Best Answer k-man, 16 July 2013 - 07:26 PM

What a great puzzle!!!

**If I pick envelope A**: I either **A1)** win nothing and lose nothing (if the computer predicted correctly) or **A2)** I win either $10 or $10,000

**If I pick envelope B**: I either **B1) **lose $1000 (if the computer predicted correctly) or **B2)** I win $1,000

**If I pick both envelopes**: I either **AB1)** lose $1010 (if the computer predicted correctly) or **AB2)** I win either $1,000 or $11,000

From the above, the safest bet for me is to pick A as I can't lose, but have a chance to win. However, since my uncle's supercomputer has such a great record of predicting my actions it definitely predicted that and put $0 in A.

If there is $0 in A then I should pick either B or A+B. It doesn't matter which - either way I should win $1000. Of course, the supercomputer predicted this next step in my reasoning too, so it didn't put $0 in A. This cyclical reasoning gets us nowhere fast, so let's try a different approach...

Let's say I decide to roll a die and depending on the outcome choose A, B or A+B with the equal probability. The computer could predict that move but couldn't predict the outcome of the roll, so it should use the expected winnings to make its decision.

If A contains $0, then my expected winnings would be $666.67,

if A contains $10, then my expected winnings would be $0.

If A contains $10,000, then my expected winnings would be $6,666.67

So, based on that expected move, the computer should put $10 in A. Since I only get to play this game once, I stand to either win $10, win $1000 or lose $1010 with equal probability.

Another approach I could consider is to decide whether I want to **maximize my winnings **with some chance of losing or **minimize my losses **with some chance of winning. In the first case, I should pick A+B. In the latter case, I should pick A.

__Conclusion:__

Ultimately, I think, it boils down to computer knowing my behavior and predicting my thinking patterns. Whichever of above methods I pick the computer must have predicted that and will make sure that my chances of winning are the lowest. So, if the computer cannot be fooled then the only safe bet is to pick envelope A and call it even.