## The Minimal Surface Equation

“The minimal surface equation somehow encodes the beautiful soap films that form on wire boundaries when you dip them in soapy water,” said mathematician Frank Morgan of Williams College. “The fact that the equation is ‘nonlinear,’ involving powers and products of derivatives, is the coded mathematical hint for the surprising behavior of soap films. This is in contrast with more familiar linear partial differential equations, such as the heat equation, the wave equation, and the Schrödinger equation of quantum physics.”

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## The Case For Doing Hard Things

We ask hard questions because so many of the problems worth solving in life are hard. If they were easy, someone else would have solved them before you got to them. This is why college classes at top-tier universities have tests on which nearly no one clears 70%, much less gets a perfect score. Theyâre training future researchers, and the whole point of research is to find and answer questions that have never been solved. You canât learn how to do that without fighting with problems you canât solve. If you are consistently getting every problem in a class correct, you shouldnât be too happy â it means you arenât learning efficiently enough. You need to find a harder class.

The problem with not being challenged sufficiently goes well beyond not learning math as quickly as you can. I think a lot of what we do at AoPS is preparing students for challenges well outside mathematics. The same sort of strategies that go into solving very difficult math problems can be used to tackle a great many problems. I believe weâre teaching students how to think, how to approach difficult problems, and that math happens to be the best way to do so for many people.

## Question 1: Calculator Permitted Multiple Choice

Answer: C

Category: Heart of Algebralinear equations and inequalities in context

**Heres how you solve it:**

1. Youll need to be able to write equations that reflect the context described in a word problem for several questions on the SAT, both linear context and non-linear contexts. If you have trouble translating word problems into math, this is a skill youll want to work on! Rewriting word problems to include words like equal to, less than, more than, sum, and so on can help you easily translate the problem into equations and inequalities.

2. Were told that Roberto wanted to sell \ insurance policies but he didnt meet his goal, which means that he sold less than \ insurance policies. Given that \ represents the number of \ policies sold and \ represents the number of \ policies sold, the sum of these two numbers is less than \. This is represented as \. We can eliminate choices **B** and **D**.

3. Next, were told that the value of all the policies sold was more than \. We need to multiply the value of the policy by the number of that type of policy, which gives us \. Choice **C** is the only answer that reflects Robertos insurance policy sales.

**Recommended Reading: What Is Sample Space In Math Terms **

## The Birch And Swinnerton

If the previous problems sounded difficult, the Birch and Swinnerton-Dyer Conjecture is recognised as one of the most difficult mathematics problems to solve. The problem is related to number theory. Clay Math **explains**, It asserts that if is equal to 0, then there are an infinite number of rational points , and conversely, if is not equal to 0, then there is only a finite number of such points.

## Strategies For Difficult Math Problems And Beyond

Here are a few strategies for dealing with hard problems, and the frustration that comes with them:

**Do something**. Yeah, the problem is hard. Yeah, you have no idea what to do to solve it. At some point you have to stop staring and start trying stuff. Most of it wonât work. Accept that a lot of your effort will appear to have been wasted. But thereâs a chance that one of your stabs will hit something, and even if it doesnât, the effort may prepare your mind for the winning idea when the time comes.

We started developing an elementary school curriculum months and months before we had the idea that became Beast Academy. Our lead curriculum developer wrote 100â200 pages of content, dreaming up lots of different styles and approaches we might use. Not a one of those pages will be in the final work, but they spurred a great many ideas for content we will use. Perhaps more importantly, it prepared us so that when we finally hit upon the Beast Academy idea, we were confident enough to pursue it.

**Simplify the problem**. Try smaller numbers and special cases. Remove restrictions. Or add restrictions. Set your sights a little lower, then raise them once you tackle the simpler problem.

**Focus on what you havenât used yet**. Many problems have a lot of moving parts. Look back at the problem, and the discoveries you have made so far and ask yourself: âWhat havenât I used yet in any constructive way?â The answer to that question is often the key to your next step.

**Also Check: Ccl4 Geometric Shape **

## Question : Calculator Permitted Multiple Choice

Answer: C

Category: Problem-solving and data analysisanalyzing graphical data, probability

**Heres how to solve it:**

**its important that you pay attention to the wording of the question so that you know exactly what youre being asked.**Once you understand that, the math needed is not necessarily complex. The question asks: If a person is chosen at random from those who recalled at least 1 dream, what is the probability that the person belonged to Group Y? It can help to rewrite the question in your own words. Heres a possible rewrite: Out of all the people who remembered at least 1 dream, what fraction were in Group Y?

