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

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0
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0answers
61 views

A room contains n people. Everybody wants to shake everyone else’s hand (but not their own).

(a) Suppose that n people require hn handshakes. If an (n + 1)th person enters the room, how many additional handshakes are required? Obtain a recurrence relation for hn+1 in terms of hn. (b) ...
2
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0answers
19 views

Can you recommend me a book about integration? [duplicate]

I'm new to this site. I'm an university student in korea and I major in engineering. Recently, I've been quite interested in calculating integration. So nowadays as my hobby, I seek many interesting ...
-1
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1answer
40 views

For what reltively prime integers $a$ and $b$ does the expression $2ab$ an even numeric palindrome? [closed]

I currently doing some research on numeric palindromes. And I am stack with the problem: For what reltively prime integers $a$ and $b$ does the expression $2ab$ an even numeric palindrome? Since ...
1
vote
3answers
65 views

Estimate the number of typos there are in a book, based on two editors' finds

This is one question from an interview I have just taken: Suppose there is a book full of typos. Tom and Jerry found $x$ and $y$ typos throughout the book, respectively. There are $z$ typos that ...
5
votes
2answers
326 views

logical problem (how long did you walk?)

My wife is very kind, she always picks me up at work by car and drives me home. Today, I finished at work 30 minutes earlier! So I decided to walk home... on the way I met my wife. She was on her way ...
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2answers
43 views

How can I know whether to round a quotient up or down (based on whether the number after the decimal point is 5+ or not) with ONLY this information?

Say I have a special calculator that, when it divides one number by another, it gives you the answer in the form of, "quotient r remainder." For example, if you divide 5 by 3, it tells you "1 ...
-3
votes
2answers
46 views

If a machine takes 3min to process 1 byte, how many machines are required to process 1000 bytes in 30min?

We have a machine that takes 3 minutes to process a byte. Now if I send 1000 bytes together the machine will take 3000 minutes to process them serially. If we want to do that in 3 minutes only we need ...
1
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1answer
203 views

Prove that if x is irrational, then sqrt(x) is irrational.

I believe the contrapositive method should be correct but i get, The contrapositive of this statement should be, (If $\sqrt{x}$ is rational, then $x$ is rational) Then I end up with ...
0
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1answer
33 views

If $n=(b_k,b_{k−1},…,b_1)_2$ where $b_i$ are the digits of n in binary, what is the binary expression of $n+1$?

I have a curiosity. If $n=(b_k,b_{k−1},...,b_1)_2$ where $b_i$ are the digits of n in binary, what is the binary expression of $n+1$? Is there a relationship that binds $n+1$ to $b_i$ (ie the digits ...
0
votes
1answer
55 views

Consider the set $Q=\{p+q \sqrt2 : p,q \in\Bbb Q\}$. Prove that if $a\in Q\setminus\{0\}$ then $1/a\in Q$

Given (For all $a,b\in Q$, $a+b\in Q$ and $ab\in Q$) This was a two part question. Part a) is to prove that $Q$ is closed under addition and multiplication. Part b) is prove that if $a\in Q$ and ...
0
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0answers
39 views

How many solutions are there to a given instance of the game *LYNE*?

Recently I've been playing a nice little puzzle game called LYNE, which involves building a (special) graph out of a given set of vertices and prescribed valencies, and I've been wondering what could ...
3
votes
1answer
82 views

Are the odds one in a million? [closed]

This is a from a card game call Magic the Gathering And my question is regarding this video during a tournament match (best of 5). One in a million. You dont need watch the video I will explain the ...
0
votes
1answer
27 views

Addition chain search tree pruning by discarding non-minimal chains

An addition chain is an ordered tuple of numbers, starting with $1$, such that each number after $1$ can be expressed as the sum of two smaller numbers in the chain. An example of an addition chain ...
8
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8answers
3k views

If yesterday were tomorrow, then today would be Friday.

