Three cards in a row
Three face-down cards carry the consecutive integers , and in a random order, with unknown. You turn one card and may keep its value, or turn a second card. After seeing the second you may keep it, or turn the third and then you must keep that one. You are paid the value you keep.
Your strategy can only use the differences between the numbers you see, since is unknown. By how much does the best expected payoff exceed ?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The first card tells you nothing on its own. So you should always look at a second card.
The second card differs from the first by or . In each case, compare keeping it with the average of the remaining card.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The first card carries no information, so always turn a second one. The difference between the two cards tells you a lot, and the right decision in each of the four cases is easy.
Step 1: the four cases (values written relative to ).
- Second is 2 higher: it is the top card, . Keep it.
- Second is 2 lower: the third card is the middle one, . Turn it.
- Second is 1 higher: the cards seen are or , equally likely. Keeping gets on average; the third card is or , also on average. Keep.
- Second is 1 lower: the cards seen are or . Keeping gets on average; the third card is or , worth . Turn it.
Step 2: the six orders. Writing the cards as above :
- : second is 1 higher, keep 1.
- : second is 2 higher, keep 2.
- : second is 1 lower, turn: 2.
- : second is 1 higher, keep 2.
- : second is 2 lower, turn: 1.
- : second is 1 lower, turn: 0.
Step 3. The average is . So the best strategy earns .
Compare. Keeping the first card earns . Looking at the differences is worth a third on average.