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Hard Math Equations and Problems – How to Work Through Them?

Oct 18, 2022

Even if you are good at math, and rank among the top 10 in your class, it is common to come across math equations that can be super hard to solve. If you have ever dealt with the same and wondered how to work through the hardest equations and problems, this post is for you.

In today’s post, we will be discussing all the strategies to get through hard math equations and problems. Along with this, we have included some of the toughest math problems ever solved by the end of the post. Let the scrolling begin!


Math Equations – What Are They?

Before we move on to the strategies for difficult math problems, let us first look for an answer to – “what is a math equation?” In simple words, it is any statement that has an ‘equal to’ symbol between two algebraic expressions. Both expressions have equal values.

For instance, 2x+9=24. Here, 2x+9 is the first algebraic expression on the left-hand side, and 24 is the second algebraic expression on the right-hand side. These expressions are equal to each other making 2x+9=24 a perfect example of a math equation.


Dealing with Hard Math Equations

Be it a long math equation or a complex one, even the best of the best can find at least some of them hard to solve. When it comes to admission to top-tier universities, there is a reason why a limited number of students can score the best. It is because numerous questions asked in these are hard. After all, if everyone can solve something, what is the need to conduct the exam in the first place?

When you appear for tests like the SAT, the whole point of it is to find students who have the capability of solving things that have never been solved. Therefore, dealing with hard math equations is the only way out. Even with consistent preparation, if you are unable to get through a problem, know that there may be something wrong, and you need to work hard enough.


Further, when dealing with hard math problems, the first step is to accept and understand them. Instead, what most students do is duck them, and move on to the easier ones. While the hard equations are difficult to solve, they teach you a lot – way more than a worksheet full of easy problems. Also, the eureka moment, in the end, is worth the effort and can give rise to brilliant ideas.

Strategies to Get Through Hard Math Equations and Problems

Now that we have discussed equation definition in math, and have given you an overview of dealing with hard math equations, let us move on to the best part of the post. Here are some of the best strategies that can help you deal with the hardest problems in math, and the frustration they bring along. Have a look:


Stop Staring, and Do something

The thing about solving a complex math equation that you have no idea how to solve is that you at least have to try. Yes, it can be hard. However, at some point, you need to overlook that part and do something about it.

While most of the methods you try to solve the problem may not work, there is a chance that somewhere down the way, you may find your answer. Initially, it may look like a waste of effort, but trust us, in the long run, all the ideas you applied to get to that one answer will surely come in handy.


Try to Simplify the Problem

The next thing you can do while trying to solve hard math equations is to simplify them. Instead of tackling the whole problem at once, break it into smaller parts. Try solving these individual parts first rather than finding a solution to the whole problem.
While doing so, do whatever suits your needs. You may try small numbers, remove restrictions, use special cases, add restrictions, or anything you want until you can tackle the whole complicated problem.

Consider the Problems You Solved in The Past

Before you come across a particular hard-to-solve problem, we are sure there may be numerous problems you have solved in the past. Chances are, some of them may have been equally harder. Ponder about how you were able to solve those problems. What methods did you use for them?


Look for any past math problem that was similar to the one you are solving now. You can even recall problems that have nothing to do with the current one. While going through all the strategies you have used for your past math problems, chances are you may find a solution for this one too.

Think About What Is Not Used Yet

Even the hardest math problems have a solution to them. If the strategies you are using are not helping you get the correct answer, chances are you are missing out on something. For instance, what if the problem that you are finding hard to solve is a geometry problem? Typically, these kinds of questions have a lot of moving parts.


In case you have spent hours solving it and still do not have the right answer, use this strategy. Make a note of all the discoveries you have made about the problem so far. Based on these, ask yourself what method you still haven’t used. Chances are, you may find a clue about solving the problem by just doing so.

