The surprise prize
You have a friend who is a logician (and he knows you're good at logic too). He presents you with ten boxes labelled 1 through to 10. He tells you that inside one of the boxes is a prize. You have to open the boxes one at a time in the order 1 to 10. He also tells you that he absolutely guarantees that you will be surprised at which the box the prize is in when you open it.
Knowing that your friend is completely trustworthy and logical, you reason that it can't be in box 10, because if you had opened box 9, then you'd know that the prize must be in box 10; but that wouldn't then be a surprise. You also know your friend will be aware of that also. You then deduce that it can't be in box 9 by similar reasoning. And so on. So it must be in box 1; but, of course, that wouldn't be a surprise either.
So which box is the prize in?
Knowing that your friend is completely trustworthy and logical, you reason that it can't be in box 10, because if you had opened box 9, then you'd know that the prize must be in box 10; but that wouldn't then be a surprise. You also know your friend will be aware of that also. You then deduce that it can't be in box 9 by similar reasoning. And so on. So it must be in box 1; but, of course, that wouldn't be a surprise either.
So which box is the prize in?
Labels: logic
posted by Chris at
6:51 AM
20 Comments:
It can be in any of the boxes, because now that you think its in none of them, it will be a surprise if you find it in any of them.
I agree shpilo.... But I'm just gonna say box 2....
The pre-announcement of the prize and the absolute guarantee of surprise are exclusive.
Now for some further analysis....
The logical deduction comes to the conclusion that no boxes could yield a surprise. One could then state that, it would then be a surprise to find them in any box, but if you are certain a prize exists then it wouldn't really be a surprise if you wound up at box 10 again.
So resorting to probability....
The probability that the prize is in each box is 1/(10-n+1) where n=1 to 10
e.g. for the first box there is a 1/(10-1+1)=1/10=10% chance the prize is in that box,for the 2nd box 1/9~=11.1%
Box1 10%
Box2 11%
Box3 13%
Box4 14%
Box5 17%
Box6 20%
Box7 25%
Box8 33%
Box9 50%
Box10 100%
So one could argue from a probability argument that, the prize could be in any box from 1-9 and be a surprise as the probability is less than certain, i.e. <100%.
One could then try to reapply the logical argument ad absurdum......
Cam
I think you all can congratulate yourselves. Despite the question insisting that it could be done, you've overridden it with your common sense(s).
This problem is a variation of the unexpected hanging (so-called) paradox. I found it in a Martin Gardner book whose title is based based on the paradox.
Follow the link to see more fully why you should be pleased with yourselves.
http://en.wikipedia.org/wiki/Unexpected_hanging_paradox
I believe that none of the boxes should come as a surprise. Whoever set it up really has no choice but to randomly select a box. I think box 1 or 2 would be the most surprising, simply because of the probabilities that Cam has provided. Put yet another way, I believe that the reasoning given in the question is simply fallacious.
HEEEELLPPP! I'm spiralling into a paradox. No matter which box the prize is in, it may or not be a surprise, or should that be the other way round? I'd be surprised if it was a decent prize though.
.
The out-of-the-box surprise prize is: The Spanish Inquisition. I bet you weren't expecting that.
it's not in any of the boxes. if im wrong, please tell me.
:P
You're right. The Spanish Inquisition isn't in any of the boxes.
I think one of the resolutions to the problem is that the term "surprise" isn't well defined.
I think Cam's approach of assigning probabilities to the boxes in the way that he has is very sound. I think I'd that my measure of "surprise" would be the converse of the probabilities that Cam gave. i.e my surprise factor would be 90% if it were in the first box. It would never be 100% surprise - that feels right to me.
I've come to the conclusion that the problem is over-rated. I doubt that I'd actually be surprised when I found the prize, no matter which box it was in.
If the prize is a cushion, with all the stuffing plumped up to one end, then the Spanish Inquisition is sure to follow.
Following the reasoning of Cam from both ends - ie the last box is no suprise, and the first box at the other end of the argument is no suprise, then it is the middle boxes - 5 or 6 that should be the most suprising.
A-HA - you didn't expect that....
Hey Karl Sharman, I did think of that... The thing is though, once you've gotten to box 5-6, it would be less of a surprise, because there's less anticipation from knowing that you would find it in the next couple of boxes...
Also with that reasoning, it would be less of a surprise... logic = box 5-6 = less surprise... so the answer is that there is no definitive answer... It's in the box the friend randomly chooses between 1-9...
haha, I just restated what I said, but expanded on the answer... silly me.... I'm feeling a bit redundant today...
The friend should have said "the box it's in will surprise you the most." The it would be in either box 1 or 2...
What if it was a really cruddy prize?
With my friend's, that wouldn't be a surprise for me.
All that stuff illustrates that you could come up with lots of expectations, so no result would be a surprise, because you'd have anticipated it.
The only surprise would be if it wasn't in any of the boxes. Doh!, I've expected that now, so it won't be a surprise when it happens.
That leaves the purple unicorn. Doh yet again!, that won't be a surprise now.
the question never states whether or not, once all boxes have been opened, if there will be more boxes, which would be a suprise in itself, making THAT the surprise. furthermore, it also fails to state that the prize is in more than one box, it may be a two part prize. lastly, if there were more than one prize, that too would be a surprize. now, stating this, some may say that it brakes the rules of the problem/question. i say, it was made by a human, is about two humans, and humans dont always follow rules. use this information as you may. thank you for reading.
I would be surprised if that was right.
Nah!, just kidding ;)
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