32 Normat 57:1, 32–33 (2009)

An Amazing Mathematical Trick with Cards

Martin Gardner

Windsor Gardens, Room 216

750 Canadian Trails Drive

Norman, Oklahoma 73072

USA

One of the ﬁnest of all mathematical magic tricks, not well known even to card

magicians, was invented by the American Howard Adams. It was ﬁrst published in

his now rare 1984 booklet with the strange title OICUFESP (Oh,I see you have

ESP).

I urge you to get a deck of cards and astonish yourself by following these simple

instructions. First prepare by removing from the deck a set of ﬁve pairs of ’mated’

cards. (A mate to a card is a card with the same value and color, i.e. the mate

of the Queen of Hearts is the Queen of Diamonds). Call the ﬁve cards ABCDE

and their mates abcde, and arrange them in order ABCDEabcde and place the

packet of cards face down on a table. Then you can cut the packet as many times

as you want. (Cutting a packet means to separate it by a cut and transposing the

two pieces, keeping the orientation of the cards.) Then split it in half (i.e. cutting

it into two pieces of ﬁve cards each) and place the halves face down on a table, side

by side. Then turn the ﬁve-card pile on the right face up.

You are now going to spell the words in the phrase ‘LAST TWO CARDS

MATCH’ in the following way. Pick up either pile and spell the letter L by moving

the top card of the pile to the bottom and put it back on the table without turning

it over. Likewise spell the second letter A of ’Last’ by once again picking a pile at

random and moving its top card to the bottom and putting it back keeping the

orientation of its cards. The spellings of S and T are done in the same way. After

the ﬁrst word is completed we have two piles, one face up, the other face down.

Now remove the top cards of each pile and set them aside, side by side, at some

vacant spot on the table. One will be face down the other face up. (You may for

convenience arrange it so that the card which is face up is on the right) Repeat

the process with the remaining piles of four by spelling the word ’T-W-O’ at each

stage picking up a random pile. After it is done, remove the two top cards and

put them side by side as bef ore, below the previously placed pair. Proceed by

spelling C-A-R-D-S and M-A-T-C-H with remaining piles of three and two cards

respectively. You will end up with two columns of cards, (face down on the left,

face up on the right), the lowermost pair being the two cards which are left in the

piles, namely the last two cards. Look at them and turn over the face-down card.

Surprise! It matches the other card! The prediction ’Last two cards match’ has

been mysteriously fulﬁlled!

Normat 1/2009 Martin Gardner 33

Is the trick over? Not by any means. There is a second, even greater climax.

Turn over the remaining four face-down cards in the column. Each of those will

match its mate!

Maybe you can ﬁgure out why the trick works. It is based on modular arithmetic.

Once you have ﬁgured out why, you may be ready to face the even bigger lingui-

stic challenge of devising a similar magic formula, in the language of your choice,

that will work for larger sets of cards, even for the entire deck!

Lösningar till uppgiften skall sändas in i sedvanlig ordning till Dag Jonsson, mate-

matiska institutionen, Uppsala Universitet, innan den 1 september 2009