This is a solution to problem 19 on the hard interview questions node. If you have not read the question, the following will make no sense to you:

You should take offer #1, because you can guarantee yourself an arbitrarily large amount of money, simply by using the following as your statement:

"I will receive neither $10 nor $1000."

If we assume the statement is true, this leads to a contradiction because you would receive $10, by the rules.

If we assume the statement is false, then to satisfy the falsity of the statement, you must receive either $10 or $1000. But since you cannot receive $10 if the statement is false, you must receive $1000.

Of course you can use any amount in place of "$1000."

I don't think that this "solution" really guarantees you that arbitrarily large sum of money that you said, because you are not making a solution possible. I think person giving the money out would refuse to give you any money in that case.

A better solution is to say "I will receive less than $1000 but not $10."

This way, if the statement is true, you should receive $10, but then the statement becomes false.

If the statement is false (which it will inevitably become), you must then receive a value other than $10 (according to the rules). However, giving you any value between $0 and $1000 (other than $10, which we already ruled out) would make the statement true again, so they are forced to give you some value above $1000 to avoid a contradiction.

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