Logic Puzzle #1
I've always been a fan of logic puzzles - ones of the "missionaries and cannibals" variety. When I was a kid, I used to do these for fun. (All you non-math types out there can shudder now!) So from time to time, I'll post one of my favorites. Here's one for today:
Suppose you're made two offers by two different people, A and B.
A's offer: You are to make a statement. If the statement is true, A promises to give you exactly $10. If the statement is false, A will give you some number (possibly zero) of dollars, either more or less than $10, but not exactly $10.
B's offer: You make a statement. Regardless of whether the statement is true or false, B promises to pay you more than $10.
Both A and B charge you $10 for their offers.
Which of these offers should you take, and what statement should you make?
Solution: Take A's offer. Your statement is: You will give me neither 10 dollars nor 1 million dollars. A is forced to give you the million, since:
- If he tries to give you $10, your statement becomes false and thus according to his offer he's not allowed to give you exactly $10.
- If he tries to give you any other amount besides $10 or the million, your statement becomes true and according to his offer he would have to give you $10!
posted by Elie @ 8/10/2005 02:54:00 PM -
8 Comments:
I say, take A's offer, and make this statement:
"You are going to give me less than $10."
I am rather tired, but if I have to pay each of them $10 to play I am going to go find C because I know that C is not such a cheapskate. ;)
Sarah: As one of my high school teachers liked to say, you're on the right track but the wrong train! I.e., this would even out the two offers but not make A's offer better.
Jack: OK; but the answer is the same even if offer is free!
And you've told me the solution. I didn't understand it then. And I'm sure that I don't remember it now! :-)
um, um, um, I have a headache...
Robert, I tried to warn the non-math people!
But just think how lucky you are - if you were a robot/computer on Star Trek, you'd be shooting off sparks and smoke by now, right? :-)
Mirty, your dad will like this one! I know mine would have.
But note I already added the solution, so you may want to cut and paste the question rather than sending the blog entry link.
Elie,
Nice puzzle. Thinking about the semantics of it, the solution depends on what might be considered a linguistic glitch.
When we say that "a statement is true" or "a statement is false", we typically assume the statement in question makes a factual claim about the past or present. "I have two arms" is true. "I ate a cardboard box for breakfast" is false.
However, a statement about the future is not, properly speaking, either true or false. It is either a promise or a prediction, and it can, after the fact, be revealed as having been a true promise or prediction. But at the time the statement is made, it is neither true nor false.
"I will give you $10" is a promise. "You will give me $10" is a prediction. The promise can be fulfilled or not; the prediction can be borne out or not. But as statements of fact, it is not proper to consider either "true" or "false" - certainly not at the time they are stated.
Incidentally, similar distinctions can be found in Hilchot Nedarim; the law distinguishes between an oath regarding the past and one regarding the future. Enough said.
Finally, regarding your proposed solution...
Let's say I'm A. You make the "winning" statement. Considering my unappealing options, I make the following statement:
"You have made a statement about my future behavior. Until I act, your statement cannot meaningfully be considered to be either true or false. (I might argue that this is so even after I act, but let me meet you halfway here.)
"I am carefully considering how to act. I may give you $10, or $1 million, or any other amount yet to be determined. Or I may give you nothing.
"However, until I give you something, I don't know whether your statement is true or false. So you'll just have to wait.
"Indefinitely."
Of course, had the promise stipulated a time limit, this solution might not work.
Finally finally: If I were A, I could simply give you $10, refunding your money and withdrawing the promise. No promise is binding under unanticipated circumstances!
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