One day it is hoped that enough mathematicians will have worked on the problem to have finally, definitively answered Steve Ballmer’s interview question. The job will be shared between them.
The "golden abacus" instead of "golden handcuffs?"
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."
With these types of trick questions, it is always interesting what is an acceptable trick and what is not. The question did not specify whole numbers as it does not specify a random selection, but one is in bounds and the other not.
> it is always interesting what is an acceptable trick and what is not
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.
> 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.
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.
Not confident but overconfident in this case. Clearly nobody has ever told him it's a crappy question (unless you're interviewing for a math PhD or something), and wrong on top of that, and he didn't have enough self-awareness to double check. Or maybe somebody did tell him, but he didn't care to listen.
>I enjoyed reading that feels in the same realm [annoying boxes]
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 don’t see why anyone would assume a link between the labels and the contents. We see this in the real world all the time. You see a tin of delicious cookies, then open it to discover disappointing sewing supplies. The sign on the door says Open, but the owner just forgot to flip it and the store is actually closed. Some joker puts a Ferrari badge on their Kia.
I assume the real point of the puzzle (which is lost in the post) is to demonstrate how not all statements have a definite truth value.
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.
No, the point is that even if both labels had said "the treasure is in the red box" then it still could have been in the green box. There is no reason why the labels should be expected to say the truth, they're just labels.
> 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.
Well, that might be the point of the parable of the dagger that he links to. It can't be the point of Mark Dominus's puzzle, because Mark Dominus doesn't understand it:
> 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.
No, that is a red herring, as he explains. It's not about the labels at all, there's no reason to assume the writer of the labels even knew anything about the actual location of the treasure. They're not part of the puzzle rules.
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”
It is. Or at least it was for some of us. I didn't care if the candidate ever got the right answer. I cared about the thinking, the questions, the strategies, and the conversation.
And if some interpretations lead to trivial solutions, but one leads to a complex problem, it's likely that their intention is the latter. A kind of tacit communication
I hate that. It turns the problem into one of those lateral thinking puzzles we were told some basic information and then the answer winds up being something totally wildly different. It wasn't being very random in the end not being very productive
Are we describing interviews here, or the process of software engineering in practice? It could be either imo.
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.
Technically true but if real numbers are allowed you don't have a puzzle. Rather, you have a silly puzzle that consists in checking if the interviewee will say, "Waaaiiiit a second, is any number allowed? Because then you can never win!" and the interviewer saying, "Nicely done!". Nevertheless, the assumption should be stated.
And that's all hinged in the expectations of the intent of the question, if it is attention to details, the real number scenario is completely reasonable. If it is not, you can pick apart these types of puzzles and seem overly hard to work with.
It also doesn't mention if Ballmer is able to change the number during the game. If he is only "thinking" about a number without writing it down beforehand, nothing stops him from doing it, as long as it is consistent with all the statements he made.
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 ;)
Those problems allow follow-up questions to better define things. In fact, as initially stated, the problems are usually so vague as to require follow-ups.
Can't this be solved to some sort of DP way of solving the sub problem?
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
It's possible it could help but it wouldn't be lead to a typical clean DP problem, because you need the full mixed strategy vectors at each level. That requires
N real numbers that sum to 1, for each player.
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.
One day it is hoped that enough mathematicians will have worked on the problem to have finally, definitively answered Steve Ballmer’s interview question. The job will be shared between them.
The real irony being that they show up to work just to discover that it was a software programming position.
But they can't quit once they taste that sweet total comp.
The "golden abacus" instead of "golden handcuffs?"
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."
You could be apply for grants all day and doing uber at the weekends you sell out!
s/stock/bond/
Rumor has it that Ballmer resigned after asking this question to a candidate named Aphyr...
Just in time for the job to be replaced with AI.
Well, Terrence Tao is trying his very best to replace himself with an AI.
I think he’s trying to replace his grad students so that he can solve even more interesting problems.
