Suppose I present you with two personal checks made out to you and offer to gift you with whichever one of them you choose.

I place the checks face down before you, labeled A and B on their backs, and stipulate that their values are 2ⁿ and 2ⁿ⁺¹ dollars for some integer value of n.

I invite you to turn one of them face up and then choose one of them to keep as a gift from me.

Suppose that the check you turn face up has value K. Then the other check is worth either 2K or K/2 and therefore by choosing the face-down check instead of the one you've revealed, you stand to gain K dollars at the risk of losing only K/2. Thus it appears that whatever the value of the check you turn over, it is to your advantage to choose the other.

Suppose now that the game is changed and you must make your choice without seeing the value of either check. Say you choose check A. Whatever its value, by the reasoning above it's to your advantage to change your mind and choose check B. But then, of course, you should change back to check A, and so forth -- but this is surely absurd.What's going on here? Why should the act of revealing the value of one check make it advisable to choose the other?

—BK