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

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1
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3answers
127 views

Olympic elementary combinatorics problem

This is a problem taken from the regional selections of the Italian mathematical olympiads: A knight is placed on the bottom left corner of a $ 3\times3 $ chess board. In how many ways can you move ...
6
votes
3answers
336 views

Secret Santa Perfect Loop problem

(n) people put their name in a hat. Each person picks a name out of the hat to buy a gift for. If a person picks out themselves they put the name back into the hat. If the last person can only ...
2
votes
1answer
163 views

Basic examples of probabilistic method

I'm looking for a truly basic example of probabilistic method proof which could be presented without a board (i.e. speaking only), that is, even moderately complicated calculations are not allowed. ...
18
votes
1answer
458 views

Infinite prisoners with hats — is choice really needed?

The problem is this (recently asked about here): A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat line up single file line. Each prisoner faces ...
5
votes
2answers
277 views

Prisoners Problem

We have an infinite number of prisoners enumerated $\{1, 2, \dots\}$, and on each prisoner there is a hat of either blue or red color. The $n$th prisoner sees the hats of prisoners $\{n+1, n+2, \dots\}...
7
votes
4answers
193 views

If $(x+\sqrt{x^2 + 1})(y+\sqrt{y^2 + 1})=p$, find $x+y$

I was given this factorization problem and I tried many things, but couldn't solve it. Can someone, please, give me a hint? If $(x+\sqrt{x^2 + 1})(y+\sqrt{y^2 + 1})=p$, find $x+y$. Here $x, y$ ...
6
votes
2answers
178 views

What is the coefficient of $x^{25}$ in $(x^3 + x + 1)^{10}$?

Here's what I have so far on the off chance that my thinking is correct... So using Vieta's the coefficient of the $x^{25}$ should be $-(r_1r_2r_3r_4r_5+r_1r_3r_4r_5r_6+...+r_6r_7r_8r_9r_{10})$ Since ...
1
vote
3answers
90 views

Combinatorial card game [duplicate]

There is a card game I've played before, where it goes as follows: You take a standard deck of cards, and shuffle them randomly. You then proceed by flipping each card and placing them down, ...
1
vote
0answers
39 views

Adding a factor to a ranking?

I have a ranking of 10 items from best to worst. Let's assume that the best is ranked 1 and the worst is ranked 10. Each item is ranked according to some rules that we cannot know so all we get is the ...
26
votes
5answers
12k views

Puzzle of gold coins in the bag

At the end of Probability class, our professor gave us the following puzzle: There are 100 bags each with 100 coins, but only one of these bags has gold coins in it. The gold coin has weight of 1....
0
votes
1answer
73 views

sequence get number in sequence from place in sequence

There is a sequence $$X = {1,1,1,1,1,1,1 \dots 2,2,2,2,2,2,2,2 \dots,3,3,3,3,3,3 \dots 4,4,4,4,4,4,4 \dots (k-1),(k-1),k}$$ So there are $(k)$ 1's, $(k-1)$ 2's and $(k-2)$ 3's and so on. Is there ...
1
vote
1answer
147 views

Tree recursive question: number of nodes and relationship with children

Suppose a given tree T has n1 nodes that have 1 child, n2 nodes that have 2 children, . . . , nm nodes that have m children and no node has more than m children, how many nodes have NO child are there ...
2
votes
3answers
69 views

Fun problem. Apparently $\prod_i(1-p_i) \geq 1 - \sum_ip_i$ with $p_i \in [0,1]$ is always true. But how to demonstrate it?

so, I want to demonstrate the validity of the following inequality: $$ \prod_i(1-p_i) \geq 1 - \sum_ip_i $$ with $p_i \in [0,1]$, it is always true, which it seems to be always the case if you test ...
4
votes
2answers
1k views

how to solve triangles count puzzle

Below is a puzzle of counting triangles.How to solve such puzzle ? source: http://gpuzzles.com/mind-teasers/how-many-triangles-challenge/?source=stackmath
0
votes
1answer
55 views

Which Snake fields can be played infinitely long?

Snake is a very old game for phones. Its a 'real time game', that means you have to make decisions fast. The rules are: You are a snake. You can move to the left, to the right or go straight ahead. ...
3
votes
1answer
54 views

How big can the deck get while still allowing this puzzle to be solvable?

Here's a classic puzzle (I think Martin Gardner talks about it somewhere, though I'm not sure exactly where): Alice and Bob are co-conspirators. Alice is dealt five random cards from a standard ...
0
votes
2answers
63 views

Combinatorics type

I have this problem: From a set of numbers, such as $\{1,2,3,4,5,6\}$, a new set is created containing all the possible single pairs. ie. $\{12,13,14,15,16,\ldots\}$. Another set contains all the ...
0
votes
1answer
73 views

Let's say you're playing ping pong. For each point you win or lose, how would you update your probability of winning the next point?