Terence is fascinating in how different his thinking is even from most other mathematicians.
Which, admittedly, seems like a much harder problem.
A world in which AI churns out amazing proofs would be pretty radical though.
With these types of trick questions, it is always interesting what is an acceptable trick and what is not. The question did not specify whole numbers as it does not specify a random selection, but one is in bounds and the other not.
> it is always interesting what is an acceptable trick and what is not
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 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.
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.
Not confident but overconfident in this case. Clearly nobody has ever told him it's a crappy question (unless you're interviewing for a math PhD or something), and wrong on top of that, and he didn't have enough self-awareness to double check. Or maybe somebody did tell him, but he didn't care to listen.
Or maybe he's testing how you deal with that situation. It's a job interview, not a school exam.
>I enjoyed reading that feels in the same realm [annoying boxes]
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 don’t see why anyone would assume a link between the labels and the contents. We see this in the real world all the time. You see a tin of delicious cookies, then open it to discover disappointing sewing supplies. The sign on the door says Open, but the owner just forgot to flip it and the store is actually closed. Some joker puts a Ferrari badge on their Kia.
I assume the real point of the puzzle (which is lost in the post) is to demonstrate how not all statements have a definite truth value.
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.
No, the point is that even if both labels had said "the treasure is in the red box" then it still could have been in the green box. There is no reason why the labels should be expected to say the truth, they're just labels.
Quite. It’s the difference between:
> 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.
Well, that might be the point of the parable of the dagger that he links to. It can't be the point of Mark Dominus's puzzle, because Mark Dominus doesn't understand it:
> 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.
No, that is a red herring, as he explains. It's not about the labels at all, there's no reason to assume the writer of the labels even knew anything about the actual location of the treasure. They're not part of the puzzle rules.
That won't stop them from being inconsistent.
Or consistent, or whatever. It doesn't matter, they're not relevant to the puzzle.
Why do you believe the content of the labels says anything about where the treasure actually is?
> any sane person would assume
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”
I always felt that part of the interview process is the candidate asking clarifying questions as well as making and stating assumptions.
It is. Or at least it was for some of us. I didn't care if the candidate ever got the right answer. I cared about the thinking, the questions, the strategies, and the conversation.
And if some interpretations lead to trivial solutions, but one leads to a complex problem, it's likely that their intention is the latter. A kind of tacit communication
Actually, it may just as likely be that the interviewer is looking to see if you over complicate things. So _ask_.
I hate that. It turns the problem into one of those lateral thinking puzzles we were told some basic information and then the answer winds up being something totally wildly different. It wasn't being very random in the end not being very productive
Are we describing interviews here, or the process of software engineering in practice? It could be either imo.
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.
Technically true but if real numbers are allowed you don't have a puzzle. Rather, you have a silly puzzle that consists in checking if the interviewee will say, "Waaaiiiit a second, is any number allowed? Because then you can never win!" and the interviewer saying, "Nicely done!". Nevertheless, the assumption should be stated.
The random selection omission is intentional.
And that's all hinged in the expectations of the intent of the question, if it is attention to details, the real number scenario is completely reasonable. If it is not, you can pick apart these types of puzzles and seem overly hard to work with.
It also doesn't mention if Ballmer is able to change the number during the game. If he is only "thinking" about a number without writing it down beforehand, nothing stops him from doing it, as long as it is consistent with all the statements he made.
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 ;)
Those problems allow follow-up questions to better define things. In fact, as initially stated, the problems are usually so vague as to require follow-ups.
Previous comments yonder:
- https://news.ycombinator.com/item?id=41434637
- https://news.ycombinator.com/item?id=41463330
Can't this be solved to some sort of DP way of solving the sub problem?
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
It's possible it could help but it wouldn't be lead to a typical clean DP problem, because you need the full mixed strategy vectors at each level. That requires N real numbers that sum to 1, for each player.
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.