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

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1
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1answer
78 views

Cyclic tower of hanoi problem [duplicate]

If I have 3 rods in a circle and it is allowed to move the disks only in the clockwise direction. How many moves is necessary to move n disks from first rod to the third rod?
1
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1answer
37 views

$[x-\frac{1}{n}, (n-1)x+\frac{1}{n}]$ contains an integer $\forall x\in \mathbb{R}$ and $\forall n\in \mathbb{N}$

For any real number x: Prove that among the numbers x,2x,...,(n-1)x ,there is one that differs from an integer by at most $\frac{1}{n}$. any hints for a pigeon solution. Non-pigeon solution ...
1
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1answer
35 views

shorter proof of generalized mediant inequality?

Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}>0$. Attempt Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$ ...
4
votes
2answers
170 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
2
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1answer
62 views

Guests at a table

Fifteen chairs are evenly placed around a circular table. On the table are the name cards of fifteen guests. After the guests sit down, it turns out that none of them is sitting in front of his own ...
2
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1answer
60 views

Ping Pong players

A and B play ping pong game multiple times. The person serving first has a probability p of winning that game. A serves the first game and thereafter the loser serves first. If p(n) = pbt that A ...
9
votes
2answers
159 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
2
votes
1answer
202 views

Sequences, sets and element position in the set.

I have a sequence Q with the length of N. This is the fragment of this sequence: 68 70 72 74 76 78 80 The sequence has been divided into the sets of 4 elements ...
0
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0answers
27 views

Probability of getting it right, when choosing answer at random. [duplicate]

So there has been this question going around on social networking sites, If you chose an answer to this question at random, what would be the probability of your getting it right? a) 25% ...
0
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2answers
167 views

Fun math riddle

In his will , a farmer left 17 horses to his 3 sons with the following instructions. 1) The eldest son is to get half of the total horses. 2) The middle son is to get one third of the total horses. ...
1
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4answers
154 views

What is the most awe-inspiring math equation you have come across [closed]

What is the favorite equation of your life? I know this might be a subjective question, and may be not-so-on-topic here, so if anyone decides to close this, could you link me somewhere I can ask this? ...
10
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2answers
317 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
4
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1answer
88 views

General approach to puzzles such as the “6 books puzzle”

Six different books (A,B,C,D,E,F) of identical size are stacked as in the figure. We know A and D are not touching. E is between two books which are both vertical or both horizontal. C touches ...
8
votes
2answers
116 views

Mathematicly Untangeling Untangle.

I have a new addiction, I play Untangle to often, and i am wondering what is the mathematics behind it. some free games: (but be warned highly addictive) Javascript: ...
-1
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2answers
120 views

Proving $1 + 1 = 2$ [duplicate]

How do you break down the theory of $1 + 1 = 2$? How do you provide a proof, please be precise. This is for one of my discrete math courses and I don't know how this is relevant to the course. And ...
2
votes
0answers
113 views

What are some cool projects that can be done in a high school math class?

I'm studying to be a secondary math teacher and will be starting student teaching next month (an algebra 2 class). It's difficult to find activities and projects that are actually informative and time ...
0
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1answer
65 views

Proving using squeeze principle

This problem sounds very confusing. Please help me solve this problem.
1
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1answer
57 views

Prove the derivative

Let $f(x) = (x^2-1)^{\frac{1}{2}}, x>1$. How do I prove that the $n$th derivative of $f(x) > 0$ for odd $n$, and the $n$th derivative of $f(x) < 0$ for even $n$?
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0answers
22 views

Convex quadrilateral

In a convex quadrilateral (the two diagonals are interior to the quadrilateral) prove that the sum lengths of the diagonals is less than the perimeter but great than one-half the perimeter.
4
votes
2answers
217 views

Careers in Mathematics?

I am a college freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. ...
0
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2answers
57 views

A question of divisibility.

Let $p$ and $q$ are relatively prime integers. Consider $S = \{\frac{p}{q}\} + \{\frac{2p}{q}\} + \{\frac{(q-1)p}{q}\}$, where $\{x\}$ is the fractional part of the real number $x$. Prove that $2S$ ...
0
votes
1answer
55 views

25 coins are arranged in a 5 by 5 array.

25 coins are arranged in a 5 by 5 array. A fly lands on one and tries to hop on to every coin exactly once, at each stage moving only to an adjacent coin in the same row or column. Is this possible?
3
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3answers
140 views

A puzzle that came when I am half awake

When I am about to wake up in the morning, a puzzle crept into my mind.It is when $\sqrt{a}$ and $\sqrt{b}$ are both non-integers where a,b are positive integers is it possible for $\sqrt{ab}$ to be ...
0
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1answer
113 views

How many squares can a knight reach in $n$ moves? [closed]

How many squares on an infinite chessboard can a knight reach, starting from a given square, in $n$ moves?
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2answers
67 views

Ball bouncing in a box, will it meet a vertex.

I have no idea upon how to solve this: A box 5cm by 3cm with a ball projected from a vertex at 45 degree angle, it reflexes at a 45 degree angle and keeps reflecting at a 45 degree angle. Will it ...
0
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3answers
87 views

How to find the planar embedding of a graph in general?

