Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

learn more… | top users | synonyms

176
votes
17answers
7k views

Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
174
votes
4answers
11k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
85
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
59
votes
12answers
13k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quantity'. The totality ...
42
votes
3answers
809 views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer multiplicity of such transforms. Is ...
41
votes
15answers
13k views

Interview riddle

On the Mathematics chat we were recently talking about the following problem @Chris'ssis had to solve during an interview : $$3\times 4=8$$ $$4\times 5=50$$ $$5\times 6=30$$ $$6\times 7=49$$ ...
41
votes
5answers
1k views

Finding the value of $\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$

Is it possible to find the value of $$\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$$ Does it help if I set it equal to $x$? Or I mean what can I possibly do? ...
34
votes
13answers
3k views

Examples of famous problems resolved easily

Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example ...
32
votes
1answer
633 views

Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
31
votes
9answers
3k views

When to give up on a hard math problem?

I practice olympiad problems from books like Putnam and Beyond. Often I come across a problem that I simply can't solve. After $\sim30$ minutes of deep thinking it feels like I'm ramming my head into ...
26
votes
3answers
449 views

How does one cut onions in a mathematically efficient way?

Perhaps a math degree and cooking don't go hand in hand, but hopefully they do. I have been thinking about this problem for some time when in the kitchen without making any real progress: How does ...
25
votes
12answers
2k views

Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$

Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)? $$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$ Using the calculator I can see that the first one ...
23
votes
14answers
2k views

How can we produce another geek clock with a different pair of numbers?

So I found this geek clock and I think that it's pretty cool. I'm just wondering if it is possible to achieve the same but with another number. So here is the problem: We want to find a number ...
23
votes
7answers
558 views

“Here's a cool problem”: a collection of short questions with clever solutions

This game will be familiar to many mathematicians, and it is always good fun to play. I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one ...
22
votes
7answers
1k views

Sum of the sum of the sum of the first $n$ natural numbers

I have here another problem of mine, which I couldn't manage to solve. Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n \\ z_n = y_1 + y_2 + \dots + y_n $$ Find ...
22
votes
10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
20
votes
4answers
333 views

Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles

I came up with this question when I'm actually starring at the wall of my dorm hall. I'm not sure if I'm asking it correctly, but that's what I roughly have: So, how many ways (pattern) that there ...
20
votes
1answer
475 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
18
votes
6answers
2k views

Helping my daughter with her homework: solving an algebra word problem.

Three bags of apples and two bags of oranges weigh $32$ pounds. Four bags of apples and three bags of oranges weigh $44$ pounds. All bags of apples weigh the same. All bags of oranges weigh the ...
18
votes
4answers
756 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
17
votes
12answers
5k views

MENSA IQ Test and rules of maths

In a Mensa calendar, A daily challenge - your daily brain workout. I got this and put a challenge up at work. The Challenge starts with.. ...
17
votes
1answer
1k views

What was Ramanujan's solution?

The wikipedia entry on Ramanujan contains the following passage: One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a ...
16
votes
3answers
269 views

Find all polynomials that fix $\mathbb Q$ and the irrationals

Problem: Describe all polynomials $\mathbb{R}\rightarrow\mathbb{R}$ with coefficients in $\mathbb C$ which send rational numbers to rational numbers and irrational numbers to irrational numbers.
16
votes
3answers
369 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...
15
votes
3answers
622 views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

i am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
15
votes
4answers
4k views

How to Improve Mathematical Thinking and General Problem Solving Skills?

I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving ...
15
votes
4answers
289 views

Find $x,y,z \in \mathbb Q$ such that $x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$

Find $x,y,z \in \mathbb Q$ such that: $$x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$$ Here is my thinking: $$x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z\\ \implies \left ( x ...
15
votes
2answers
382 views

Why are braid numbers of the form $Q_h^2$ or $2 \times Q_h^2$?

