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?