Ethics when solving problems on Brilliant

The cool thing about Brilliant problems is that your answers are checked instantly, because just a numerical answer is required. There is a downside to this though: you don't need to prove something in order to use it to solve a problem, which is different than actual olympiads, where you do have to prove it. This has raised some ethical issues:

If a problem is about some big number, and you first try smaller numbers $(1,2,3,\dots)$ and find a pattern, do you try to prove (probably by induction) that that pattern continues? Or do you just assume the pattern will continue and get the right answer without actually solving the problem completely?

For example: The table is empty. Every minute, John doubles the amount of money that's on the table and adds one more dollar. So, after one minute, there is just one dollar on the table. If John has $15000, how long can he keep this up?

Trying several small cases, you will find that there is 1 dollar after 1 minute, 3 dollar after 2 minutes, 7 dollars after 3 minutes, etcetera. You could now conjecture that after $n$ minutes, John will have put $2^n-1$ dollars on the table. If you assume this pattern is no coincidence and it will continue, you will quickly find out that John can keep doing this for 13 minutes, no more. Actually proving that this pattern will continue will take you some more effort, which you do not have to go through, as only the answer is required.

If a problem is about a quite general case and what's asked (apparently) does not depend on which case it is, do you try to solve the problem for the general case, or do you just pick a special case that is very easy?

For example: A group of friends is celebrating Christmas together by giving each other presents, but without knowing who they got their present from. So, each one has written down his/her name on a piece of paper, and put it in a box. After the box has been shaken a few times, everyone takes out a piece of paper. However, you obviously can't give yourself a present. What is the expected number of people to take out a piece of paper with his/her own name on it?

As it is unknown how large the group of friends is, it apparently doesn't matter. What you could do is just try a group of 2 friends, and you will find out that the answer is 1. You then know the answer to the question, but did not prove that this is the answer for any number of friends, which is the hard part of this problem.

What are your thoughts on this? Do you try to always solve it for the general case, or do you give the answer whenever you think you got it right?

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Ultimately, the purpose of brilliant is for students to learn. If a student decides to "cheat" in this manner, he may be robbing himself of a chance to pick up a new technique. Of course, he can answer first and read the rigourous proof the following week, but the motivation is somewhat lessened and a bit of frustration (at failing to solve the problem) can actually motivate him to remember the solution.

Bottom line is: we're all in charge of our own education.

That being said, testing and guessing isn't a bad start: in fact that's how most interesting conjectures and theorems are born. I certainly hope that students won't just stop there, that they'd at least put in some effort to prove the general theorem.

C Lim - 7 years, 11 months ago

I sometimes do assume a few things when solving problems, but I try to go back the next week to see the solution and fill in the holes.

Michael Tang - 7 years, 11 months ago

I have to admit, I usually find the pattern or look for and solve one specific case in the problem. Every once in a while I don't though. But I think the reason I do this is 1) because its easier 2) some Olympiads in the USA are in this answering style and 3) the general case is not that hard to prove if you've got the hang of it with a special case, and the pattern is usually easy to prove too. I don't think it's necessarily horrible to do these things, but if I ever get bored, I have a plethora of things to prove and write up.

In school we are always thought to tell how the questions hints toward the answer. Work smarter, not harder.

Bob Krueger - 7 years, 11 months ago

Last week, I had a difficult geometry problem which was about a figure which could be drawn in many different ways, and some area was asked. I just drew it in a way that some lines were on top of each other, which made it a very easy problem. That obviously wasn't the point of the original problem.

In my examples, the patterns were not too hard to prove, but I did once have a problem in which I found a pattern of which I had no clue why it was there, but it did lead me to the right answer.

I felt like I cheated in those situations. True, some olympiads seem to be designed to encourage this behaviour, but I did not learn anything from it, I just answered something of which I knew that it was true, not why it was true (which -I think- is the whole point).

Would you say that I shouldn't have answered before I proved it, or that I did the right thing, because after all, I did find out the correct answer myself?

Tim Vermeulen - 7 years, 11 months ago

Heh, check out first solution: http://www.artofproblemsolving.com/Wiki/index.php/2009AIMEIProblems/Problem4

Jan J. - 7 years, 11 months ago

Why would it be cheating? Brilliant has the system set up that way; it doesn't ask you to prove your answer for a reason (granted, it's partly because it would be impossible to check all of the answers that way) and many other math competitions have it like that. It's important to know both how to prove your answer and how to get to an answer quickly, as many of the olympiads which do not require you to prove your answer compensate by heavily restricting you on time.

Michael Tong - 7 years, 11 months ago

And it's often difficult to prove something on a computer. The day this happens, I'll stop using Brilliant. Simple reason: I have the proof right here in my copy. Come to my house and I'll show you. But if a sum took 3 pages to do (which in many cases, happens!), seriously I'm not going to write down the entire proof on a computer page. That's ridiculous, like so much of formatting and all those. And first you are proving this on your copy, and then you are putting it up on the computer...it's double an effort. And to be precise, I like more numerical answers than long proofs. The website is to teach you, not to help you cheat. If you're doing it, you may have an increase in points but wait. What'll happen if you get 20000 points by cheating? ABSOLUTELY NOTHING! You're not going to become world-famous, these 20000 points will not help you in anyway. Rather, you may boast of your score in public and you'll be immediately approached by people who'll consider you as a maths wizard. They'll give you tough problems to solve and there you are, unable to solve them. Rather, the better thing is, you go slow but straight. You'll learn then. Otherwise, what's the point of solving? You're not DOING it, you're just a middleman. And as for the second part of your question, I guess we should be careful while adding our answers as the website approves only 3 solutions, and hence there should always be a proof ready so that, somehow if your answer doesn't match with the answer given (after all, we are all humans), but you consider your answer to be perfect, you may request clarification or dispute and then you may be able to back your answer.

