Preferably with hints or answers. I've been studying quite a bit of probability theory, but I ...
2
votes
1answer
523 views

Puzzle on the triangle.

In triangle top four figures that have to be repositioned to form the "triangle" without a unit square. How to explain this? Thank's.
8
votes
3answers
201 views

Group of sphere transformations, impressing friends

Ok, so here's the story: I am reading a book on algebra and, via some exercises, discovered that in any group $G$, the order of $x \cdot y$, written $o(x \cdot y)$, equals $o(y \cdot x)$. Now, this is ...
0
votes
2answers
199 views

Mathematical doodle games

Vi Hart's doodling videos and a 4 year old son interested in mazes has made me wonder: What are some interesting mathematical "doodling" diversions/games that satisfy the following criteria: 1) They ...
2
votes
2answers
208 views

(Theoretic) probability greater than 1

I am not expecting a "realistic" answer to my question, since it is based on an impossible scenario. What I'm waiting for is a theoretic explanation/interpretation so that I can sleep at night :) ...
1
vote
2answers
186 views

100 roads in a city, 1 is closed

In a certain country, 100 roads lead out of each city, and one can travel along those roads from any city to any other. One road is closed for repairs. Prove that one can still get from any city to ...
5
votes
1answer
308 views

67 67 67 : use 3, 67's use any way how to get 11222

I need to get 11222 using three 67 s (Sixty seven) We can use any operation in any maner 67 67 67 use 3, 67's use any way but to get 11222.
0
votes
2answers
1k views

What is the significance of the mirror numbers?

I'd like to hear insights and theory of the mirror numbers and their possible significance in mathematics and geometry. With mirror numbers I mean these four examples: ...
6
votes
1answer
391 views

“8 Dice arranged as a Cube” Face-Sum Problem

I found this here: Sum Problem Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same. $\hskip2.7in$ Here is one of 20 736 ...
7
votes
2answers
287 views

Mean and Median in a Classic River Crossing Problem

Consider the following classic problem: Four people on the west side of a river wish to use their single boat to get to the east side of a river. Each boat ride can hold at most two people, and the ...
2
votes
4answers
516 views

Speechless mathematical proofs.

Do you have proofs without word? Your proofs are not necessary has zero word, you may add a bit explanations. As an example, I has a "Speechless proof" for ...
1
vote
6answers
663 views

Queens in the chess board

A curious question. a) How many different ways exists to put "n" queens on a chessboard and they all can't capture the others? b)If you put one queen in the chess board knowing that exist one other ...
0
votes
1answer
42 views

Role of a difference in generating symmetric palindromes

An example : 923456781-123456789=799999992 Now divide it by the difference between the terminal digits, i.e. 9-1=8 So 799999992/8=99999999 Another example : 52314780-02314785=49999995/5=99999999 ...
3
votes
1answer
494 views

Super palindromes

Can anybody be kind enough to explain what exactly is a super palindrome? Also consider the following example : $923456781-123456789=799999992:9=88888888$ The largest prime factor of $88888888$ is ...
3
votes
3answers
302 views

9 hidden in every number regardless of number of digits?

Let us consider a number, e.g. 1234 Now reverse the positions of the terminal digits, so we get 4231 4231-1234=2997 which is divisible by 9 i have seen this for n-digit numbers, where n ranges ...
1
vote
1answer
421 views

Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
17
votes
7answers
1k views

Properties of the number 50

I will shortly be engaging with my 50th (!) birthday. 50 = 1+49 = 25+25 can perhaps be described as a "sub-Ramanujan" number. I'm trying to put together a quiz including some mathematical content. ...
1
vote
1answer
177 views

Winning single-pile, variable limits Nim [duplicate]

Possible Duplicate: Winning strategy for a matchstick game The rules of this variant of Nim are as follows: Starting at zero, each player counts up between 1-N numbers. The person that ...
0
votes
1answer
139 views