Consider two piles of $h$ playing cards each, all distinct. Repeatedly take one of the cards on top of one of these two piles and move it on top of one of two new piles, until both of the new piles ...
15
votes
4answers
357 views

Maximize $x_1x_2+x_2x_3+\cdots+x_nx_1$

Let $x_1,x_2,\ldots,x_n$ be $n$ non-negative numbers ($n>2$) with a fixed sum $S$. What is the maximum of $x_1x_2+x_2x_3+\cdots+x_nx_1$?
14
votes
5answers
446 views

Find the value of $3^9\cdot 3^3\cdot 3\cdot 3^{1/3}\cdot\cdots$

Find the value of $3^9\cdot 3^3\cdot 3\cdot 3^{1/3}\cdot\cdots$ Doesn't this thing approaches 0 at the end? why does it approaches 1?
14
votes
6answers
2k views

List of problem books in undergraduate and graduate mathematics

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
14
votes
3answers
451 views

Decipher the greeting card $( X^2 +Y^2 -1 ) ^3 - X^2 Y^3 = 0$

A friend of mine just got a rather weird congratulations card through the door They have zero idea what it means, I have tried graphing it and nothing spectacular comes out. Is there a standard ...
14
votes
2answers
158 views

Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$

How do I simplify: $$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$ Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
14
votes
2answers
276 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
14
votes
2answers
269 views

Does every “balloon” (dragon, tadpole, canoe paddle) admit a graceful labeling?

Post-Bounty Edit: Still no answer; will gladly accept if someone can provide a reference. Earlier Edit: It appears that the answer is "yes," either by an already existent publication or by ...
14
votes
3answers
382 views

How to prove this innocent inequality?

I need to show that $$ (a-d)^2+(b-c)^2\geq 1.6 $$ if $a^2+4b^2=4$ and $cd=4$. I was told that there exist nice and sweet elementary solution.
14
votes
2answers
1k views

Asking 2011 Putnam B6

I wish to ask today's Putnam problem B6: Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ number of $\sum^{p-1}_{k=0} k! n^{k}$ is not divisble by $p$. ...
14
votes
3answers
300 views

On a Putnam's 2009 problem [duplicate]

Find all even natural numbers $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any regular $n$-gon $A_1...A_n$, $f(A_1) + ...
14
votes
2answers
466 views

The Farmyard problem

Problem: There is a farmer who has a $1\text{ mile}\times 1\text{ mile}$ square piece of land. He knows that there is a completely straight pipe underneath some part of his property, but it could ...
13
votes
4answers
454 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
13
votes
3answers
497 views

'Fixed Point' Irrationals

I found this interesting problem which turns out to be more difficult than it first appears: Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that $f(f(x))=x$ for all $x \in ...
13
votes
1answer
305 views

How many $N$ of the form $2^n$ are there such that no digit is a power of $2$?

How many $N$ of the form $2^n,\text{ with } n \in \mathbb{N}$ are there such that no digit is a power of $2$? For this one the answer given is the $2^{16}$, but how could we prove that that this ...
12
votes
5answers
251 views

Issues with text problems

When I tutor, I often see people who kind of know the stuff they cover in school at the moment and succeed at straight problems like: Find the derivative of $f(x) = \frac 12 x^2$ But when it ...
12
votes
4answers
1k views

Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially

I have to give a combinatorial proof of $$\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}.$$ I find it difficult to solve such problems. I'm not a brilliant person and never will be so I need to have ...
12
votes
3answers
808 views

Improving concentration and stamina when solving difficult problems.

I am trying to improve my problem solving skills by solving olympiad problems (Putnam, IMO, etc). So far, I have discovered that problem solving is somewhat like panning for gold: you think of all the ...
12
votes
1answer
209 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
12
votes
1answer
179 views

Request for a proof of the following continued-fraction identity

I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close): $$ ...
11
votes
4answers
1k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
11
votes
3answers
616 views

An Integral involving $e^{ax} +1$ and $e^{bx} + 1$

For fun, I was looking at the following Putnam-Style problem the other day on this page: (It is problem B2) Evaluate the integral ...
11
votes
3answers
404 views

Penguin Brainteaser : 321-avoiding permutations

There are $k$ penguins, $k\ge 3$. They are all different heights. How many ways are there to order the penguins in a line, left to right, so that we cannot find any three that are arranged tallest to ...