Self-reference
A while ago I mentioned the self-referential aptitude test by Jim Propp. Propp's test is really fun, and can be solved pretty straightforwardly because we know that it's possible to get 20 out of 20 correct. Therefore, you should always try to answer each problem correctly. We have no such assurance for another self-referential test by Don Woods -- which I haven't solved yet. In Woods's test, the complexity comes from the fact that "[y]our goal is to achieve as high a score as possible" (without necessarily trying to get any individual problem right).
Woods includes this remarkable question:
20. The maximum score that can be achieved on this test is: (A) 18 (B) 19 (C) 20 (D) indeterminate (E) achievable only by getting this question wrong
When I first looked at this, I thought, well, the answer (D) can't be right, because surely there is some number that is the highest possible score. Then I remembered that you're trying to get the highest possible score, not to get every question right. So (D) actually could be the answer you should give, even though it would be incorrect.
Then I thought, well, (E) couldn't be the answer because that would create a paradox. If the highest possible score can be achieved only by getting question 20 wrong, then an answer to (E) would be right, and it would no longer be true that the highest possible score can be achieved only by getting question 20 wrong. Thus, the answer to question 20 must not be (E).
But that reasoning is just as bad as my previous reasoning about the answer (D). The goal of the test is not to avoid creating a paradox. The goal of the test is to get the highest score possible. If the way to get the highest score possible is to answer (E) for question 20, then you should answer question 20 with the answer (E). If this is so, then the best answer to question 20 is an answer that effectively claims that it is wrong (or is not the best answer).
I think Don Woods is just devious enough that the answer to question 20 might well be (E) even though (E) would be neither the right nor the wrong answer to question 20.
