(by Colm Mulcahy)
Projective Geometry (The Fano Plane)
"I'm thinking of two different simple geometric shapes, one inside the other,"
you say to a volunteer from the audience. "I want you do to the same thing.
Have you thought of something? Please sketch what you imagined, I won't look
yet. Earlier, I made a sketch of my own thoughts, I'll show it to you in a
"Are you finished? I really did want you to do the same thing as me: I tried to
project my geometric image to you. We call this projective geometry. Let's see
if it worked."
Before the volunteer reveals his sketch, ask other audience members what shapes
they had imagined. Several will say a circle inside a triangle, which is what
you hope the volunteer has sketched. With surprisingly high probability the
volunteer will indeed have opted for that combination; this curious fact is
fairly well-known, e.g., see Banachek's Psychological Subtleties (Magic
Inspirations, 1998). If this is the combination drawn, turn over a piece of
paper on which you've also sketched such an image, to the amazement of all.
Should the volunteer name a triangle inside a circle instead, you can still claim
victory. Say, "It's fascinating, you and I thought of the same shapes, but with
their roles reversed. I think we have complementary personalities. Such duality
is very common in projective geometry." If your luck is out, and the volunteer
has the bad taste to mention a square or something equally vulgar, you can
always say, "It seems that I was projecting in the wrong direction," indicating
somebody who had earlier named the desired circle inside a triangle.
In any case, draw attention to your picture, and continue, "This shape plays an
interesting role in a randomization procedure I'd like you to try with me."
Re-draw the image on a larger piece of paper or cardboard. This time, inscribe
the circle in an equilateral triangle, with seven points and lines marked as
The image needs to be large enough to accommodate the placement of a card at each
relevant intersection point. (Below, these points are marked with empty circles
to help the reader to imagine the possibilities in what comes later.)
Remark that the image depicts the Fano Plane, a delightful mathematical entity
representing the smallest finite geometry, of which you are a big fan.
Produce two decks of cards, one blue-backed, one red-backed, and explain that the
volunteer is going to determine some random placement of blue-backed cards at the
points of the figure, which will in turn determine a particular card from the
The blue-backed deck is shuffled, and eight cards from it are dealt out into a
face-down pile. The volunteer is asked to make choices to determine which four
cards he gets, and which four you get. Each of you pickk up and looks at your
cards, and arranges them in order (by value) in a fan. Whichever of you has the
value card sets that aside face-down. This requires comparing high values;
one way to pull this off without revealing much is for each of you to hold up
your highest card, while turning away, and have the audience indicate which of
you has the higher one. You can joke that this card represents the "point at
infinity" to a geometer.
The person with the second highest card sets it face-down at the apex of the
bounding triangle in the Fano plane. The other person then places any one of
the remaining cards in hand at the vertex in the centre of the diagram. This
card is also face-down, like all subsequent placements. You each take turns
placing cards at the other five points in the figure, until all eight cards have
been used up.
Next, the volunteer is asked to push to one side the three cards forming any
side of the triangle, after which you push together the remaining four cards.
Say, "Which would you like, the three you have indicated, or the four I have
here? It's your choice."
No matter which groups of cards is selected, gather up the remaining cards and
shuffle them back into the blue-backed deck, which is now removed from the scene
to avoid the possibility of confusion later.
Pick up the red-backed deck for the first time, shuffling it openly, and have the
volunteer sum the values of the three or four selected blue-backed cards along
with that of the ``point at infinity'' card which was set aside earlier.
Turn away and ask that the volunteer look at and remember the red-backed card in
the position corresponding to the total just calculated. For instance, you can
explain, if he got a total of fifteen, he would look at the fifteenth red-backed
card. Have the deck squared up again and turn back.
Do an elimination deal until just one card remains. Stress that it would be a real
miracle if the one card left after elimination from a shuffled deck would be the
same card the volunteer had looked at earlier. Ask what red-backed card was
noted, and have the "last card standing" turned over. They match.
You can now turn over the piece of paper or cardboard to reveal on the other side
a written prediction to the same effect.
"Just as projected," you can add in closing. "When one uses the Fano Plane,
everything aligns perfectly."
Explain a Ton
Here is a breakdown of the above effect, performance-wise, along with an
explanation of why it works.
At the outset, the top eight cards of the blue-backed deck are a 3, 5, 6 and 7,
followed by an Ace, 2, 4 and 8. The suits are irrelevant, but it's a good idea
if they appear random. This stock of cards can be maintained at the top of the
deck throughout some convincing looking shuffling. When eight cards are dealt
out, this puts the Ace, 2, 4 and 8 on top of the resulting pile.
