A cool magic trick for you to try!
You will need an assistant and an ordinary pack of 52 playing cards, with
four suits §, ª,
¨, © and 13
differently ranked cards 2, 3, 4, 5, 6, 7, 8, 9,10, J, Q, K, A. The cards
in your pack are ordered by 2<3<4< 5<6<7<8<9<10<J<Q<K<A
and if two cards have the same rank then they are ordered by the suits §<ª<¨<©.
This ordering is important as your assistant will later pass you three cards
and you need to be able to decide which is the “smallest’, the ‘middle’
card and the ‘largest’. For example, given the three cards J§,
2ª, Aª, you
will know 2ª is the smallest, J§
is the middle and Aª is the largest, or given
A©, K§, Aª,
you will know K§ is the smallest, Aª
is the middle and A© is the largest. Another
important idea is: Take any 13 cards of different ranks and place them in
a circle as shown below. Now pick any two cards. Notice that you can always
start at one of the two cards (nominated the start cards) and move in a clockwise
direction at most six steps until you reach the second card. For instance,
if you pick a 2 and a J, then on the circle you can start at the J (nominated
the start card) and move five steps in a clockwise direction to reach the
2. Finally note that if you pick any five cards there will be two of the same
suit.
Now, to dazzle your maths teacher, let them select any five cards from the pack and hand them, unseen by you, to your assistant. Your assistant chooses two cards of the same suit, identifies the ‘start card’ and places the other ‘secret’ card in their pocket. Your task is to identify the secret card. They hand the start card to you, and you immediately know the suit of the secret card. Now they must use the remaining three cards to tell you how many steps you must take in a clockwise direction around the circle (given below) before you reach the rank of the secret card. Remember, these three cards can be classified as ‘smallest’, ‘middle’ and ‘largest’. So your assistant looks at the table below, and finds the ordering which corresponds to the number of steps you must move around the circle, and hands you the cards in that order.
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So if the 5 cards are A§, 4ª, J§, 4¨, 3©, the assistant takes the A§, J§ and notes that we can start at the J and move three steps in a clockwise direction around the circle to reach the A. Therefore the start card is the J§ (handed to you) and the A§ is the secret card (placed in their pocket). Now your assistant must tell you to move three steps further on than the J. So they hand you the remaining cards in the order, ‘middle’, ‘smallest’ then ‘largest’; that is 4ª, followed by 3©, followed by 4¨. Now you know the secret card is a club and after consulting the above table you know to move three steps around the circle in a clockwise direction from the J§ to the A§, which is the secret card.