Try Working Backwards

The next strategy we have for you is working backward to solve a math problem. This can come in handy if the problem you are stuck on needs you to prove something or discover the proof. Here instead of moving from the part that tells what you know to the part that you want, reverse the order. Start the part of the equation that you want, and look for what you need to do to get there easily.

Do Not Hesitate to Ask for Help

Be it math or a life problem, asking for help seems like the best solution. However, not everyone can do it. When it comes to outstanding students who are known to solve even the hardest of questions, asking for help may be super hard. After all, they are so accustomed to completing things on their own and even getting them right.

So, when you are one everyone comes to when they do not understand a problem, how could you admit that you may need help with one? However, it is always okay to ask others once in a while. Nobody is going to judge you for that, especially not your teachers and friends. So, if you cannot get through a question, make sure to ask out rather than struggle along with it.

Along with asking your teacher, there are numerous ways to get help. For instance, you can readily find a math equation solver online, and look for the solution. Also, numerous study threads can help. Certain forums also allow you to post your question anonymously. Whatever method suits you, make sure to use it to ask for help, and solve that hard math problem quickly.

Always Start Early

While this strategy may not work with timed tests, it is sure to come in handy with your school and college coursework. For all the assignments and end-term exams, do not wait until the last minute. Instead, start your preparation early.
This way, when you come across a hard problem, you will not have to deal with time pressure. Also, you will have a lot of time to understand the complex equations well, and put them into practice.

Get Away for A Bit

If nothing works out, and you still find yourself stuck at the problem, get away from it for a bit. Yes, at this time, when you are highly frustrated, thinking of a solution is not possible. Therefore, taking a break from the math problem is the best idea. Then, when you come back after a while, you will find yourself fresher.

Also, by taking a break, you are not getting away from the problem entirely. Subconsciously, your brain is still in the loop about thinking about the solution. With the time it gets to process all the tricks you have already applied; chances are you may get closer to the solution.

And now, since you are well-rested, you can solve it easily. So, after you have put all your math work aside, to start over again, begin from the scratch. Take a fresh sheet of paper, write down the questions again, and try solving them from the beginning.

It Is Okay to Give Up

What if even after you go through various math equations examples, and are still not able to solve that one problem? In such a case, it is okay to give up. Yes, you read that right. There is no way that even an exceptional student can solve all the problems given to them. An ideal student will know when to cut losses, and move on from a question.

This is especially true if you are just starting your preparation. At this stage, you are just learning new equations and formulas. So, it is okay if you are unable to get through a single difficult problem. Instead of getting hopelessly stuck at it, it is best to quit solving the question, pick up your solution book, and go through the answer.

Be Reflective When Going Through the Solution

When you give up and decide to go through the solution for the problem, make sure to do so well. Instead of just reading it passively, understand every step actively. With every step, think about how you were not able to reach it, and what could have led you to the step. Make a mental note of all you did wrong, and make sure not to repeat it in the future.

Also, if there is something in the solution you do not understand, look it up. For it, search the internet, ask a friend, or consult your teacher – do whatever is best for you. All these tiny steps that you take notice of at this stage will act as lessons that will serve you later. With these, you will be able to solve the problem yourself, the next time you come across it.

Come Back a Few Weeks Later

Whenever you give up on a complicated math equation and go through its solution, do not just stop there. After a few weeks have passed, come back to it, and try to solve it again. If you can solve it well this time, it means you have understood it properly. If not, make sure to go through it once again.

No, what if you cannot find a solution to a problem at all? What will you do in such a case? According to us, if you have given up on the problem, it is okay. However, despite it, do not just forget it completely. Keep a note of the question somewhere. Then, every time you come across a new concept, try applying it to the problem to see if it works.

Now that you know about all these strategies to solve hard math equations and problems, let us tell you about the toughest math problems ever solved. We hope these will help motivate you the next time you come across a difficult math question.

Some Toughest Math Problems Ever Solved

When it comes to math, there is always an answer for everything. Sure, it may take years, decades, or even centuries to find these answers sometimes. However, they are still there waiting to be explored by someone. Who knows, that someone might be you?