Leonardo DiCaprio - 7 years, 11 months ago

It is true that most of the highest level level olympiads require you to prove your answers (e.g. IMO, USAMO, USAMTS), there are many other competitions (which are of a level such that many more people can participate in them, such as AMC and AIME) which do not; in fact, the AIME competition also asks for an integer answer between 0 and 999, so I don't think it's really too much of a travesty that you don't have to prove your answer. However, a good way that brilliant has you practice actually proving your answer is when you have to write a solution because then you must prove your answer or else it is marked incorrect because you use assumptions in getting your answer. So all in all, I don't really see a problem with the method brilliant has it now as the olympiads most people are participating in don't require you to prove your answer and it would alienate many users if you had to prove your answer because it certainly is a lot harder to do so.

For your last question, when I solve problems I sometimes assume things. For example, in your example with the money, if I saw those partial sums I would assume the same thing and report my answer. However, since I am an inquisitive person I would not just stop there and be done with it, I usually try to prove / figure out why that happens, and I'm sure other people do something like that, too (if they have time).

I'm not saying that everyone should write down a complete and rigorous proof before submitting an answer. Of course that might be too hard for some people on this website. But if you see a pattern of which you don't know why it is there and it just happens to be correct, you didn't really solve the problem. If some people are able to see the pattern but not to prove it, then it might be because they are in a too high problem set.

\overline{abc}

Anmol Gupta - 7 years, 11 months ago

The table is empty. Every minute, John doubles the amount of money that's on the table and adds one more dollar. So, after one minute, there is just one dollar on the table. If John has $15000, how long can he keep this up?

Trying several small cases, you will find that there is 1 dollar after 1 minute, 3 dollar after 2 minutes, 7 dollars after 3 minutes, etc. You could now conjecture that after n minutes, John will have put 2^n−1 dollars on the table. If you assume this pattern is no coincidence and it will continue, you will quickly find out that John can keep doing this for 13 minutes, no more. Actually proving that this pattern will continue will take you some more effort, which you do not have to go through, as only the answer is required.

Finding a pattern and assuming that the pattern is true is wrong.

Let the amount of money after n minutes be $f(n)$ . Then $f(n+1)=2f(n)+1$ and $f(0)=0$ . $2^n - 1$ satisfies this function so it is the correct answer.

This approach uses 'finding a pattern and guessing the general term' technique, but it is not wrong, as we have made no false assumptions.

Eklavya Sharma - 7 years, 11 months ago

If you prove that $2^n-1$ actually satisfies the function, then there's nothing wrong with that. But if you see that the function satisfies for the first few values, and then just assume that it will continue to satisfy (without proving it), then you are making assumptions.

This precise thing happened to me in a nice polynomial problem in level 4 algebra this week. I guessed the answer correctly. And then took a couple of days to prove it o_0. I am waiting for a simple solution next week. I am aware I solve problems in a very complicated manner :(

A part of mathematics is guesswork. Once I know my answer is correct, I can prove the claim that leads to the answer. Surely at the time of entering the answer, I am unethical. But on the long run, I am not. It is a part of guessing and proving.

Another interesting issue of ethics is using a computer code to answer questions. I generally write a computer code to verify my answer (if I have only one try left). Clearly if I just plug the answer in, I am unethical. But using a computer allows me to guess patterns faster and formulate better hypotheses.

I also discuss the problems with my friends (who dont have brilliant account and they do not want one). Clearly this might be considered a breach of ethics too. But it is a part of problem-solving culture. In teens, students are encouraged to be independent, but some of the best research arises from collaboration. Further one learns faster by discussing with like-minded individuals.

The main point is that I lurk in Brilliant for the joy of problem solving and to learn new techniques. Using a computer, and guessing the answer first on brilliant are some techniques I use. These help me learn the subject better. I do not think my score reflects my intelligence for the mentioned reasons.

Sri Kanth - 7 years, 11 months ago

This is good point I think. But I don't see any remedy over it. For example if problem asks to find out last three digits of 23! then one can use mathematica and write the correct answer instead of using mathematical weapons. Do you suggest some solution to this problem?

Snehal Shekatkar - 7 years, 11 months ago

It depends whether it is in Math Olympiads or Computer Science. If it is in Math Olympiads, there is apparently a way do solve it mathematically and I'd say it's cheating when you use a computer.

When we think about it, the ethics of problem solving here stoop much lower when people begin to use things like calculators for large numbers, or worse, Wolfram|Alpha for complicated expressions. (I sheepishly admit that I've used a calculator on some problems due to frustration.)

Tanishq Aggarwal - 7 years, 11 months ago

hey friend are you on facebook??? would like to interact with you and have some discussion regarding questions,,

Ravi Prakash Roy - 7 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$