# Card Trick 2

Here are the instructions to a second card trick. This is also mathematical. Try and explain how it works.

Volunteer selects any three cards and places them face down on the table.

Volunteer then shuffles the pack of cards and returns them to me.

You are going to find a card in this pack which depends on the three cards you have already chosen. I am going to try and predict what card that will be. I am not going to alter the order of the cards, I am just going to remove a card which points to the card you will find.

Check the fourth from bottom card and remove the card of the same numerical value and colour and place it face down on the table.

Volunteer turns over each of the three cards in turn and counts onto 15 for each one. Remove the counted out cards each time. (Jack =11 Queen =12 King =13)

The numbers of the three cards are added, and that number of cards is counted out, the last one being kept.

Turn over that card and the predicting card.

You may have found that this trick does not work if the fourth card from the bottom is the same numerical value and colour as one of the 3 chosen cards or as the first, second or third card from the bottom, a probability of 6/45. When the 'magician' looks at the cards and sees that this has happened the best thing is to carry on with the trick but first to say that the cards should be shuffled again and give some convincing reason!

SUGGESTED SOLUTION

The card which the volunteer keeps will always be the fourth
card from the bottom of the pack which has the same numerical value
and colour as the predicting card. This is because, whatever 3
cards are selected by the volunteer, with these 3 cards and the
predicting card, 4 cards are removed from the pack. Then 45 cards
are counted out, and this leaves the last 3 cards to make up 52
altogether. Suppose the 3 cards selected have values *x* ,
*y* and *z* then the number of cards counted out is
(15 - *x* ) + (15 - *y* ) + (15 - *z* ) +
*x* + *y* + *z* = 45.

Correct solutions were sent in by:

**Sarah** - Archbishop Sancroft High School