Let's say I start out believing that my probability of winning the next point is in the interval $[0.25, 0.5]$ with 50% confidence. If I win the next point, what is an intuitive or "good" way to ...
0
votes
2answers
137 views

Birthday problem, the hard way(not using 1-unfavourable outcomes).

How would you go about calculating the chance of two people having the same birthday in a room of 3 people and a year consisting of 365 days?
5
votes
3answers
469 views

Is the square-wheeled tricycle at MoMath stable?

My question has to do with the geometry of the square-wheeled tricycle ride Pedal on the Petals at the National Musuem of Mathematics in New York (MoMath). The tricycles ride on a circular track (...
1
vote
0answers
58 views

sum of integers of two exponents equal

For what values of n, such that $n \in \mathbb{Z}^+,$ does the sum of digits $(214)^n$ and $(2014)^n$ equal? So I found $1$, which is fairly obvious, there are supposed to be more?
0
votes
2answers
133 views

How do you solve “sum of ages” puzzles?

Ma and Pa and brother and me. The sum of our ages is eighty-three. Six times Pa’s age is seven times Ma’s age, and Ma’s age is three times my age. What is Pa’s age? What is Ma’s age? What is my ...
0
votes
1answer
56 views

Placing bricks on Board

Situation: I have a $8\times 8$ board (sic), but two squares from it's one diagonal are removed (Black colored squares are removed) I'm given with plenty of(Rectangular) bricks having dimensions $...
0
votes
3answers
138 views

The fly flying between two trains

I know this question has been posted many times, but I don't understand it. Two trains travel on the same track towards each other, each going at a speed of 40 kph. They start out 180km apart. A fly ...
13
votes
1answer
315 views

Combinatorial prime problem

Update As Barry Cipra noted in the comments, a better framing of the question might be that I'm looking at absolute differences $|a−b|$ or totals $a+b$ for $5$-smooth numbers $a$ and $b$ satisfying ...
0
votes
1answer
75 views

Combinatorial prime puzzle

Is it true that no prime larger than $241$ can be made by either acting or subtracting $2$ coprime numbers made up out of the prime factors $2,3,$ and $5?$ Update Above example is clearly wrong, as ...
1
vote
1answer
63 views

looking for specific recreational math puzzle book

Long time ago, I read a (recreational) math puzzle book and I remember was that in the pocket book there was a puzzle where the parents of a worm were deciding how big the blanket for their baby ...
3
votes
2answers
118 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
2
votes
1answer
759 views

John von Neumann- Exercise about a Fly and two Trains [duplicate]

A fly is flying between two trains, each travelling towards each other on the same track at 30 km/h. The fly reaches one engine, reverses itself immediately, and flies back to the other engine, ...
1
vote
2answers
221 views

Two math professors problem

My friend asks me a question from internet. The question is as follows Two math professors, professor Uno and professor Dos, play chess at the park while reminiscing about their past. Prof. ...
0
votes
6answers
178 views

How to know a number is divisible by a given number without using a calculator?

My question is simple and comes from my curiousity during studying math. How to know a number is divisible by $7$ or $13$ without using a calculator? For example, how do we decide intuitively that ...
8
votes
1answer
82 views

Is every point on a Menger Sponge visible from the outside?

Choose an arbitrary point on the surface of a Menger Sponge. Can you find a straight line starting at that point and extending beyond the sponge that doesn't intersect the sponge anywhere else? That ...
0
votes
2answers
85 views

About the rationality of $1.1010010001\dots$ [duplicate]

Let's define $\rho=1.1010010001\dots$ which can be expressed by: $$\rho=\frac{1}{10^{0}}\underbrace{+\frac{1}{10^{1}}}_{\text{power}+1}\underbrace{+\frac{1}{10^{3}}}_{\text{power}+2}\underbrace{+\frac{...
1
vote
1answer
59 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.
2
votes
0answers
96 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
76 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
0answers
216 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 ...
1
vote
3answers
79 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 ...
4
votes
3answers
125 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
1answer
72 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
vote
4answers
551 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 ...
0
votes
1answer
139 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) ...
20
votes
1answer
494 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. ...
1
vote
0answers
168 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 ...
4
votes
2answers
175 views

If $abc=1$, then $\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} $

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 ...
6
votes
2answers
97 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, ...
4
votes
0answers
97 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
306 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
47 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 ...
7
votes
0answers
156 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in $\...