A Craps Tutorial

Other Topics -- Gardner's Card Trick

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A Card Trick by Martin Gardner ( An Application of Base 3 Arithmetic )

Although this is a self-working trick it can be impressive if well executed.
The audience will think at first, "Ho-hum -- I've seen this trick before."
But the surprise is that the magician not only finds the secretly chosen
card but also forces that card to appear at a pre-determined position in the
packet of cards being used.

The major difficulty in performing this trick is that the magician has to know
how to convert an integer from base ten to base three.   Or he must have some
way of secretly consulting a table that shows the base 3 values of the integers
that range from 0 thru 26.

If You Need Help Converting Base 10 To Base 3

The Presentation

The magician asks his friend to pick a number between 1 and 27.
The friend selects any integer X in the range   1 <= X <= 27 and
tells the magician what number he picked.
The magician writes this number down on a piece of paper in case the
friend later forgets what number he picked.

The magician then removes any 27 cards from a deck and deals them out face up
into 3 columns of 9 cards each. ( The remaining 52 - 27 = 25 cards are never used. )
He always deals from the top of his packet of cards that he holds face down.
He starts by dealing 3 cards face up to the table, forming the 1st row.
Then he deals a 2nd row overlapping but not hiding the 1st row.
When he finishes dealing,   the faces of all 27 cards are visible to both the
magician and the friend.

The magacian asks the friend to mentally select a card and then asks him which
column the card is in.

The magician gathers up the cards one column at a time, being careful to not disturb
the order of the cards within any column. He then deals out the packet of cards
for a second time into 3 columns and asks the friend which column the card is in.
The magician gathers the columns and deals out the cards for a 3rd time and asks
again which column holds the chosen card.

Finally the magacian gathers up the cards one column at a time into a packet and asks
the friend,   "Now, what was that number you selected ?"
( If the friend forgot then consult the scrap of paper the number was written onto. )

The magician then peels off the cards one at a time from the top of the packet,
counting out loud as he deals the cards face down to the table. When his count reaches
the chosen number he deals that card face up to the table, and it turns out to be the chosen card.

How The Magician Finds the Card

Each time the magician picks up the 3 columns of cards he must be careful to place the
column containing the chosen card either on top or in the middle or on the bottom relative
to the other two columns.
He uses this encoding scheme:
       0   means   top
       1   means   middle
       2   means   bottom
Let X be the number chosen by the friend.
Even though the magician doesn't really know what the selected card is, he can force it
to appear at a location X cards down from the top of the final packet as follows.

We want to have X - 1 cards above the selected card.   Convert X - 1 to base 3 and
express the result as a 3-digit number. Then read this number "backwards" from
right to left.

If, for example,   X = 6,   then X-1 = 5 = (012)₃   and reading backwards we get   2-1-0.
This would mean that the pickup order must be   "bottom then middle then top".

Example

Instead of using cards from an ordinary deck,   let's assume we have cards that
are simply numbered from 1 thru 27.
Let N = the secretly chosen card = 20
Let X = the chosen number we write down = 12
Then X - 1 = 11 = ( 102)₃
So the pickup order is 2-0-1   meaning   "bottom then top then middle".

In this example the magician must remember   "BOTTOM   then   TOP   then   MIDDLE"

For simplicity,   let's assume that the cards are initially sorted in numerical order 1 thru 27.
The secretly chosen card ( N = 20 ) is shown in red.
The magician will force this ( unknown to him ) card into position X = 12.

initial deal out

There are two ways of putting the column containing N = 20
at the bottom of the packet formed by gathering up the 3
columns of cards.
One way is to first pick up the leftmost column then the
rightmost column and then the middle column.
Let's do that and then deal out the cards a second time.

2nd deal out

Now there are two ways of putting the column containing N = 20
at the top of the packet formed by gathering up the 3 columns of cards.
One way is to first pick up the leftmost column then the middle
column and then the rightmost column.
Let's do that and then deal out the cards a 3rd time.

3rd deal out

Finally,   there are two ways of putting the column containing N = 20
at the middle of the packet formed by gathering up the 3 columns of cards.

Let's first pick up the leftmost column then the rightmost column and then
the middle column.

packet arrangement after gathering in the 3rd deal out

Thus the top 12 packet cards in order are:

Notice that the chosen card   N = 20   is at location   X = 12.

Last Update: 24 May 2006