June 2015

# Solution

Well, one thing is certain: Among the three cards he has pulled out there is an ace, a king, and a queen (this becomes apparent after reading the clues). Among the suits is a club, a spade and a heart (this, too, is apparent after reading the clues). That is some progress made. We have only to determine the suit of each rank. Sounds easy; but is it?

Let's consider the first clue: A king is to the right of an ace, but not necessarily next to it.

This clue gives rise to three possibilities: a) The ace is the left-most card, the queen is the card in the middle and the king is the right-most card, or b) The ace is the left-most card, the king is the card in the middle and the queen is the right-most card, or c) The ace is the middle card with the king immediately to the right of it and the queen is the left-most card. The ace cannot be the right-most card because in that situation the king cannot be to its right.

Let's consider each of these options separately:

a) The ace is the left-most card, the queen is the card in the middle and the king is the right-most card. Let me represent this as: A Q K. Clue 3 says: A queen is to the left of a heart, not necessarily next to it. The queen, in this situation, is to the left of the king and, therefore, the king has to be the king of hearts. Clue 4 tells me: A heart is to the left of a spade, not necessarily next to it. But here the king of hearts is the right-most card and, so, cannot be to the left of any card. This option, therefore, does not hold true.

Let's try option b: The ace is the left-most card, the king is the card in the middle and the queen is the right-most card. Let's represent this situation as: A K Q. Clue 3 says: A queen is to the left of a heart but not necessarily next to it. But in this situation the queen is the right-most card and is not to the left of any card. This option, too, does not hold.

That leaves us with option c: The ace is the middle card with the king immediately to the right of it and the queen is the left-most card. Let's represent this as: Q A K. Let's go back to Clue 3: A queen is to the left of a heart but not necessarily next to it. So, either the ace is of hearts or the king is of hearts. Clue 4 says: A heart is to the left of a spade but not necessarily next to it. This means that the king cannot be of hearts because it is not to the left of any card (there is no card to its right). So, (ace, hearts) is the only option.

That leaves us with two options each for the queen and the king: (queen, club) or (queen, spades) and (king, club) or (king, spades). Let's consider (queen, spades). Clue 2 says: A club is to the left of a spade. But the queen of spades is the left-most card and has no card to its left. Therefore this option cannot hold; the only choice left to explore is that the queen is the queen of clubs. So what we have is queen of clubs is the left-most card, ace of hearts is the middle card and the king is the right-most card and its suit has to be spades. In such a situation all the clues are satisfied: The king is to the right of an ace but not necessarily next to it, a club is to the left of a spade but not necessarily next to it, a queen is to the left of a heart but not necessarily next to it, and a heart is to the left of a spade, not necessarily next to it.

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in