When I was taking AP Physics in my Junior year, I always had fun trying to leave all my work in terms of variables until the end, and avoid substituting in values in intermediate calculations. When I finished the following problem it gave me much joy to see that the equation was correct.

There are many, actually. That depends on how you define "the best problem". Some problems (particularly in Physics and Chemistry) are beautiful since they are composed of applications of a variety of techniques, while Mathematics problems are based on mechanical reasoning, mostly. But you can't deny the joy that lies herewith. I have solved many good problems but let me recollect a small incident. I was then a ten years old child, and I didn't know the sum of first n natural numbers. In our school, it was taught much, much later, say, 3 years later, while performing statistical experiments. I curiously took up a problem of summing, and actually DEVISED this method of n(n+1)/2. And my joy knew no bounds when I could solve each and every summation using this formula. Those three years were the best in my life, as I felt I had invented some ingenious formula, and never told it to anyone. And you can expect how shattered I was when I saw how my INVENTED formula was right there on a page of a book, staring back and laughing at me. Well, this seems funny now, but I was only a kid back then.

I know the feeling. Just over a year ago I basically proved the multinomial theorem, taking an n-termed polynomial to the m-th power. And then no more than three months later I stumbled upon the multinomial theorem on wikipedia. Dreams crushed.

I know how that feels. I spent 2 months of my summer proving a theorem, which hinged on a crucial fact. When I presented my work, no one in the audience knew about it. 3 years later, someone told me that "Oh, it's a special case of another theorem".

But, that's also the great thing about maths. It's the ability for you to figure out what patterns exist in the data that you're given, make conjecture about the validity of certain statements, and then prove those statements, or find a counter-example to show why you are wrong.

When I was about five or six years old, I discovered something I thought was extraordinary: if you take a number and square it, it it one more than the product of the two adjacent numbers to the original number. For example, $5^2=25$ is one more than $4\cdot 6=24$. I did not know algebra to proof it at that point, but I tried it many times (even for negative numbers) and I was sure it would be true for every number.

Only much later (in 8th grade) I continued my investigations. What if you would not take the product of the two adjacent numbers to the original number, but numbers that are further off? For example, $5^2=25$ and $3\cdot 7=21$, that's a difference of $4=2^2$! The difference would always be the square of the original difference. I had algebra back then to prove it, and I felt like a real mathematician. Literally a week later, it was taught at my high school, as just a little trick. I was very angry for the fact that my school had demoted such a wonderful identity to just a little trick.

Later on, quite recently, before being exposed to the awesome world of olympiad mathematics, I thought the following: if $4$ times $8$ does not get me to $6^2$, what do I have to multiply $4$ with to get to $6^2$? It turned out to be $9$. I tried to generalise. I always seemed to get three elements of a geometric series, which I thought was quite fascinating. I proved it with simple algebra.

Those were my own discoveries I'm very proud of, especially the first one. I know that it is not really a problem I have solved, but you said 'anything will do', so… ;-)

I am not able to remember many, if I will recall it I would really tell you. But the most difficult problem for me on Brilliant was the "primed triangles".

@Siddharth Kumar
–
If anyone other than you follows the link you provided, they'll be redirected to their challenge page with the heading 'problem record not found'. Try posting the 'share this problem' link.

Siddharth, if you want to replace it with your own, let me know and I'd pull my link down. You can then see how many people have worked on the problem through clicking your link.

Well..The best problem I have ever solved is coming up with my username...jk
Actually, in my opinion, every problem is a little part of a bigger "whole". We are just coming in closer and closer to this "whole" , but never quite reaching it, like how $\sum_{i=1}^{\infty} \frac{1}{2^i}$ approaches but never touches two.

I won't claim that it's the "best" problem I've ever solved, but one of my favorite problems (in the sense that I enjoyed the process of coming up with a solution) is one from extremal graph theory:

"Prove that any simple graph on $n$ vertices and $e$ edges has at least $\frac{e}{3n}(4e-n^2)$ triangles."

When I was in 9th grade, I solved the following problem:

Find all natural number triples (x,y,z) such that $2^x + 3^y = 4^z + 1.$

I solved this problem without sleeping; it took me an entire day. This was my first exposure to interesting problems.

In retrospect, I think it was a simple problem. The joy of getting an answer after spending hours of thinking is a thrill. I still seek that thrill and solve problems. I run local math competitions for high school students and have also introduced a weekly undergraduate problem solving seminar in my university.

