A "friend" offers to play the following game: you throw a die, and he throws two dice. If both his dice are either higher or lower than yours, he wins; otherwise, you win.

First, you reason: out of three dice, one will always be the "middle" one, and only one out of three times it will be mine, so my odds are just 1/3 -- I shouldn't play.

After a while, you realize that you forgot about duplicate numbers. About 50% of the time, all three dice will be different, and then you have 1/3 chance of winning. But on the other 50%, you assuredly win, so the game stands 2/3 in your favor.

It's clear that BOTH lines of reasoning cannot be right, if any. Should you play, or shouldn't you?

Note: you can solve this mathemathically, or you can use "lateral thinking"; can you find both ways?

(In reply to easy math but laterally??? by David)

With the kind of "friend" that is alluded to in this problem, which I would take to be my own evil twin as a typical model, one can be sure that the game offered will favor that "friend." I don't know if this thinking qualifies as lateral, but it seems pretty sound to me.

Googling "lateral thinking" gives some interesting results, but I couldn't find anything very much like this problem.

Thanks, by the way, for confirming my "Counting the Ways" result with a different approach.

Posted by Richard on 2004-07-09 14:08:47