### Brain Teasers

# Four Queens, Four Kings, and One Ace

Twenty-one cards were dealt to Queenie, Rachel, and Sara. These cards were four nines, four tens, four jacks, four queens, four kings, and one ace. The nines, tens, and jacks were discarded. Now the combined hands of the three women only had nine cards: four queens, four kings, and one ace. Here are some facts:

1. Queenie had two cards, Rachel had three cards, and Sara had four cards.

2. The woman with the most singletons did not have the ace.

3. No one had more than two kings.

Who had the ace?

1. Queenie had two cards, Rachel had three cards, and Sara had four cards.

2. The woman with the most singletons did not have the ace.

3. No one had more than two kings.

Who had the ace?

### Hint

A card is a singleton if it is the only card of its rank (ace, two, etc.) in its hand.### Answer

Rachel had the ace.If the woman with the most singletons had three singletons, then they had to be the king, the queen, and the ace, contradicting fact 2. The ace itself would be a singleton, and therefore anyone who had it would have at least one singleton. This means the woman with the most singletons had to have more than one, which could only be two: a queen and a king.

If Rachel had one queen and one king, her third card had to be an ace, contradicting fact 2. If Sara had one queen and one king, her other two cards had to both be aces. Therefore, Queenie had one queen and one king.

Since Queenie had the most singletons, Rachel and Sara could each have only had one at most. If Sara had no singletons, then they had to be two queens and two kings, leaving Rachel with one of each rank, contradicting fact 2.

This means Sara must have one singleton and three other cards, all of the same rank. Fact 3 tells us that three of Sara's cards must be queens. If Sara's fourth card was the ace, then Rachel had three kings, contradicting fact 3. This means that Sara's fourth card was a king, leaving only Rachel with an ace.

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