I just want more people to appreciate the thrill of problem solving. I like this site.

Coming back to the thread, the most recent best question I tried was:

Prove that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line

It's called the Erdos-Anning theorem. It has a wonderful proof.

https://brilliant.org/i/OUN6fq/
I had to spend a lot of time on this problem.I thought over the problem so much at times that I couldn't focus on anything else.Oneday after a nap I don't know what happened but I grabbed my copy and started attempting the problem,at that moment everything seemed so clear,had to do approximation while getting a root of a quadratic which i really don't like,but the experience was totally awesome.

There are other ways to approach the problem, that doesn't require you to find the root of a quadratic (which is typical in such questions). Here's a start.

Hint: For any ride, there are at most 3 people who didn't ride it.

I think that I have way too many favorites but in geometry I really really like this problem. I am very bad at geometry but this problem was so neat I actually took the time to go through it.
I specially like it because it involves so many different ideas at the same time.
http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2456654&sid=27636296f39c4ce8533cb20c72415848#p2456654

This problem is absolutely wonderful: Lazy Liz's Escape, currently in the level 4 Geometry and Combinatorics set ( https://brilliant.org/i/YzpBhp/ ). I'm pretty sure I missed the proper way to solve it, and I can't wait to see the worked solution next week. How I did it: just try some lower values, making a conjecture without even proving it… I found the fibonacci numbers, triangle numbers, binomial coefficients and other things I do not understand yet in a problem stated so simply. Amazing.

$</code> ... <code>$</code>...<code>."> Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in $</span> ... <span>$ or $</span> ... <span>$ to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestWhen I was taking AP Physics in my Junior year, I always had fun trying to leave all my work in terms of variables until the end, and avoid substituting in values in intermediate calculations. When I finished the following problem it gave me much joy to see that the equation was correct.

Problem: https://imgur.com/Aui8aR2

$v_3=\sqrt{2gy_2+\left(\frac{\left(m_{G+S}\right)\sqrt{2gy_1+\left({\frac{v_1 m_G}{m_{G+S}}}\right)^2}}{m_{G+S+B}}\right)^2}=15.6 \, \mathrm{\frac{m}{s}}$

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I do that too, not using a calculating till the very end :D

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There are many, actually. That depends on how you define "the best problem". Some problems (particularly in Physics and Chemistry) are beautiful since they are composed of applications of a variety of techniques, while Mathematics problems are based on mechanical reasoning, mostly. But you can't deny the joy that lies herewith. I have solved many good problems but let me recollect a small incident. I was then a ten years old child, and I didn't know the sum of first n natural numbers. In our school, it was taught much, much later, say, 3 years later, while performing statistical experiments. I curiously took up a problem of summing, and actually DEVISED this method of n(n+1)/2. And my joy knew no bounds when I could solve each and every summation using this formula. Those three years were the best in my life, as I felt I had invented some ingenious formula, and never told it to anyone. And you can expect how shattered I was when I saw how my INVENTED formula was right there on a page of a book, staring back and laughing at me. Well, this seems funny now, but I was only a kid back then.

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I know the feeling. Just over a year ago I basically proved the multinomial theorem, taking an n-termed polynomial to the m-th power. And then no more than three months later I stumbled upon the multinomial theorem on wikipedia. Dreams crushed.

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I know how that feels. I spent 2 months of my summer proving a theorem, which hinged on a crucial fact. When I presented my work, no one in the audience knew about it. 3 years later, someone told me that "Oh, it's a special case of another theorem".

But, that's also the great thing about maths. It's the ability for you to figure out what patterns exist in the data that you're given, make conjecture about the validity of certain statements, and then prove those statements, or find a counter-example to show why you are wrong.

Log in to reply

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When I was about five or six years old, I discovered something I thought was extraordinary: if you take a number and square it, it it one more than the product of the two adjacent numbers to the original number. For example, $5^2=25$ is one more than $4\cdot 6=24$. I did not know algebra to proof it at that point, but I tried it many times (even for negative numbers) and I was sure it would be true for every number.

