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

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4
votes
0answers
25 views

Inequality: $(a^3+3b^2+5)(b^3+3c^2+5)(c^3+3a^2+5) \ge 27(a+b+c)^3$

Proving inequality for positive real $a,b,c > 0$: $$ (a^3+3b^2+5)(b^3+3c^2+5)(c^3+3a^2+5) \ge 27(a+b+c)^3$$
-3
votes
1answer
108 views

SEVEN - NINE= EIGHT [on hold]

Things are not always what they seem. What is true from one point of view may be false from another and vice-versa, and here is a puzzle to prove it. Despite the fact that every arithmetic teacher in ...
1
vote
3answers
56 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 ...
5
votes
3answers
63 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 ...
0
votes
0answers
10 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. ...
0
votes
0answers
21 views

Point of most theoretical potential moves in a game of Scrabble

I was recently playing a game of scrabble with a friend and the point difference all but ensured that I was going to lose (100+ points with one rack of tiles left, and no more in the "pot" and I ...
0
votes
0answers
24 views

finding distinct digit [on hold]

Each letter stands for a distinct digit as shown in the figure. What are all the digits represented by alphabetical letters? FIVE+ONE+TWO=EIGHT
12
votes
1answer
220 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
109 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, ...
6
votes
4answers
149 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 $x, y$ ...
3
votes
0answers
43 views

Help in deriving a formula [on hold]

Background I am working on a vocabulary building application under which I am trying to build an adaptive test for the student. The test would be adaptive to the user's response: When the student ...
12
votes
0answers
129 views

Sheldon Cooper Primes

On the $73^{\text{rd}}$ episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons $(1973 - \stackrel{\text{hopefully}}{2073})$ revealed his favorite number to be ...
6
votes
2answers
139 views

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

Working on some contest problems and came across this question. 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 ...
1
vote
3answers
52 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
25 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 ...
0
votes
0answers
24 views

making /adjusting a perodic signal given an equation

I know if I have the sin wave equation Asin(2pif*t+phase) I can increase/decrease/adjust the periodic frequency of the signal by changing f. But if I have the following equation below how can I also ...
15
votes
5answers
1k 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 ...
0
votes
1answer
57 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
39 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
58 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 ...
3
votes
2answers
88 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
44 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. ...
2
votes
1answer
40 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
58 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
52 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
39 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?
4
votes
3answers
126 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
49 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
43 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 ...
0
votes
1answer
47 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
71 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 ...
12
votes
1answer
256 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
55 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
34 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
84 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
375 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
votes
2answers
25 views

City A is 800 miles from city B. City B is 1500 miles from city C. Which of the following could be the distance from city A to city C? [closed]

City A is 800 miles from city B. City B is 1500 miles from city C. Which of the following could be the distance from city A to city C? a) 600 b)1000 c)1200 d)1500 e)2000 f)2500
-1
votes
1answer
46 views

Using the difference equation to find the problem. [closed]

let $a_1 = 2\sqrt 2$ and for any $n>1$ define $a_n = 2^{(n+1)/2}\sqrt{2^n - \sqrt{4^n - (a_{n-1})^2}}.$ Find $a_n$ in closed form and evaluate $\displaystyle\lim_{n\to\infty} a_n.$
1
vote
2answers
115 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
73 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 ...
7
votes
1answer
45 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 ...
-2
votes
2answers
119 views

I need to fill in all the operations to that 123456789= 1998

i got a problem that 123456789=1998, and i need to fill in the operations between each digit. all of the numbers are single digits so they can not be combined like 2 3 equaling 23. between each digit ...
0
votes
2answers
69 views

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

Let's define $\rho=1.1010010001\dots$ which can be expressed by: ...
1
vote
1answer
37 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
56 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
63 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
54 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
55 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
94 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
50 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. ...