You’re trying to solve a puzzle. Maybe it’s a jigsaw puzzle, maybe it’s a Sudoku puzzle, maybe it’s an interesting math problem. In any case, it’s one of those puzzles where you know a solution when you see it, and once it’s almost solved, everything falls into place.
At the moment, you’re kind of stumped. You’ve been unable to figure out any more facts using deductive reasoning, so now it’s time to resort to trial and error. You have three independent hypotheses about the puzzle. Hypothesis A seems to have an 80% chance of being right, hypothesis B a 50% chance, and hypothesis C a 20% chance. You’re going to pick a hypothesis and investigate what seems to happen if this hypothesis is true.
Do you:
choose hypothesis A, since it’s probably right and so it’s likely to lead to the right answer,
choose hypothesis B, since this one will yield the most expected information if you figure out whether it’s true or false, or
choose hypothesis C, since it’s probably wrong and so you’re likely to find a disproof, thereby giving you more information?
(Of course, assuming that a hypothesis is false is equivalent to assuming that one of these hypotheses is true.)
The answer, of course, is probably “it depends”. But what does it depend on, and what’s the most likely choice overall?
My first thought is that if it’s a really big puzzle, then you’ll pretty much only make useful progress by establishing things with certainty (since if you make an assumption you think has a probability of 80% five times, there’s only a 33% chance that you were right every time), so your best bet is to assume the hypothesis that is most likely to be falsified, i.e. C. The desire for certainty overrides all other concerns.
But for a really big puzzle, it also makes sense to pick hypothesis B, and try to prove it and to disprove it, because, although you’re less likely to come up with a proof, a proof will end up being more useful in this case.
If it’s a small puzzle, assuming C is likely to be a waste of time. You might be best off picking hypothesis A, since this is likely to lead you straight to a solution. Or perhaps B is a better option, since now there are four ways it could be useful. You could find that there’s no solution where B is true; you could find that there’s no solution where B is false; you could find a solution where B is true; or you could find a solution where B is false.
At one extreme is the weekly sudoku puzzle that is completed for my own enjoyment and I am content with my mastery level. At that extreme I pick A, it is not really a goal to improve my sudoku skill, negative results contribute little.
At the other extreme are your hard coding problems, complex engineering problems, and those little bent steel puzzles that you have to take apart and reassemble in apparently impossible ways; Here the long term goal is not to solve the single problem but to learn and solve the problem type. These problems are either without an A solution, or you want greater mastery of the field so that next time only A or B solutions are on the table instead of the B and C solutions. Eventually your peers start coming to you with their A, B, C dilemmas and you can give them the right answer with 100% confidence. After that you will be known as a guru in your field and reap all the prestige and profit that comes with that (results may vary depending on field).
In short it depends on your goal, A to solve the problem, C to solve the problem type. B is a compromise if you want C but the budget doesn’t allow for it.
I’d select B for the “hard coding problems”, as that would give me the most information. (I’m already relatively sure that C won’t work, but I may have absolutely no idea whether B would work).
Do you mean “hypothesis” as something that solves the problem?
If yes, then there’s a problem. Either your beliefs are inconsistent (as they don’t add up to 100%), or the hypotheses cannot be independent. Assuming your beliefs are consistent, your best best would be to figure out what’s the correlation between these hypothesis is between choosing one to test. For example, C could be correlated with A. Choosing to pursue A (or C) would then give information about C (or A) as well.
If no, the amount of the information you’ll get from testing them is incomparable. For example, B could be about a relatively minor thing, while A about 99% of the solution.
You’re trying to solve a puzzle. Maybe it’s a jigsaw puzzle, maybe it’s a Sudoku puzzle, maybe it’s an interesting math problem. In any case, it’s one of those puzzles where you know a solution when you see it, and once it’s almost solved, everything falls into place.
At the moment, you’re kind of stumped. You’ve been unable to figure out any more facts using deductive reasoning, so now it’s time to resort to trial and error. You have three independent hypotheses about the puzzle. Hypothesis A seems to have an 80% chance of being right, hypothesis B a 50% chance, and hypothesis C a 20% chance. You’re going to pick a hypothesis and investigate what seems to happen if this hypothesis is true.
Do you:
choose hypothesis A, since it’s probably right and so it’s likely to lead to the right answer,
choose hypothesis B, since this one will yield the most expected information if you figure out whether it’s true or false, or
choose hypothesis C, since it’s probably wrong and so you’re likely to find a disproof, thereby giving you more information?
(Of course, assuming that a hypothesis is false is equivalent to assuming that one of these hypotheses is true.)
The answer, of course, is probably “it depends”. But what does it depend on, and what’s the most likely choice overall?
My first thought is that if it’s a really big puzzle, then you’ll pretty much only make useful progress by establishing things with certainty (since if you make an assumption you think has a probability of 80% five times, there’s only a 33% chance that you were right every time), so your best bet is to assume the hypothesis that is most likely to be falsified, i.e. C. The desire for certainty overrides all other concerns.
But for a really big puzzle, it also makes sense to pick hypothesis B, and try to prove it and to disprove it, because, although you’re less likely to come up with a proof, a proof will end up being more useful in this case.
If it’s a small puzzle, assuming C is likely to be a waste of time. You might be best off picking hypothesis A, since this is likely to lead you straight to a solution. Or perhaps B is a better option, since now there are four ways it could be useful. You could find that there’s no solution where B is true; you could find that there’s no solution where B is false; you could find a solution where B is true; or you could find a solution where B is false.
Any thoughts?
At one extreme is the weekly sudoku puzzle that is completed for my own enjoyment and I am content with my mastery level. At that extreme I pick A, it is not really a goal to improve my sudoku skill, negative results contribute little.
At the other extreme are your hard coding problems, complex engineering problems, and those little bent steel puzzles that you have to take apart and reassemble in apparently impossible ways; Here the long term goal is not to solve the single problem but to learn and solve the problem type. These problems are either without an A solution, or you want greater mastery of the field so that next time only A or B solutions are on the table instead of the B and C solutions. Eventually your peers start coming to you with their A, B, C dilemmas and you can give them the right answer with 100% confidence. After that you will be known as a guru in your field and reap all the prestige and profit that comes with that (results may vary depending on field).
In short it depends on your goal, A to solve the problem, C to solve the problem type. B is a compromise if you want C but the budget doesn’t allow for it.
I’d select B for the “hard coding problems”, as that would give me the most information. (I’m already relatively sure that C won’t work, but I may have absolutely no idea whether B would work).
Expanded reading on why I favor C
Do you mean “hypothesis” as something that solves the problem?
If yes, then there’s a problem. Either your beliefs are inconsistent (as they don’t add up to 100%), or the hypotheses cannot be independent. Assuming your beliefs are consistent, your best best would be to figure out what’s the correlation between these hypothesis is between choosing one to test. For example, C could be correlated with A. Choosing to pursue A (or C) would then give information about C (or A) as well.
If no, the amount of the information you’ll get from testing them is incomparable. For example, B could be about a relatively minor thing, while A about 99% of the solution.