A world in which AI churns out amazing proofs would be pretty radical though.
Or as my undergrad mathematics advisor described friends who left academics for finance, "they spend their days clipping (stock) coupons instead of solving math problems - not as interesting but more interest."
Those clarifying questions and some of the thinking through consequences are the only really topical part of SE interviews, the rest is just math you won't use on the job 99.99% of the time anyway.
I have not found the type of person who asks trick questions to be the type of person who finds it interesting to have the trick questions they've posed to be prodded.
Completely tangential, but something I enjoyed reading that feels in the same realm: https://blog.plover.com/math/logic/annoying-boxes-solution.h...
I had to reread that a few times to figure out what he was saying. All that comes down to is the fact that in his presentation technically there's nothing linking the propositional value of the box labels to the box contents. In most puzzles this linkage specified "outside the puzzle world" but in this case it's specified "inside the puzzle world" and so nothing can be deduced from it. But any sane person would assume the two align (especially in the setting of a puzzle), and so there's the gotcha.
Seems very different from the kind of "trick" questions in interview which are closer to one-way questions where the problem is trivial with some key insight but quite hard otherwise.
I disagree, and when I first encountered it it seemed pretty obvious to me, but maybe I’m just used to question where the answer can be “not enough information”
If we assume that the label on the red box must be either true or false then we can prove that the treasure is in the red box. We’d be wrong though, since the treasure is in the green box.
> So if you said the treasure must be in the red box, you were simply mistaken. If you had a logical argument why the treasure had to be in the red box, your argument was fallacious, and you should pause and try to figure out what was wrong with it.
He doesn't really elaborate on this, because he doesn't know the answer:
> Here are some responses people commonly have when I tell them that argument A is fallacious:
> "If the treasure is in the green box, the red label is inconsistent."
> It could be. Nothing in the puzzle statement ruled this out. But actually it's not inconsistent, it's just irrelevant.
This is an unfortunate point in an otherwise good essay. The problem in the puzzle is precisely that the red label is inconsistent, in the ordinary sense that no matter what you assume about it, a contradiction will result. Its truth implies its falsity, and its falsity implies its truth. Holding the location of the treasure fixed, no Boolean model exists in which the red label has a truth value at all.
The puzzle is an example of the Cretan paradox; there's not much more to it than that. It catches more interest because it's presented as if it were a different kind of puzzle than it is.
Why do you believe the content of the labels says anything about where the treasure actually is?
> You wake up in front of a box which has a label that says “treasure inside”. Should you assume there is treasure inside?
to which the answer is clearly “maaaaybe?” and
> You wake up in a world with accurately labelled boxes, in front of a box which has a label that says “treasure inside”. Should you assume there is treasure inside?
Where the answer is yes, and if the person setting the problem says “hahah no!!!” you can say “well look, that wasn’t a fair puzzle”.
It is primarily a “do you have sufficient information” problem, like in the GMAT, with a level of misdirection thrown in.
I find it depends entirely on whether the person is asking a trick question to try prove themselves smart (and are sensitive about it), or as in this case, are confident in their own intelligence, and want to assess yours.
The random selection omission is intentional.
This trick actually makes the problem easier, you always need 7 tries, and the payoff, and not just the expected value is $940. Of course, if these are not whole numbers, the payoff is $0.
Now you can continue the interview whether you would cheat if you are guaranteed not to get caught, as it is the assumption you made for Ballmer here ;)
Do the payout between 0 and 1 as the percentage of the amount.
With a range of 1 to 1 to pay off is obviously one
With a range of 1 to 2 the payout is .5
At three values it becomes more interesting. There are two strategies for the candidate either a binary search for the endpoints.
At four values you still have one level of binary search possible but after that it devolves down to the two value problem.
At five values. If the interviewer thinks the candidate would choose binary search and it becomes too too value problems on each side after removing the middle element.
There's definite problems with this but I wonder if he's already possible pay off matrix
Assuming you've found such a strategy for N, when you go to N+1 you still need to find the (N+1) element vector representing the probability that you select each number as your first guess, and you likewise need to know your opponent's probability vector for adversarially choosing a number. Once you guess at those vectors you can use your recursively built up DP sub-solutions to get the value of the game, but you are still stick with solving the optimization problem of finding those mixed strategy vectors for N+1, and will probably need something like CFRM or a similar technique to find them.