Gradually rising or falling numbers

I'm looking for a number series I can use for gradually rising or falling numbers. The number series should not be linear and should converge to a number at some point. (Sorry I'm really scared of ...
1
vote
1answer
143 views

Lemonade, Sandwiches, and Biscuits

Here is the example case. You might recognize this to be from Lewis Carroll's "A Tangled Tale". If you're already familiar, you can scroll down to the question. Example Given that one glass of ...
3
votes
3answers
530 views

An interesting problem of tossing a fair die successively

A fair die is tossed successively. Let $X$ denote the number of tosses until each of the six possible outcomes occurs at least once. Find the probability mass function of $X$. I'm also given this ...
2
votes
0answers
78 views

Tilting sealed drums for fun

A sealed cylindrical drum of radius r is filled with 9% of water. Now if the drum is tilted to rest on its side, show that the fraction of the curved surface area (not counting the flat sides) ...
2
votes
4answers
88 views

Solve for $x$ in this equation

How do I solve for $x$ algebraically? $$\dfrac{x^2(x^2-1)}{x+3} = 12$$
2
votes
0answers
137 views

For which positive integers n does there exist a prime whose digits sum to n?

Motivated by this earlier question, I thought of this problem: Question: For which positive integers $n$ does there exist a prime whose decimal digits sum to $n$? We can make two "easy" ...
1
vote
1answer
82 views

Unique representation of reals by (infinite) application of the (+,-,/,*,^) operations to elements of $\mathbb Q$?

There are many ways to express $\pi$ by infinite application of some simple operation (+,-,/,*,^) . Is there a method that represents all real numbers uniquely? By method, I mean a restriction to ...
3
votes
1answer
411 views

Mathematics of Tetris 2.0

Based on the question The Mathematics of Tetris, I was wondering if it is possible to have a series of tetris blocks that is impossible to clear. For example, getting the string TTTSS.. forces the ...
13
votes
1answer
283 views

What hexahedra have faces with areas of exactly 1, 2, 3, 4, 5, and 6 units?

I tried for a while, not very hard, to construct a polyhedron with exactly six faces, whose areas were respectively 1, 2, 3, 4, 5, and 6 units. I did not meet with any success. Still, it seems that ...
3
votes
1answer
99 views

How Much Should I Make?

A little background: I thought about this problem yesterday, while I was helping a friend out at a particular competitive event where he was to cook a certain amount of food for guests, who would then ...
2
votes
2answers
563 views

How do I find the center of a circle on a rectangular piece of paper?

I have a rectangular piece of paper with a circle printed on it. I also have a handy-dandy writing utensil. How can I locate and mark the center of the circle? Here's some technicalities: The ...
3
votes
1answer
134 views

Finite groups, where the number of subgroups is equal to the number of elements

I am looking for finite groups $G$ such that the number of subgroups is equal to $|G|$. Examples are: the trivial group $\mathbb{Z}/2\mathbb{Z}$ $S_3$ Does anyone know some more examples or can ...
3
votes
2answers
253 views

Rotating the graph of a function

I think I have answered the following puzzle: Rotations Under what conditions can you rotate the graph of a function about the origin, and still have the resulting graph being the graph of a ...
1
vote
2answers
112 views

Differing by $2$ “puzzle”

I've been thinking about the following: For (1) I drew $y= 0$, $y=2x$ and $y = 2x$, $y = 4x$. The expression I wrote down if $y = mx + b$ is one of the lines then $y_2 = (m \pm 2)x + b$ is the ...
1
vote
3answers
80 views

Looking for an equation

This is kind of a reverse question. A few years back I was presented with a functional equation problem, I don't remember it completely, and now I would appreciate the help of the math.SE hivemind to ...
7
votes
1answer
162 views

“Multi-facets” rope puzzle

I've done the following, can you tell me if it's correct? If $n$ is the number of sides of the rope and $k$ is the number of rotation, e.g. $k=0$ for glue each side to itself then I think the ...