The choices given to the volunteer in selecting his four cards utilize the Bill
Simon Sixty-Four Principle, as considered in the
April 2006 Card Colm, and recently reviewed in the
December 2007 Card Colm. Surprisingly, but crucially, the volunteer
ends up with the power of two valued cards: Ace, 2, 4 and 8 (in some order).
Alternatively, if the power of two values originally alternate with the other
ones at the top of the deck, then the principle explained in "Martin Gardner's
Coins to Cards Effect" in the
2006 Card Colm can be used to ensure that the volunteer gets the
Announce that each of you must now arrange you cards in order in a fan, keeping
the faces close to your respective chests. There is a subtlety here: it's not
so important that your cards are in order--although it's a good idea to put the
7 on one end so that you appear above board a little later--but you want to know
which card is which in the volunteer's hand. Since you don't officially have
any idea what cards he has, this ordering should not raise any suspicions.
Each of you now picks out your highest valued card--this is where you need to
watch to see which end the volunteer picks his card from--and shows them to an
audience member who indicates which is the higher. That card, the 8, the
volunteer sets aside face-down. You, having the 7, place it face-down at the
apex of the bounding triangle.
Next, the volunteer places any of his cards in the centre of the figure. You
need to know which card that is, which is easy if his cards are indeed in order
in his fan and you saw which end he plucked the 8 from. Place a card below his
choice so that yours and his total 7, e.g., if he places the 4, you place the 3.
Invite him to fill in the middle of either the left or right side of the
bounding triangle, by placing a card there. Again, "match" his choice by secretly
using the complement in 7 principle. The other side of the triangle is completed
in the same way. Note that the volunteer has filled in the middle "V row" of the
figure, and you have filled in the bottom side accordingly.
Amazingly, no matter which side of the triangle the volunteer now selects, the
three values in question sum to 14, as do the four cards left behind.
In fact, up to left-right reflection, one of the following cases must arise.
As a result, adding that total to the value of the card set aside earlier (the
8) leads without fail to 22. The red-backed deck can be shuffled casually if
only the top third and bottom half are moved around, without jeapordizing the
outcome. Having the volunteer count down to and note the card corresponding to
the total arrived at ensures that the original 22nd card is the one remembered,
and this is the one whose name you have written in advance as a prediction.
The elimination deal with the red-backed deck works like this:
Have the volunteer deal the cards into two piles, A to your left, B to your right,
starting with pile A. Have pile B picked up, and again dealt into two piles, by
first adding to the existing pile A, while creating a new pile B beside it. Have
this repeated, always dealing to pile A first, until just one card remains.
As Stewart Judah published in 1936 (see the
December 2004 Card Colm for more information on this
deal), that final card is always the one which started in position 22.
If you wish, you can ask the volunteer which card was his highest valued one,
and when he indicates an 8, you can turn one sideways and say, "I guess I was
correct when I said it represented the point at infinity."
The possible arrangements of 1, 2, 3, 4, 5, 6, 7 above correspond to the three
essentially different ways that ants in these quantities can be distributed in
the seven spaces in the figure below, to that the total number of ants on either
side of any of the three lines shown is the same.
Indeed, this month's column was inspired by a puzzle which asked for such
arrangements ("Ant-ics" in Ivan Moscovich's wonderful Leonardo's Mirror
& Other Puzzles, Sterling, 2004).
While the projective geometry tie-in claimed in the suggested presentation
is a bit of a stretch, there is a duality between that last figure above and the
original Fano plane. The seven regions inside the above circle correspond to
the seven points in the first figure considered, and two of those regions are
adjacent iff the corresponding points in the first figure are connected by a
Furthermore, if the volunteer names a triangle inside a circle as his thought-of
combination, you could produce the last figure--saying "in mine I continued the
triangle sides till they met the circle"--and adapt the patter (and mathematics)
It goes without saying that an actual performance of every step as outlined
above is not for the faint of heart. Many shortcuts are possible, from
streamlining the "free choice of four card from eight" part to the total
elimination of the elimination deal (and the second deck). The latter option
suggests turning the trick into a book force: you can predict ahead of time
the first word on page 22 of a book of your choice.
Many thanks to Scott Hudson (of Direct Data Communication) for providing all of the images
used above, and to magician
Joe M. Turner for the Banachek reference.
Colm Mulcahy (firstname.lastname@example.org)
completed his PhD at Cornell in 1985, under
F.T.W. Rosenberg. He has been in the department of mathematics at Spelman
College since 1988, and writing Card Colms---the only MAA columns to
actively encourage lying on a regular basis---bi-monthly since October 2004. For
more on mathematical card tricks, including a guide to topics explored in previous
Card Colms, see
"Fan actions" is one of several interesting anagrams of "Fano antics."