Problem 173 Solution

1. To the right of the King there's a Queen or two.
2. To the left of a Queen there's a Queen or two.
3. To the left of a Heart there's a Spade or two.
4. To the right of a Spade there's a Spade or two.

From clues 1 and 2 we can infer that a king is on the left and a queen on the right and the middle card is either a queen or 2.

First consider the possibility that a 2 is in the middle. If there are two consecutive spades (from clue 4) then then three cards must be the king of spades, 2 of spades, queen of hearts. If to the right of a spade there is a 2 (again from rule 4) then the three cards are the kings of spades, a two of unknown suit, and the queen of hearts. So if we assume a 2 in the middle it is not certain what the suit of the 2 is.

Next consider the possibility that a queen is in the middle. Clue 4 tells us there must be two consecutive spades, since there is no 2. Both queens can not be spades since they are from the same deck, thus the left and center cards must be spades. Clue 3 tells us the right queen is a heart. So the three cards would be the king of spades, queen of spades, and queen of hearts.

So a 2 in the middle does not lead to a definitive answer, but a queen in the middle does. The question implies there is just one possible answer. Therefore there is a queen in the middle, leading to the final answer of king of spades, queen of spades, and queen of hearts.

Credit for this problem goes to Martin Gardner who published it in the November 1965 Scientific American.