(S) If yesterday were tomorrow, then today would be Friday. Question: What day is today? This seems to be an old puzzle, and depending on the interpretations, the answers are Wednesday or ...
0
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2answers
33 views

Given $A \subseteq \mathbf{Z}$ and $x\in \mathbf{Z}$, we say that $x$ is $A$-mirrored if and only if $−x\in A$. We also define…

Sorry if this question seems kind of long but I am confused for part C. My proof for part C that $M_a$ is closed under addition is as follows: The set $M_a$ is closed. Let $x$ be in $M_a$ and ...
57
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0answers
7k views

“The Bachelorette Problem” (slightly adapted from Tao's Google+ account) [duplicate]

The following puzzle being very much recreational for me, I couldn't resist myself from sharing it with my fellow MSE user friends. Let's have a look at it. You are the most eligible bachelorette ...
7
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5answers
1k views

Hank and his old car

I'm sort of struggling with this riddle told to me by a friend: Hank owns a car. He has been taking good care of his car; In fact, he has been taking such good care of it that the age of Hank, ...
5
votes
0answers
119 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
0
votes
0answers
14 views

finding percentge given number range

I have a range from 2.3566e-19 to 0.0010997 I'm trying to get the bottom 10% and the top 10% the formula / numbers I used is below but the answer doesn't look right how can I fix this. ...
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0answers
63 views

Is the solution to this elementary number theory problem correct?

Problem: A natural number $n$ is called nice if the following properties hold: • The expression is made ​​up of 4 decimal digits; • the first and third digits of $n$ are equal; • the second and ...
0
votes
2answers
75 views

Arranging identical balls in a circle

In how many ways can 4 identical red balls and two identical white balls be arranged in a circle? This is an elementary problem, but many tries have not yet yielded results. I tried by taking the ...
1
vote
0answers
116 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
0
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0answers
57 views

What are some interesting, atypical mathematical topics that a student who has taken an introductory calculus sequence can learn about?

I understand that usually the next step after $3$ semesters of calculus and $1$ semester of ordinary differential equations (plus one semester of linear algebra, for some) is something like an ...
2
votes
2answers
91 views

Explaining a Pattern in a Matrix Generated by Minimum Excluded Number in Rows & Columns

I have been given the following math puzzle: You are given a matrix that is filled by the following rule: Every cell i,j is evaluated by taking the lowest non-negative number that is not ...
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0answers
24 views

Find all points on the line 9x-21y=6

For this equation we are suppose to use the Euclidean Algorithm. But I run into a problem For the GCD (9,-21)= i tried 9=(-21)(0)+9 -21=9(3)+6 9=6(1)+3 6=3(2) +0 which gives a gcd of 3 and the ...
49
votes
10answers
1k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
0
votes
2answers
61 views

Probability of getting 6 letters right [duplicate]

A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 ...
1
vote
0answers
52 views

Placing $4n$ non-attaking queens of in a $4n \times 4n$ chessboard.

Is it possible to place $4n$ non-attaking queens of in a $4n \times 4n$ chessboard?? I have found that it can be done for $4 \times 4$ chess board and trying to extend it to $8 \times 8$ chessboard ...
7
votes
3answers
665 views

Is every arrangement reachable by shuffling this way?

Suppose we have a vertical stack of $n$ distinguishable coins, each of which is either heads-up or tails-up. Let a shuffle be the following procedure: divide the stack at will into a top- and ...
45
votes
10answers
6k views

Is there something special about 2015?

Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ...
1
vote
2answers
35 views

Changing the state of coins and finding the minimum number of steps to do it

I have $N$ coins all showing heads. At each turn, I change the state (i.e., a head is changed to a tail, vice versa) of $N-1$ coins. Prove that all the coins can end up showing tails if and only if ...
1
vote
1answer
32 views

Rigorous proof for a maximization problem

Problem: Eight players entered a round-robin tennis tournament. At the end of the tournament, a player who wins $N$ sets will take home $N^2$ dollars. The entry fee is $17.50 per player. Why is this ...
0
votes
1answer
69 views

The definition of “Dhuruva Numbers” [closed]

From my readings I encountered this number called "Dhuruva Numbers" Dhuruva Numbers are defined as follows: Definition. The numbers which do not change when performing a single operation or a ...
14
votes
6answers
588 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
0
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1answer
20 views

Covering deficits with values with different weights

SO I have a couple of assessments with specific weights as follows: Assignment 1: 5% => Mark 60% Assignment 2: 5% => Mark 53% Assignment 3: 5% Assignment 4: 5% Test 1: 30% => 47% Test 2: 30% ...
2
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0answers
70 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
0
votes
1answer
28 views

proving onto function of composite functions.

