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

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36
votes
2answers
972 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
95 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
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 ...
8
votes
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.
0
votes
2answers
95 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 ...
1
vote
1answer
50 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 ...
2
votes
1answer
164 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 ...
61
votes
24answers
12k 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
68 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
105 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 ...
33
votes
5answers
1k 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
836 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
121 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
233 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
105 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
973 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
1k 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
413 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!
7
votes
0answers
109 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
79 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
181 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
89 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
247 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
58 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
172 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
185 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
173 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 ...
4
votes
1answer
72 views

Odd marble out in $m$ weighings on a balance

Thinking that this may be as poorly known as it was $35$ years ago when I first knew the truth - based on How can you pick the odd marble by 3 steps in this case? and others, I pose two closely ...
0
votes
3answers
445 views

Ways to select donuts

Wanted to share this puzzle: A restaurant offers choice of six different types of donuts, each available in unlimited quantity. How many ways can you select three donuts? You can pick any number of ...
2
votes
1answer
89 views

Alligators and Creepy Crawlers

I got this math question wrong, but I'm not exactly sure why. Here's the question: If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? ...
4
votes
1answer
202 views

How can I find the smallest set such that all even integers $4≤x≤N$ can be written as the sum of two elements in the set?

Given a positive integer N, I want to create a set of positive integers such that any even number $4,6,8,...N$ can be written as the sum of two elements in the set. I also want the set to be as small ...
1
vote
1answer
3k views

How to greet a mathematician on his birthday with an excitement

Someone told me this, but I dont get it. I hope this is on topic. Otherwise please tell me and I will delete this question. $$\Gamma (Happy Birthday + 1)$$ Why is this the way to greet a mathematician ...
2
votes
3answers
1k views

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end.

Prove that in a parabola the tangent at one end of a focal chord is parallel to the normal at the other end. Now, I know prove this algebraically, and that's very easy, but I am not getting any ...
1
vote
0answers
102 views

Probability of occurrence of games in a football league

This question just came to me as I was watching a football game. There is a football league with 20 teams. Each team has to play every other team at home and away, which means each team will play a ...
1
vote
1answer
50 views

Sum of largest two angles

All the inner angles of a 7 sided polygon are obtuse, their sizes in degrees being distinct integers divisible by 9. What is the sum (in degree) of the largest two angles?
-3
votes
2answers
1k views

Egg problem-Brain Teaser-Amazon Interview Question

A lady from the chicken farm gathers the eggs and brings it to sell it in the market. She sells the eggs but few eggs are left over. The 2nd day the left over eggs was doubled. Yet she sells the ...
3
votes
8answers
437 views

simple theorems that non-mathematicians can understand and appreciate.

For example, I stated this fact/theorem at a dinner to some friends and they were pretty impressed. Given any sequence of n integers, positive or negative, not necessarily all different, some ...
0
votes
1answer
65 views

Non-uniform scaling

I have 10 numbers $x_1, x_2, \dots, x_9, x_{10}$ which sum to a total of $1,000$. I want to scale these numbers so the total is equal to $10,000$, however I don't want them to scale exactly. I'm ...
10
votes
4answers
795 views

Recommended survey of mathematics [closed]

What do you see as the most explanatory and beautiful survey of mathematics book...For short is there a book like Feynman lectures but for math?I've looked at Elementary Mathematics from an Advanced ...
2
votes
3answers
869 views

There are 81 trees with {1,2,3,…81} apples on them respectively. Distribute among 9 people. Each get equal apples

It is clear. There are 81 apple trees. 1st tree has 1 apple, 2nd tree has 2 apples, 81st tree has 81 apples. Distribute the "trees" (not apples) among 9 people so everyone gets equal amount of apples. ...
0
votes
3answers
1k views

The largest number that cannot be made using a combination of $5$ and $11$?

Using just the numbers $5$ and $11$, what is the largest number that can not be made? An example of a feasible combination: $5 \cdot 20 - 11 \cdot 9 = 1$. An example of an unfeasible number is 13 ...
5
votes
2answers
343 views

A puzzle on game theory

Bob and Alice are playing a game. They will start with an integer $n$. Alice goes first, in each turn, a player can choose an integer between 1 and 13 and that number is to be subtracted from $n$. ...
5
votes
4answers
15k views

Find the pattern - puzzle

I have recently encountered a reasoning question that I have solved half , but I can't solve one part of it. Question : ...
10
votes
1answer
997 views

Maximally touching toruses

7 identical cylinders can mutually touch each other, if sufficiently long. For cylinders of different sizes, 8 can touch each other. What is the maximal number of mutually touching toruses? I ...
24
votes
4answers
1k views

Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it in a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
2
votes
2answers
224 views

Probabilities associated with negatively marked questions

First of all: not a native english speaker, and not a mathematician. Please explain as you would to your 10 years old son. I have 120 questions to answer True or False For each right answer, i ...
3
votes
3answers
634 views

number of points on two circles

(sorry I don't know how to add pictures) Two friends argue if larger circles have more points than smaller circles Friend number 1 (a well known argument) Say the circles are concentric. you cannot ...
0
votes
1answer
173 views

Discrete mathematics for someone from a non-mathematical background

I have been a software programmer for over six years and I'm from a non-mathematical background. Though I had some limited exposure to discrete mathematics in my college years it didn't leave any ...