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

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How many multiples of 3 are between 10 and 100? (SAT math question)

In the figure above, circular region A represents all integers from 10 to 100, inclusive; circular region B represents all integers that are multiples of 3; and circular region C represents all ...
11
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2answers
7k views

What is the best strategy for Cookie-Clicker-esque games?

Today, I stumbled across the game Cookie Clicker, which I recommend you avoid until you have at least a few hours of time to waste. The basic idea behind the game is this: You have a large stash of ...
2
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3answers
654 views

A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013

On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and ...
2
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2answers
132 views

When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
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1answer
64 views

Guessing Game Stochastic Optimization

This is part of another post I did, but I think it has interest in its own right: Let $Y =\{X_{1},X_{2}...X_{N}\}$ be a set of $N$ random quantities with assocated set of distributions ...
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2answers
186 views

Interesting problems using group/representation theory

I've been going through this representation theory lecture notes, and I've found the ''hungry knights'' problem very interesting. Do you know any interesting problems similar to that one? To define ...
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3answers
295 views

Find all the integral solutions to $2x+3y=200$

What's the best way of going about this? $$2x+3y=200.$$
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4answers
516 views

Interesting Medieval Mathematics Lecture/Activity Ideas?

Recently I have been invited to give a talk about Medieval Mathematics or mathematics in the 500 AD - 1500 AD time frame. I have been researching the time frame for the past week and have found ...
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0answers
71 views

What are all possible numbers gotten by an digit-exchange operation?

A friend of mine taught me a number game. Supposting that $a_na_{n-1}\cdots a_1$, which satisfies $a_n\gt a_1$, is a natural $n$-digit number with decimal representation, let's consider the ...
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2answers
1k 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...
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1answer
185 views

Filling 4l, 5l bottles from two 10l bottles

There are two bottles of 10litre each filled with water. Now two persons having empty bottles of 4litre and 5litre want to take 2litres of water each from the previous 10litre bottles.. Now you ...
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1answer
484 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
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0answers
95 views

Conway's Game of Life

Is there a mathematical way to directly calculate iteration n from the first iteration skipping calculating the iterations in between in Conway's Game of Life? I would assume, if it is possible, it ...
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1answer
82 views

A persistent difference

Here's a fun math problem. I wasn't able to get it - am curious what you guys have to say. Pick a four-digit number whose digits are not all the same. From its digits form the smallest four-digit ...
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2answers
91 views

Simple question that I can't solve [duplicate]

Here is a relatively simple question that I'm unable to solve :/ There are $10000$ closed lockers in a hallway. A man begins by opening all $10000$ lockers. Next, he closes every $2^{nd}$ locker. ...
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3answers
216 views

Can we identify the time if we know every angle between three hands of a watch?

Let $M, H, S$ be the minute hand, the hour hand, the second hand of a watch respectively. Also, let $A_{MH}, A_{MS}, A_{HS}$ be the angle between $M$ and $H$, $M$ and $S$, $H$ and $S$ respectively. ...
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1answer
71 views

A neat application of Chebyshev's inequality

An interesting little fact I noticed today while problem-solving: Show that, for any positive reals $x_1, x_2, ... x_k$ and any positive integers $m, n$ with $n >m$, $$ x_1 x_2 \cdots x_k \ge 1 ...
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2answers
384 views

The next number that has this property?

I noticed that $1/8 = 0.125$ and the sum of the digits of the number $0.125$ is $0+1+2+5=8$. It's lovely. I searched other numbers who have that propriety : I only found $1$, $3$ and $8$. Is there ...
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6answers
2k views

answer to iq test with colored squares

What is the best (whatever this means) answer and why?
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1answer
104 views

How to calculate x in formula $ y =\left( \frac{\frac{17000}{x+400}+8.5}{100}+ 1\right) x $?

I am not even sure how it's officially called (so not sure with tag to give it). As an example if you have a math problem $y = x + 1$. You have a $y$ value, but not $x$. To you revese the problem $y - ...
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1answer
66 views

given a positive integer $n\geq 2$, we have a positive integer $m$ such that $m+2,m+3,\dots m+n$ are composite. (TIFR exam $2012$)

Question is to prove that : given a positive integer $n\geq 2$, we have a positive integer $m$ such that $m+2,m+3,\dots m+n$ are composite. I tried checking for small numbers to see if there is any ...
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0answers
84 views

Existence of a Vampire number on the form $v = xy = a^bb^a$?

A number $v = xy$ with an even number $n$ of digits formed by multiplying a pair of $n/2$-digit numbers (where the digits are taken from the original number in any order) $x$ and $y$ together. ...
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19answers
2k views

Literary statements that are false as mathematics [closed]

I recently wanted to use the title of the famous short story "Everything that Rises must Converge" in a poem of mine. However, the mathematician in me insisted on changing it to "Everything that ...
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1answer
167 views

Surprising limit (probability of no two coinciding pairs)

I stumbled upon this question by random chance. The motivation is kind of long, the question is pretty short; if you're just here for the limits, feel free to skip to the break. I'm taking five ...
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1answer
340 views

How many $n$-disk legal configurations are there for the Tower of Hanoi?

This question comes from this homework assignment from ECS20 at UC Davis. How many $n$-disk legal configurations are there for the Tower of Hanoi? A "legal configuration" means that no disk is ...
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1answer
135 views

A puzzle related to three cars which leave a town and reach another.