**C**.

## Simple Math Problems No One Can Solve

Easy to understand, supremely difficult to prove.

Mathematics can get pretty complicated. Fortunately, not all math problems need to be inscrutable. Here are five current problems in the field of mathematics that anyone can understand, but nobody has been able to solve.

Pick any number. If that number is even, divide it by 2. If it’s odd, multiply it by 3 and add 1. Now repeat the process with your new number. If you keep going, you’ll eventually end up at 1. Every time.

Mathematicians have tried millions of numbers and they’ve never found a single one that didn’t end up at 1 eventually. The thing is, they’ve never been able to *prove* that there isn’t a special number out there that never leads to 1. It’s possible that there’s some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1. But no one has ever been able to prove that for certain.

So you’re moving into your new apartment, and you’re trying to bring your sofa. The problem is, the hallway turns and you have to fit your sofa around a corner. If it’s a small sofa, that might not be a problem, but a really big sofa is sure to get stuck. If you’re a mathematician, you ask yourself: What’s the largest sofa you could possibly fit around the corner? It doesn’t have to be a rectangular sofa either, it can be any shape.

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## Can You Give An Example Of A Complex Math Problem That Is Easy To Solve

I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. Id like to make the point by presenting three math problems that on the surface look equally challenging. But

- One is simple to solve
- One is complex to solve
- And one is impossible

So if a mathematician cant simply look at a problem and say, I can solve that in a day, or a week, or a month, how can anyone else that is truly solving a problem? The very nature of problem solving is that we dont know where the solutions lies and therefore we dont know how long it will take to get there.

Any input or suggestions would be greatly appreciated.

This isn’t exactly what you’re asking for, but it should serve the same purpose very nicely.

Hilbert gave a talk in 1920 or so in which he discussed the difficulty of various problems.

He said that great progress had been made in analytic number theory in recent years, and he expected to live to see a proof of the Riemann Hypothesis.

Fermat’s Last Theorem, he said, was harder maybe the youngest members of his audience would live to see a proof.

But the problem of determining whether $2^$ is transcendental was so hard that not even the children of the youngest people in the audience would live to see a solution to that one.

With the benefit of hindsight, we can see that Hilbert had it exactly backwards.

$2^$ was settled in 1929 – Hilbert lived to see it.

## Read And Reread The Question

They say theyve read it, but have they *really*? Sometimes students will skip ahead as soon as theyve noticed one familiar piece of information or give up trying to understand it if the problem doesnt make sense at first glance.

Teach students to interpret a question by using self-monitoring strategies such as:

- Rereading a question more slowly if it doesnt make sense the first time
- Asking for help
- Highlighting or underlining important pieces of information.

**Recommended Reading: Angle Addition Postulate Worksheet Answer Key **

## Fermats Famous Last Theorem

Maths is an ancient discipline, around 4,000 years old, and there are some theories that have remained unresolved for hundreds of years.

The longest-standing unresolved problem in the world was Fermats Last Theorem, which remained unproven for 365 years. The conjecture was established by Pierre de Fermat in 1937, who famously wrote in the margin of his book that he had proof, but just didnt have the space to put in the detail.

In 1995 Andrew Wiles published his own proof and famously said hed need more than a margin to prove his point.

## Lists Of Unsolved Problems In Mathematics

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

List | |
---|---|

Thurston’s 24 questions | 24 |

Jair Minoro Abe, Shotaro Tanaka | 2001 |

2007 |

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## Deceptively Simple Maths Problems That No One Can Solve

We all know that maths is really hard. So hard, in fact, that there’s literally a whole Wikipedia page dedicated to unsolved mathematical problems, despite some of the greatest minds in the world working on them around the clock.

But as Avery Thompson points out at *Popular Mechanics**,* from the outset at least, some of these problems seem surprisingly simple – so simple, in fact, that anyone with some basic maths knowledge can understand them… including us. Unfortunately, it turns out that proving them is a little harder.

Inspired by Thompson’s list, we’ve come up with our own list of deceptively simple maths problems to frustrate you.

Prime numbers are those magical unicorns that are only divisible by themselves and 1. As far as we know, there’s an infinite number of primes, and mathematicians are working hard to constantly find the next largest prime number.

But is there an infinite amount of prime numbers pairs that differ by two, like 41 and 43? As primes get larger and larger, these twin primes are harder to find, but in theory, they should be infinite… the problem is no one’s been able to prove that as yet.