For instance, there is a popular math puzzle called the Diophantine Equation. For decades, this summing of three cubes equation remained unsolved but was finally solved in 2019. For this puzzle, you need to find x, y, and z such that x³+y³+z³=k for each k from 1 to 100. For some, this may seem easy. But is it so?

Let us find out. Do you know what the value for x, y, and z should be to fulfill the equation x³+y³+z³=8? Simple, to complete this x should be 1, y should be -1, and z should be 2. Now, what if the equation is x³+y³+z³=42? It is hard, isn’t it? Well, we think it is since no one was able to get the value of x, y, and z here for 65 years.

After these years, a supercomputer found a solution. Do you know what it was? For x³+y³+z to be 42, x should be equal to -80538738812075974, y should be equal to 80435758145817515, and z should be equal to 12602123297335631. Amazing, isn’t it? Similar to this problem, here are some of the other toughest math problems ever solved. Have a look:

The Poincaré Conjecture

The next example of the toughest math problem ever solved is the Poincaré conjecture. This was among the 7 questions the Clay Mathematics Institute asked for people to solve. If anyone could even crack one of these, they offered $1,000,000 to them. While the other 6 problems are still a mystery, the Poincaré conjecture is decoded.

For this, the foundational work or the topology was done around the 20th century by Henri Poincaré. As per this, topologists needed mathematical tools for differentiating between abstract shapes – especially shapes in 3D space. Poincaré took this a notch up to 4-dimensional stuff. All in all, everything summed up to a simple statement – the simplest 4D shape is the 4D of a sphere.

After almost a century in the year 2003, Grigori Perelman posted proof of this. The work done by the Russian mathematician was groundbreaking, yet diffident. Then, after years of research on verifying whatever Grigori Perelman came up with, he was offered the million-dollar Millennium Prize. Along with this, he was also offered the Fields Medal which is the Nobel Prize for Math.

While Perelman was able to solve the problem correctly, he rejected both the prize money and the award. He believed that whatever he did was only for the benefit of the subject and not himself. Inspiring, isn’t it? While the other 6 questions are still unsolved, we are sure they will get their solutions one day.

Fermat’s Last Theorem

Fermat’s Last Theorem by a well-known 17th-century French lawyer and mathematician – Pierre de Fermat is again one of the toughest math problems ever solved. Math was more of a hobby for Pierre. Therefore, most of the theorems proposed by him were only mere claims with no proof. Years later, mathematicians were able to decode Fermat’s Last Theorem. When you look at this theorem, it is super simple to write.

As per it, many integer trios satisfy x²+y²=z², where x, y, and z are the said integers. Some examples of these Pythagorean Triples are 3,4,5 and 5,12,13. Now, let us see if the same trios can satisfy x³+y³=z³. If you look closely, it cannot, and this exactly is Fermat’s Last Theorem. Surprisingly, Fermat wrote it by hand, and that too in the margin of a textbook. While he had proof of the same, he could not fit it into the margin.

Thus, the proof went away with his death making the entire math world wonder if there was actual proof for it or not. After 330 years of his death in 1995, a British mathematician was finally able to find the proof. This mathematician, Sir Andrew Wiles even got a unique honorary plaque for it. He was also knighted by Queen Elizabeth II for the achievement.

A reason why Wiles was able to solve this was that he combined new research with the concept. Elliptic Curves, which was earlier not known by many, finally helped him get his answers. It proves that if you apply every concept you find to a hard math equation you are unable to solve, chances are you might end up finding the answer to it someday.

The Four-Color Theorem

This hard math equation to be ever solved is easy to state yet hard to prove. As per it, if you take 4 crayons of different colors and any map, you can color each state/country on the map without using the same color for those who share a border. That’s true, take any map out there, and the result will be the same as per the four-color theorem.