I'm trying to solve the following puzzle: $C_1, C_2$ and $C_3$ are three cars that leave town $T_1$ and reach town $T_2$. For a car, say $C_k, k$ is considered to be the car number. The car number ...
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1answer
147 views

Lattice Squares; Basic Interesting Facts and Problems

I'm going to write an article in an educational magazine for middle school students, about the game Square It. The purpose of the game is to make lattice squares: I want to introduce the game, and ...
3
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1answer
116 views

Find the values of $(a,b,c)$ such that $a^{2013}+b^{2013}=c^{2013}$ and $a^2+b^2=c^2$.

My professor likes to give our class some questions for fun every once in a while. He posed the following problem in class yesterday, and I've been stuck. Find the values of $(a,b,c)$ such that ...
6
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1answer
426 views

Hilarious Comic … DiffyQ and infinity ensue…

I ran across this comic, and it's gold. It is orginially published here If I am correct, the first panel alone defines a self-referential loop if not a differential Equation: $X$: Amount of Black ...
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2answers
101 views

Is $\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}?$, for $a,b,c,d>0$?

Is $$\frac{a+b}{c+d}<\frac{a}{c}+\frac{b}{d}$$ for $a,b,c,d>0$ If it is true, then can we generalize? EDIT:typing mistake corrected. EDIT, WILL JAGY. Apparently the real question is Is ...
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3answers
143 views

Rating changes and probability calculations for chess world championship

I have some interesting questions that have to do with the rating changes and calculations for the Anand-Carlsen world championship. Most of this has to do with solving a system of linear equations. ...
2
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1answer
39 views

Error correcting binary partition

Let's say I have a collection of $2^n$ labeled objects, and I want to find one of them. If I can ask yes-no questions about it, binary partition would immediatly lead us to the desired object in $n$ ...
0
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1answer
116 views

finding the total number of ways of arranging bulbs!

In a grid of size $3n$, each tile is $1$ unit long and $1$ unit wide. There are bulbs beneath the tiles. Each bulb is responsible for lighting exactly two tiles. No tile should be lighted by more than ...
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0answers
28 views

Question about Horses [duplicate]

For P(n) being the assertion that in every set of n horses, all horses have the same color: P (n) ⇒ P (n + 1), when n > 1. How do I prove this problem, can someone give me explanation or a headstart? ...
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3answers
106 views

Finite number of points inside a disk

Let $n\ge 2$ and suppose that $z_1, z_2, \ldots, z_n$ are distinct points in the interior of some disk $D$ in the plane. Why is it true that there exists a smaller disk $D'\subseteq D$ such ...
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2answers
347 views

The rotation of dice on a grid

Imagine you have a plane, flat surface with a square grid drawn on it. You have a standard cubic die which is placed flat on the surface. Its length is the same as the length of side of each grid ...
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11answers
2k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
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1answer
2k views

How to arrange a pile of coins into two piles such that both piles have equal number of heads up?

You have a pile of 100 coins with 90 tails up and 10 heads up. You have to divide this pile of 100 coins into two piles (may be of unequal size). You are blind-folded and you have to divide it such ...
2
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0answers
59 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
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1answer
83 views

Confusing DE concept question

This is the question: A reduced copy of a painting by Kandinsky is placed on the top of original. Is there a point of the painting covered by a point of the copy which has the same color? If it is ...
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3answers
155 views

All Humans have the same gender

This was actually a homework assignment in a math lecture in Germany. We prove with mathematical induction that all humans have the same gender. So consider a room with $n$ people. For $n=1$ the ...
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2answers
524 views

Numbers Brain Teaser

Alice and Bob have two positive integers, x and y respectively, glued to their foreheads, so that each can read the other’s number but not their own. They also know that |x − y| = 1. The following ...
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3answers
256 views

The game of hand cricket.

(If you could add a more suitable title, I'd be very grateful to you) My friend gave me this question: In the game of hand-cricket (hand baseball, if you like it :D) two players $X$ - the batsman ...
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2answers
928 views

Possible distinct positive real $x,y,z \neq 1$ with $x^{(y^z)} = y^{(z^x)} = z^{(x^y)}$ in cyclic permutation?

Can we have distinct positive real $x,y,z \neq 1$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutation? It does not work well if any ...
36
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2answers
986 views

Proof that $123456789098765432111$ is prime?

The mathematician Charles Weibel asks on his home page the following "fun question": How can you prove that 123456789098765432111 is a prime number? (He notes the fact $$12345678987654321 = ...
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1answer
96 views

How to derive a generating function for the following series

Given an integer n how would derive a function fn that is without conditional statements, does not use ...
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2answers
1k 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 ...
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2answers
1k views

How to create models for solving logic puzzle

this is not about getting an answer for this problem, but to create models for solve them, below I will show a simple puzzle in order to demonstrate what I mean: Lets say two friends just left a bar ...
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2answers
195 views

Question on triangle with heights

Prove that there exists no triangle with heights 4,7, and 10 units. I am completely puzzled.
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2answers
98 views

Selection minimum out of $n$ different objects!

Suppose we have $n$ differnt persons and $n$ different objects.They have to select from the the objects such that every pair of person has at least one uncommon object. What is the minimum number of ...