Claudio Rocchini

This is something most of us have struggled with before – you’re moving into a new apartment and trying to bring your old sofa along. But, of course, you have to maneuver it around a corner before you can get comfy on it in your living room.

Friends/NBC

If Ax + By = Cz

## Use Concepts You’re Less Familiar With

Another reason the questions we picked are so difficult for many students is that they focus on subjects you likely have limited familiarity with. For example, many students are less familiar with algebraic and/or trigonometric functions than they are with fractions and percentages, so most function questions are considered high difficulty problems.

*Many students get intimidated with function problems because they lack familiarity with these types of questions.*

**Also Check: Eoc Fsa Warm Ups Algebra 1 Answers **

## Whats The Simplest Unsolved Maths Problem

**Asked by: Emily French, Newbury**

If by simplest you mean easiest to explain, then its arguably the so-called Twin Prime Conjecture. Even schoolchildren can understand it, but proving it has so far defeated the worlds best mathematicians.

Prime numbers are the building blocks from which every whole number can be made. All numbers are thus either prime themselves, or can be made from a unique combination of primes multiplied together. When the prime numbers are written down two patterns emerge. First, they become progressively rare: while 25 per cent of numbers between 1 and 100 are prime, this falls to just 5 per cent between 1 and a billion. But while they thin out, there still seems to be an endless supply of twin primes like 3 and 5, 29 and 31, 41 and 43, which differ by 2. But do these twins ever run out?

Over 2,300 years ago, the Greek mathematician Euclid proved that primes themselves go on forever. So it seems possible that twin primes might do so too. Thats not proof, however and this remains elusive. Currently, all that mathematicians have managed to prove is that theres an infinite supply of primes differing by no more than 246.

* to BBC Focus magazine for fascinating new Q& As every month and follow on Twitter for your daily dose of fun facts.*

## Identify Important And Extraneous Information

*John is collecting money for his friend Aris birthday. He starts with $5 of his own, then Marcus gives him another $5. How much does he have now?*

As adults looking at the above problem, we can instantly look past the names and the birthday scenario to see a simple addition problem. Students, however, can struggle to determine whats relevant in the information thats been given to them.

Teach students to sort and sift the information in a problem to find whats relevant. A good way to do this is to have them swap out pieces of information to see if the solution changes. If changing names, items or scenarios has no impact on the end result, theyll realize that it doesnt need to be a point of focus while solving the problem.

**Recommended Reading: Segment Addition Postulate Color By Number Worksheet Answer Key **

## Navierstokes Existence And Smoothness

In three space dimensions and time, given an initial velocity, does there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the NavierStokes equations?

The Navier-Stokes equations are two *non-linear partial differential equations *that describe fluid motion in 3D. It is a set of two equations that link the velocity vector field and its rate of change with the pressure field, external forces applied to the liquid. The equations are written as such:

We wont delve into what each term means but essentially, the first equation is the fluids version of Newtons *F=ma *the forces contributing are the sum of pressure, viscous stress and external forces. The second equation is very simply a conservation of mass and requires that the fluid is incompressible.

For a solution to be valid we have two conditions:

**v**and scalar field p are globally defined and continuous over the whole space.

So the Navier-Stokes problem is boiled down to proving one of two cases:

**The affirmative**: given **f **= **0 **and an initial velocity field there exists a velocity and pressure field that satisfies and .

**The breakdown: **There exists an initial vector field and external force field where there is no solution that satisfies and .

## Im Not Interested In Money Or Fame

Much has been made of the fact that a woman has been awarded one of this years four awards for the first time in the history of the prize. Maryam Mirzakhani works in the field of geometry and the description about her work reads: Because of its complexities and inhomogeneity, moduli space has often seemed impossible to work on directly. But not to Mirzakhani. She has a strong geometric intuition.

But the British community of mathematicians has also been delighted about Martin Hairers award only the eighth Briton to win the prize ever. He works in the field known as stochastic analysis. So far, so impossible to understand. But Professor Lyon saidhis the area builds on a field that has had a huge impact on everything from mobile phones to the stock market.

And experts have even speculated that his work could shed some light on another of those so-far unsolvable problems the Navier Stokes problem, which is one of six remaining unsolved Millennium Prize Problems, which include the Riemann hypothesis.

But then again, not all mathematicians are in it for the glory. The Russian genius Grigori Perelman famously rejected the Fields Medal and an additional $1m in 2010 for proving the Poincare conjecture, then one of the worlds most difficult problems.

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