Only much later (in 8th grade) I continued my investigations. What if you would not take the product of the two adjacent numbers to the original number, but numbers that are further off? For example, $5^2=25$ and $3\cdot 7=21$, that's a difference of $4=2^2$! The difference would always be the square of the original difference. I had algebra back then to prove it, and I felt like a real mathematician. Literally a week later, it was taught at my high school, as just a little trick. I was very angry for the fact that my school had demoted such a wonderful identity to just a little trick.

Later on, quite recently, before being exposed to the awesome world of olympiad mathematics, I thought the following: if $4$ times $8$ does not get me to $6^2$, what do I have to multiply $4$ with to get to $6^2$? It turned out to be $9$. I tried to generalise. I always seemed to get three elements of a geometric series, which I thought was quite fascinating. I proved it with simple algebra.

Those were my own discoveries I'm very proud of, especially the first one. I know that it is not really a problem I have solved, but you said 'anything will do', so… ;-)

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All i can say is that moment is exhilarating!!! No one around only you and the problem that just got solved.

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I am not able to remember many, if I will recall it I would really tell you. But the most difficult problem for me on Brilliant was the "primed triangles".

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Could you either post a link to that problem or put it into words yourself? You've made me curious, but Google isn't able to find it. :)

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This is the link to the problem "primed triangles" :- https://brilliant.org/assessment/s/algebra-and-number-theory/351092/

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This is a link to the Primed Triangles question.

Siddharth, if you want to replace it with your own, let me know and I'd pull my link down. You can then see how many people have worked on the problem through clicking your link.

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Just solved 'Primed Triangles'. Nice problem!

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Well..The best problem I have ever solved is coming up with my username...jk Actually, in my opinion, every problem is a little part of a bigger "whole". We are just coming in closer and closer to this "whole" , but never quite reaching it, like how $\sum_{i=1}^{\infty} \frac{1}{2^i}$ approaches but never touches two.

Log in to reply

I won't claim that it's the "best" problem I've ever solved, but one of my favorite problems (in the sense that I enjoyed the process of coming up with a solution) is one from extremal graph theory:

"Prove that any simple graph on $n$ vertices and $e$ edges has at least $\frac{e}{3n}(4e-n^2)$ triangles."

Log in to reply

When I was in 9th grade, I solved the following problem:

Find all natural number triples (x,y,z) such that $2^x + 3^y = 4^z + 1.$I solved this problem without sleeping; it took me an entire day. This was my first exposure to interesting problems.

In retrospect, I think it was a simple problem. The joy of getting an answer after spending hours of thinking is a thrill. I still seek that thrill and solve problems. I run local math competitions for high school students and have also introduced a weekly undergraduate problem solving seminar in my university.

I just want more people to appreciate the thrill of problem solving. I like this site.

Coming back to the thread, the most recent best question I tried was:

Prove that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight lineIt's called the Erdos-Anning theorem. It has a wonderful proof.

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The proof is great. It makes good use of the fact that 2 conics can intersect in at most 4 points.

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https://brilliant.org/i/OUN6fq/ I had to spend a lot of time on this problem.I thought over the problem so much at times that I couldn't focus on anything else.Oneday after a nap I don't know what happened but I grabbed my copy and started attempting the problem,at that moment everything seemed so clear,had to do approximation while getting a root of a quadratic which i really don't like,but the experience was totally awesome.

Log in to reply

There are other ways to approach the problem, that doesn't require you to find the root of a quadratic (which is typical in such questions). Here's a start.

Hint:For any ride, there are at most 3 people who didn't ride it.Log in to reply

I think that I have way too many favorites but in geometry I really really like this problem. I am very bad at geometry but this problem was so neat I actually took the time to go through it. I specially like it because it involves so many different ideas at the same time. http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2456654&sid=27636296f39c4ce8533cb20c72415848#p2456654

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This problem is absolutely wonderful:

Lazy Liz's Escape, currently in the level 4 Geometry and Combinatorics set ( https://brilliant.org/i/YzpBhp/ ). I'm pretty sure I missed the proper way to solve it, and I can't wait to see the worked solution next week. How I did it: just try some lower values, making a conjecture without even proving it… I found the fibonacci numbers, triangle numbers, binomial coefficients and other things I do not understand yet in a problem stated so simply. Amazing.Log in to reply

Well https://brilliant.org/assessment/s/geometry-and-combinatorics/191578/

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That link is unique to you. Try posting the 'share this problem' link.

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sorry bro forgot bout that..

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That gives me an error.

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