Let $X, Y, Z$ be arbitrary sets. Suppose $\alpha$ is a function from $X$ to $Y$ and $\beta$ is a function from $Y$ to $Z$ such that $\beta\circ\alpha$ is an onto function. How do I prove that $\beta$ ...
2
votes
2answers
93 views

What is the most appropriate book for teaching, not the content but skills of mathematics

Hello Everyone I am a high school student currently doing Extension 1 Mathematics at my school. I am currently looking for a high quality mathematics book. Although I am not looking for a book, like ...
4
votes
0answers
60 views

Algorithm for finding “fact families”

My friend's 3rd grader encountered the following question regarding "fact families" on her math homework: I was in 3rd grade sometime in the 1980s, so I don't believe I ever encountered this term ...
1
vote
1answer
196 views

Is the Center of Math Wrong?

The center of math retweeted the following problem: I surmised the answer is 22, using the following reasoning: Odd entries increase by 2, whereas even entries increase by 1 $a_{1}=16, \, ...
2
votes
0answers
37 views

Is there an intuitive reason why hippopede, the intersection curve of a sphere and a cylinder, is traced by composing two rotational motions?

The hippopede is historically famous because Eudoxus used its properties in the first mathematical model of planetary motion. He nested concentric spheres rotating at different inclinations to each ...
1
vote
2answers
66 views

Probability of the 'big guns' staying apart until final?

It is a non-rigorous discussion on probability. I am reading the book 'How long is a piece of string?' by Rob Eastaway and Jeremy Wyndham. In one of the chapters it talks about sports games and why ...
10
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4answers
266 views

Linear Combinations of Fibonacci Numbers (integer coefficients)

While working on problem #2 on Project Euler, I came across the need to express $F_n$ as a linear combination of $F_{n-3}$ and $F_{n-6}$. This is relatively simple to do: $$\begin{align} F_n &= ...
4
votes
0answers
92 views

Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
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0answers
38 views

2D poisson equation

solve the following 2D poisson equation d2w/dx2 + d2w/dy2 = a boundary conditions y=o we have dw/dy = 0 y=bx we have cdw/dx+d dw/dy = 0 y=ex+f we have w = 0 a,b,c,d,e,f are constants its a triangle ...
0
votes
2answers
80 views

How many routes are there that pass through at most one congested intersection

I am trying to solve the following problem, but i am not quite sure how to attack. Problem Description A taxi drives from the intersection labeled A to the intersection labeled B in the grid of ...
1
vote
1answer
34 views

Tower of Hanoi solutions for non-legal initial configuration

I just found an Towers of Hanoi game (see http://en.wikipedia.org/wiki/Tower_of_Hanoi) messed up by a someone to one tower not obeying the rules, eg. large and small disks where interleaved. I just ...
6
votes
2answers
75 views

How can I prove whether a $9\times 9$ square can be filled with L-shaped pieces in a completely “regular” way?

There are a great many ways to fill a $9\times 9$ square with L-shaped pieces. One of them is below. Now, note that there are eleven $2\times 3$ rectangles that are formed, as well as a larger L ...
3
votes
2answers
117 views

Transforming a matrix A into a zero matrix using finitely many steps.

Let $A$ be a $m\times n$ matrix whose entries are positive integers. A step consist of transforming the matrix either by multiplying every entry of a row by $2$ or subtracting $1$ from every entry ...
5
votes
2answers
192 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...