Earlier in the 19th century, the colors for this were considered fiver rather than four. By 1976, they came down to four. This was done by two mathematicians at the University of Illinois. These were Kenneth Appel and Wolfgang Hakan, and they were able to do so with computer assistance. After nearly checking 2,000 cases, they came to this conclusion that came to be known as Appel and Hakan’s proof.

Apart from these hard math equations to be ever solved, there are plenty of other equations that you can easily read about online. Some such names include:
● The Classification of Finite Simple Groups
● (The Independence of) The Continuum Hypothesis
● Gödel’s Incompleteness Theorems
● The Prime Number Theorem
● Solving Polynomials by Radicals
● Trisecting an Angle
● Euler’s Identity
● The Euler Product Formula
● The Pythagorean Theorem
● The Analytic Continuation of the Factorial
● The Cardinality of the Continuum
● The Gaussian Integral
● The Explicit Formula for the Fibonacci Sequence
● The Basel Problem
● The Harmonic Series
● The Explicit Formula for the Prime Counting Function

Final Thoughts

So, this is all you need to know about hard math equations and how to get through them. The examples in the post prove that even the toughest equations have a solution. Therefore, to get through any hard math problem, all you need is the right resources, a little dedication, and if needed, some help. Also, once while preparing for any exam that has tough math questions, make sure to practice them as much as you can to avoid any silly mistakes. All the best!

Frequently Asked Questions

1. What are the top five formulas in math?

1. Arithmetic mean(average) = Sum of values/Number of values.  

 It is used to calculate the mean value of a given set of numbers.  

2. Probability = target outcomes/Total outcomes.  

It is used to calculate the chances of something occurring from a set of possible outcomes  

3. Quadratic Formula: x= −b ± √b²-4ac/2a.  

It is used for determining the x-intercepts of a quadratic equation 

4. Distance Formulad=√ (x₁ – x₂)² + (y₁ – y₂)²  

It calculates the distance between two points on a coordinate plane. 

5. Slope Formula: Slope = y₂ – y₁ / x₂ – x₁ 

It is used to calculate the slope (angle) of a line that connects two points on a plane. 

2. How to work through hard Math problems? 

  1. Here are some strategies to tackle hard problems and the frustration that sets in.  
  2. Get going. Make a start even if you do not know how to solve it.  
  3. Problem simplification. Go for smaller numbers, remove restrictions or add some. Lower your sight, then look higher after tackling the easier one.  
  4. Ponder over successes.  Think about strategies employed to solve similar and tough problems in the past. 
  5. Focus on the unused elements. Consider what can be used constructively. This often shows the way.  
  6. Seek help. Do not hesitate to reach out for guidance if you are stuck with a problem. Even if you have a strong base, this situation is unavoidable. It brings you on track without causing heartburn.  
  7. Work backward. This is useful while trying to discover proofs. Start from what you want and ask yourself what you need to get there. 
  8. Begin early. Hard problems need to be tackled on a priority basis where the tests are not timed. It is ideal for hard and complex problems as they need more time to comprehend fully.  
  9.  Take a break. Always effective to divert your mind instead of lingering over it and then begin afresh.  
  10.  Be introspective.  Think about where you went wrong and pore over the clues in the problem. Contemplate the routes explored to get the solution and how to avoid them.  
  11.  Come back. If you gave up and looked at the solutions, then come back and try the problem again later.

3. What is x3y3z3k?

he diophantine equation is known as the summing of three cubes, and it has puzzled mathematicians for decades. K represents all the numbers from one to 100. The challenge is to find integers that work for all equations or the numbers for x, y, and z that will all equal.  

4. Is there any unsolved math problem? 

There are many unsolved problems in mathematics.  The key ones include The Goldbach conjecture, The Riemann hypothesis, Hadamard matrix, twin prime conjecture, Colatz problem, and NP problems as P-problems.  

5. What is the formula of a³-b³? 

An expression of the form a³-b³ is called the difference of cubes.

a³-b³ = (ab)[(ab+3ab]



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