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A Controversial Tactic

At least once, I have seen a posted solution to a problem on Brilliant in which the following argument appears to have been used:

"The problem would not have been set unless it has a unique answer. In order to have a unique answer, the answer must be true for <a specific case of the problem>. Therefore, the solution to the problem [in its general form] is the same as that for <its specific case>."

This idea isn't a new one. Take a look at the Hole in the Sphere problem, which I originally came across in Martin Gardner's book Mathematical Puzzles and Diversions. Both the web page and the book actually solve it properly, but the book also notes this shortcut as a way to solve the problem.

I have used the same technique myself. For example, I solved this problem by noting that whatever the answer is will be true for an equilateral triange, and so was able to reduce the problem to this specific case and thereby solve it.

Once upon a time, there was an exchange on the solution board for a problem that went something like this:

  • Person A: <a solution that appears to use the shortcut>
  • Person B: Your solution isn't valid, as it makes up specific values for the variables for which no values have been specified.
  • Me: I think this person has reasoned that the problem wouldn't have been set unless it has a unique answer, in which case the answer must be true for these values of the variables.
  • Person B: So Person A gets the marks for the correct answer, but no marks for the method.

This begs the question: the method for doing what? Arriving at the answer, or proving that the problem is valid in the first place? Must one untie the Gordian knot, or will it do to grab one's sword with both hands and cut said knot?

Here's another example of solving a problem by taking advantage of the fact that there must be a unique solution. Every Sudoku has a unique solution. Some advanced strategy guides mention a 'unique rectangles' tactic. In this tactic, you eliminate a possible value for a square by realising that it will necessarily lead to a non-unique solution. By applying this technique, you lose the rigour whereby the solution process proves that that is the solution, but at the end of the day it's still a way of reaching the solution. The same goes for problems in algebra, geometry or potentially any branch of mathematics.

Of course a complete, rigorous solution would prove that the answer holds regardless of the values of the variables present in the problem. But at the same time, the skill at solving mathematical problems lies in arriving at the answer in the most efficient way. Seeing shortcuts is part of this skill. As such, I personally think that this shortcut is a legitimate tactic.

Note by Stewart Gordon
9 months, 1 week ago

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Getting the right answer and knowing why it is the right answer is two completely different things.

Take the following 3 problems as an example:

  1. For this problem, if you've submit 123 as your answer, does that means that the limit exist? No! You still need to explain why the limit must exist.

  2. For this problem, plugging in values of 1,2,3 quickly solved this problem, but that doesn't guarantee that it's always true for all values of \(n\). As a solution writer, you need to confirm the fact that the problem statement is correct.

  3. Similarly, you can also easily solved this problem by some quick guesses but proving that the answer is unique is the hard part.

Pi Han Goh · 9 months, 1 week ago

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@Pi Han Goh Of course they're different things. This is the whole basis of my essay.

Anyway, looking at your examples:

  1. I think you've misread it - the problem says to enter 123 if the limit does not exist. But the problem as stated is ambiguous. By "the limit", does it mean:

    • the limit to which the formula always converges, regardless of the value of \(x\)?
    • the limit to which the formula always converges for those values of \(x\) for which it converges at all?

    By the first interpretation, you would have to prove whether it converges to the same limit for all values of \(x\) in order to solve the problem. If you use the second interpretation, on the other hand, then it suffices to:

    • prove that there's only one possible limit
    • find that limit
    • realise that there's one value of \(x\) for which it is bound to converge to this limit
  2. Does it say this somewhere in the house rules? I've never noticed it - indeed, I would have thought it the responsibility of the setter, rather than the solver, to make sure the problem statement is correct.

  3. Exactly. It's nice to have a proof that there's only one solution, but as you say that's the hard part. Plugging in the obvious coefficients is an efficient way of arriving at a solution, and indeed being certain that it's a correct solution, if that's what you're trying to do.

Stewart Gordon · 9 months, 1 week ago

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@Stewart Gordon No I didn't misread the question. What I've said was "if I tried to submit 123 as the answer, and the system tells me that I've inputted the wrong value, does that mean that I can conclude the fact that the limit exist?"

Since they didn't give any value of \(x\), you shouldn't assume that it only applies for a certain value of \(x\).

It is both the responsibility of the setter of the problem and the person who wrote the solution to verify that the problem statement and the (final) answer to be correct.

So what are you trying to accomplish here? That it's alright to post solutions (on Brilliant) like "oh I tried 123 and it doesn't work, so the limit must exist!" ? By doing so, you've made a lot of problems trivial, like assuming the triangle is equilateral, or the angle can be 90 degrees, "I've found two numbers, one is an integer, the other is not, but since this problem only accepts integer, I know what value to submit!" Pi Han Goh · 9 months, 1 week ago

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@Pi Han Goh Uh, half of your comments have nothing to do with what I've written.

"It is both the responsibility of the setter of the problem and the person who wrote the solution to verify that the problem statement and the (final) answer to be correct." Can you refer me to the official statement of this policy? In the absence of an official statement, it's just your opinion. Stewart Gordon · 9 months, 1 week ago

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Due to the need for a simple evaluation system, we lose the advantages of a detailed marking scheme that allows differentiation of the extent to which a solution is deemed correct/complete. We do not have the resources of the GCSE board, which hires thousands of teachers to grade all of the incoming O/A level scripts. As such, "Have a challenge master review your solution" is currently a B^2 feature, because it can take significant time to work through a proof solution with no clear marking scheme.

It is already hard enough to get started on problem solving, and I do not want to enforce "No guessing / estimation / hunches / assumptions / ... allowed". In fact, doing so runs counter to the way that most people approach a problem. We look at the given information, think about what it implies and what are some things that are "nice to have", before ultimately proving it for ourselves. Of course, not everyone is at the final stage of being able to prove their statements concretely, in all of the topics that they are interested in.

At the end of the day, you decide how you want to use Brilliant, and what is the right amount of rigor / interest you want to display. Given the diverse range of interests that we have, there is certainly no hard and fast rule to what is desired / allowed / implied. Calvin Lin Staff · 9 months, 1 week ago

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