I need to find the planar embedding of a graph in general if one exists and specifically want to solve the problem for the graph in the figure below. I am acquainted with the graph algorithms but have ...
1
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1answer
35 views

matrix row/col mapping

Many square matrices are symmetric. i.e. $a_{i,j}=a_{j,i}$ For such matrices, we can only store the upper triangle elements, i.e. any $a_{i,j}$ for which $i<=j$. Assume a 10x10 matrix. Using this ...
5
votes
1answer
83 views

Hockey Classics at Matheletics '13

I'm trying to solve a challenge from Matheletics '13: Micheal Nobbs is organizing a training camp for identifying new talents in Indian Hockey. The camp witnessed a total of ($3K+1$) players. Each of ...
4
votes
1answer
297 views

BINGO Probability: Controlling average game duration

I wandered over here from StackOverflow and my understanding of advanced mathematics is limited, so bear with me... A standard, BINGO game card has 24 numbers arranged in a 5x5 format. The center of ...
6
votes
3answers
164 views

What went wrong?

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
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1answer
126 views

Integer coefficient polynomial - values as powers of 2

Does there exist a polynomial f with integer coefficients such that $f(0) , f(1) ... f(n) $ are all distinct powers of 2 ? I have no clue about how should i start thinking about this problem but ...
2
votes
1answer
73 views

Functions which satisfy $\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w)$

Let $\mathrm{f}$ be a complex-valued function with the following property: $$\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w) $$ for all $w,z \in \mathbb C$. Necessary conditions are that ...
73
votes
12answers
8k views

Logic puzzle: Which octopus is telling the truth?

King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always tell the truth. One day, four servants met. ...
0
votes
3answers
1k views

How do you find the altitude in a pyramid? (SAT math question)

The pyramid shown above has altitude h and a square base of side m. The four edges that meet at V, the vertex of the pyramid, each have length e. If e = m, what is the value of h in terms of m? A) ...
9
votes
0answers
157 views

Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
1
vote
1answer
141 views

Primes created by “n + digital-root(n)” sequences

I've looked at the sequences created by repeatedly adding the digital root of a number to the number until it becomes prime. This is the pseudo-code for the program I've used:   n = 0 ...
2
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0answers
67 views

How much advantage would a Blackjack player gain by being able to see the underside of cards?

In the novel Spaceland by Rudy Rucker, the protagonist Joe Cube is grafted with an eyestalk that sticks vout into the fourth dimension. This lets him see under and inside three-dimensional objects ...
6
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3answers
233 views

Why are the surreals considered “recreational” mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
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0answers
37 views

On a certain type of card game

Suppose two players are playing a card game, which is described as follow. Each player is allowed to construct their own decks of exactly $n$ cards with an additional repeatable card, where each ...
2
votes
0answers
103 views

Inserting +/- into 123456789…

I'm looking at a generalization of the problem of inserting + and/or - into the blocks $123456789$ and $987654321$ to create a formula for $100$, like this: $$123 - 45 - 67 + 89 = 100$$ $$9 - 8 + 7 ...
2
votes
2answers
76 views

A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?

I did not know what to search to see if this existed elsewhere. But, I could not find it. Here's the question, does there exist a continuously differentiable function, $f: [0,1] \rightarrow ...
4
votes
1answer
77 views

Expected number of clusters on chessboard

N distinct squares are selected uniformly at random on an MxM chessboard, what is the expected number of clusters? A cluster is a collection of squares which are connected sideways, not cornerwise.
2
votes
1answer
110 views

Fifteen pennies lie on the table in the shape of a triangle

Fifteen pennies lie on the table in the shape of a triangle, with five pennies on each side. For some reason, the pennies are painted either black or white. Prove that there exist three pennies of ...
2
votes
1answer
92 views

Homotopy of a (non-spherical) cow.

I heard once that, from a topologist, that a cow and a doughnut ($\mathbb T^2$) are the same thing. It wasn't hard to believe that, since food enters by the snout and, well, goes out somewhere else. ...
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2answers
289 views

How to choose between an odd number of options with a fair coin

It is possible to choose between three equally desirable outcomes by tossing a fair coin as follows: Choose option 1 if the first head appears on an even toss Choose option 2 if the first tail ...
5
votes
1answer
76 views

Definite Integral that Evaluates to Teacher's Initials: TAA

My school's calculus teacher's birthday is in a couple of days, and our class decided to give him a surprise birthday card that has a definite integral which evaluates to his initials (TAA). So far ...
1
vote
1answer
40 views

Uniformed Distribution - Recap

I have divide the interval $[0,1]$ into $k$ equal sub-intervals, which I call classes, and generated $n$ observations from a uniform distribution. The number $X_{1}$ of the $n$ observations that fall ...
2
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2answers
92 views

Articles on matchstick puzzles

There are many ingenious puzzles involving matchsticks that are arranged as squares, rectangles or triangles, and can be moved under some restrictions (for a lot of examples see ...
5
votes
2answers
103 views

Can you incentivise competitors to handicap accurately, and also try to win?

A problem I ran into for real. A group of friends of widely differing abilities wants to hold a handicap cycling race, so that if everyone does about as well as expected, there would be a perfect dead ...
0
votes
1answer
482 views

How many triangle can be drawn with those points? [duplicate]

There are 7 points on the circumference of a circle.How many acute triangle can be drawn with those points. please help me to solve this problem.