Which offer is better?

Problem

1) You are to make a statement. If the statement is 

true,you get exactly $10. If the statement is false, 

you get either less than or more than $10 but not 

exactly $10.
2) You are to make a statement. Regardless of whether 

the statement is true or false, you get more than $10.

Answer: 1

Solution Approach:

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.”