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

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5
votes
1answer
179 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 ...
2
votes
1answer
413 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 ...
2
votes
1answer
169 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 ...
0
votes
1answer
185 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
votes
1answer
119 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
votes
1answer
490 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 ...
1
vote
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 ...
1
vote
3answers
163 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
votes
1answer
41 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
votes
1answer
155 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 ...
2
votes
0answers
30 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? ...
1
vote
3answers
109 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 ...
9
votes
2answers
488 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 ...
25
votes
11answers
3k 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. ...
0
votes
1answer
3k 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
votes
0answers
68 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. ...
1
vote
1answer
86 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 ...
5
votes
3answers
204 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 ...
3
votes
2answers
631 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 ...
2
votes
3answers
362 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 ...
43
votes
2answers
1k 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 ...
41
votes
2answers
1k 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 = ...
2
votes
1answer
119 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 ...
24
votes
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 ...
1
vote
2answers
2k 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 ...
8
votes
2answers
202 views

Question on triangle with heights

Prove that there exists no triangle with heights 4,7, and 10 units. I am completely puzzled.
0
votes
2answers
107 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 ...
2
votes
1answer
54 views

Underlying Connections (The Tower of Hanoi)

It is well established that the least number of steps of solving the Hanoi tower problem for $n$ disks is $2^{n}-1$. Now consider the following problem: Player 1 picks a natural number between 1 and ...
3
votes
1answer
189 views

Where can I find Putnam competition questions and solutions online?

Math people: Until recently, at least, there existed at least one Web page containing complete Putnam competition problems and solutions from the past twenty years or so. In retrospect, I see that I ...
70
votes
24answers
15k views

Blue eyes: a logic puzzle

Today I read the Blue Eyes puzzle here. I also read the solution which I find quite interesting. But there are three follow up questions which I don't know the answer to: What is the quantified ...
0
votes
1answer
102 views

Man walking along a circle falling in a ditch.

Consider a circle as in the figure. It has a small ditch of width $L$. A man is walking around the circle with step length $\alpha$ (measured along the circumference). $\alpha$ is irrational. We need ...
0
votes
1answer
111 views

Is it appropriate to use conjectures in contest?

Is it appropriate to use conjectures in math contest? I've been to fair amount of math contest and as far as I know the judges want a solid proof for every step take to make to solve the problem. But ...
39
votes
5answers
2k views

How much weight is on each person in a human pyramid?

After participating in a human pyramid and learning that it's very uncomfortable to have a lot of weight on your back, I figured I'd try to write out a recurrence relation for the total amount of ...
8
votes
3answers
894 views

What is the probability that GCD of $(a,b)$ is $b$?

My question is quite simple. I have been googling a lot lately trying to find a solution to this: Given a sequence of n integers $[1,2,...,n]$. If we pick two numbers randomly from the set say, a and ...
3
votes
5answers
1k views

Riddles with a mathematical twist

I am looking for riddles that are understandable for everyone(so especially non-mathematicians) but require mathematical knowledge or deep abstract ideas to be solved. The best answer will be the ...
4
votes
1answer
142 views

Fictional math proof = prime return function

I am trying to write a piece of future fiction where one of the characters is famous for proving an important truth related to primes. I want to make it as realistic as possible, but i'm not a math ...
0
votes
2answers
351 views

Math Riddle Problems

My numerator AND denominator are both prime numbers less than 10. I am more than 1/2. I am less than 1. I am nit equivalent to 8/12 or 10/14. -What number am I? If you double me, I'm more than 1. If ...
-1
votes
2answers
118 views

Topic on Proofs without words [closed]

There's a mathematical presentation to be done in college of 15 minutes! I've intended to show two proofs without words! Can you suggest me any two of them?
1
vote
1answer
2k views

How do you calculate work (KJ) and Power (W) when jogging on a treadmill?

The following is known... The Weight of the Person, The Angle/grade of the Treadmill, The Time that they are on the treadmill, The velocity of the treadmill. How do I calculate Work in Kilojules and ...
4
votes
1answer
2k views

“The Original IQ Test” — A peg puzzle

Many of you may be familiar with a puzzle game that consists of a 15-peg triangle and is dubbed as "The Original IQ Test". The idea behind the game is that you fill 14 of the 15 holes with pegs, and ...
2
votes
6answers
641 views

Neat expressions that equal 1

I would like to see beautiful and elegant expressions involving elementary and non-elementary functions, transcendental numbers, etc. that equal 1. Be creative!
9
votes
0answers
128 views

Generalization of Gray codes

A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ...
3
votes
1answer
81 views

Basic number theory curiosity

Let $a,b $ be positive integers, and consider the set $D := \{am + bn: m,n \in \mathbb{Z} $ and $am + bn > 0 \}$. Let $d$ be the minimum integer in $D$. Is it true that $D = \{kd : k \in ...
2
votes
2answers
218 views

General and Simple Math Problem. [duplicate]

Three friends brought 3 pens together each 10 dollars. Next day they got 5 dollars cash back so they shared each 1 dollar and donated 2 dollars. Now the pen cost for each guy will be 9 dollars (\$10 ...
0
votes
1answer
94 views

Variation on Birthday Problem - Probability that 47 of 191 students have birthdays on two conditions.

It's my birthday, and I figured I will create a problem based on birthdays that I myself am unable to solve! Assuming time is denoted by HH:MM:SS, MM/DD/YYYY, what is the probability that in a class ...
4
votes
2answers
297 views

Filling a bag with fruits of four types, with constraints

Here's a diabolical math problem that I found. In how many ways can we fill a bag with n fruits subject to the following constraints? • The number of apples must be even. • The number of bananas ...
1
vote
0answers
61 views

Linear algebra function that creates decreasing product vector of original vector

For vector $y=[y_1,y_2,\dots y_n]$ , let $\gamma = \sum_{i=1}^n \gamma_i$ , and $\gamma_i(n-i+1)=y_n*y_{n-1}*\dots y_i$ so that $\gamma$ looks like $[y_1*y_2*\dots y_n, y_2*\dots*y_{n}, \dots y_n]$ ...
2
votes
3answers
205 views

How many zeros are there in $1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}$?

How many zeros are there at the end of $1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}$ ? I tried it by grouping all the $2$'s and $5$'s and $5$'s and $6$'s but cant get my answer...
3
votes
1answer
188 views

the ratio of the following two areas

Suppose you have the following triangle $ABC$: with the following properties: $|AB|=4\cdot |AA'|$, $|AC|=4\cdot |CC'|$, $|BC|=4\cdot |BB'|$. I have to find the ratio of the total area of the triangle ...
1
vote
1answer
211 views

Show me where I have made a mistake in interpreting or solving this number puzzle

The following number puzzle was published in the Sunday Times on 25th August this year. It is credited to Danny Roth. George and Martha have a book of puzzles numbered from 1 to 30. The solutions ...