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Projective Plane Games – Fire and Ice

By Gary Gordon and Liz McMahon

We are living in a golden age for games. In previous posts, we looked at the card game SET and the closely related game EvenQuads. These games share lots of attractive properties, and in those posts, we connected those properties to an underlying geometric structure. Our goal is to convince you that this close connection to (finite) geometry helps explain the appeal of these and other games – Games that Exist Because of Geometry.

Both SET and EvenQuads make use of Affine Geometry, which is a kind of geometry that includes parallel lines. Affine geometry should be familiar – it’s the usual Euclidean geometry of the plane. Euclid's Fifth Postulate (also called the Parallel Postulate) is valid in affine geometry:

Parallel Postulate: Given a line and a point not on that line, there is a unique line through the point parallel to the given line.

Is it possible that there are no parallel lines at all? This question led to the development of the field of projective geometry.

Projective geometry underlies the game we are considering in this post, Fire and Ice. Designed by Jens-Peter Schliemann, the game board and play use multiple copies of the Fano plane (shown in Fig. 1). The Fano plane is a finite geometry named for Gino Fano (1871-1952), an Italian pioneer of finite geometry. It appears throughout discrete mathematics, with connections to coding theory, design theory, topology, linear algebra, and much more. Bud Brown describes many such connections in his award-winning article The Many Names of (7,3,1).

Figure 1: The Fano Plane.

In the Fano plane, there are seven points, labeled A to G, and seven lines: ADB, BEC, AFC, AGE, BGF, CGD, and DEF – it's important to remember that lines don't have to be straight.  Here's the most important thing to notice in this drawing: there are no parallel lines!  Axiomatically, this collection of points and lines satisfies these properties:

A. Every pair of distinct points determines a unique line, and

B. Every pair of distinct lines meets at a unique point.

Additionally, there are at least four noncollinear points (this is often taken explicitly as an axiom to eliminate trivial cases).

The Fano plane is the smallest projective plane, a projective geometry of dimension 2. Each of its points is on 3 lines, each line has 3 points, and the total number of points (and lines) is 7. Since there are no parallel lines, if you color each of the 7 points either red or blue, then there must be (at least) one line with all 3 of its points the same color, and there cannot be a line with three points of the other color. You can think of this as a tic-tac-toe game where ties are not possible.

Fire and Ice

Figure 2: The game Fire and Ice.

The Fire and Ice game board consists of 7 Fano planes (the game calls them islands) arranged so that each island sits on a “point” in a larger Fano plane. Two players, Fire (red) and Ice (blue) alternate turns, placing markers in one of the 49 slots on the board. The goal is to control 3 islands that form a “line” in the larger Fano plane.

To control an island, you must have 3 of your pegs on a line in that island. As mentioned above, the absence of parallel lines ensures that only one person can control an island. And this same comment applies to the large Fano plane; the game cannot end in a draw.

Figure 3: In the middle of a game of Fire and Ice.

In the play of the game, there’s a wrinkle in how the pegs get placed. The game begins with a red (fire) peg in the center hole of the center island; Fire will then make the first move. A move comes in two steps:

  1. Choose one of your pegs on the board and move it either to a vacant spot on the same island or to the same (vacant) spot on a different island
  2. Place one of your opponent's pegs in the place that your peg vacated. 

This game play is rather unusual, because each move adds one of your opponent's pegs in a spot you once held.

This game appears to be out of print, but it’s not hard to make your own version of the game: At MoMath’s Equilibrium Game Night, a monthly event we host, we made our own boards on (rather large sheets of) paper by drawing a game board consisting of seven small Fano planes arranged in a large Fano plane, as shown in Fig. 4. We used dried beans for the pegs. So, make yourself a board, find a partner, and play a bit right now. Can you find any winning strategies?

Figure 4: The game board for Fire and Ice.

Projective Planes

Projective planes (which are 2-dimensional projective geometries) can trace some of their history to the introduction of perspective that artists discovered during the Renaissance: a scene looks much more realistic if you project your 3-dimensional scene onto a plane (the canvas).  Then train tracks seem to meet “at infinity” (meaning at the horizon), so for each family of parallel lines, an artist may add a point at infinity. This new point will be on each line in the parallel class. Moreover, the collection of all the points at infinity form a line, a “line at infinity”. 

Figure 5: A small affine plane.

We’ll demonstrate how this “adding points at infinity” process works to construct the Fano plane. We begin with the smallest affine plane (see Fig. 5). Then it’s easy to check that the three affine plane axioms hold: (1) Two points determine a unique line, (2) There are at least three non-collinear points, and (3) Given a line and a point not on the line, there is a unique line through the point parallel to the given line.  (Important: the place where the two diagonal lines intersect is not a point in this geometry.)

To construct the Fano plane, we need to add a point “at infinity” for each of the three classes of parallel lines in our little affine plane. There are three classes of parallel lines: the horizontal ones (AD and BC), the vertical ones (AB and CD), and the two diagonals (AC and BD: recall these do not intersect, so they must be parallel). To add the points at infinity, we extend the lines as necessary to force intersections. That means we'll put a point E where AB and CD intersect, a point F where AD and BC intersect, and a point G where AC and BD intersect, where “intersections” are understood to take place on extensions of the three 2-point lines if needed. Finally, we insist that the three new points E, F, and G are collinear. See Figure 6a.

Figure 6a: Adding line EFG “at infinity”
Figure 6b: Redrawing Fig. 6a to see the Fano plane

The picture in Figure 6a doesn't look quite right, but if we move the line EGF to the outside and straighten it, then make BGD into a circle, voilà! It's the Fano plane! (See Figure 6b.)  You may also notice that our little affine plane ABCD is still there, looking a lot like it did originally (squished a bit); the remaining points E, F, and G form the “line at infinity”. This picture has a lot of symmetry, and in fact, you could choose any of the seven lines and designate that line as the line at infinity, and you'll see that the four points that remain form a little affine plane.

This gives a big hint about how to prove what we claimed earlier: If you color each of the 7 points of the Fano plane red or blue, there must be at least one monochromatic line by the Pigeonhole Principle, but it’s not possible to have both a red line and a blue line. There cannot be lines of both colors because every pair of lines intersects—the point of intersection would need to be colored both colors.

Strategies for Playing

What kinds of strategies can you use? Must there be a winner in the game? Yes! We know that when all the pegs have been placed, each island will have an owner. This was the point of the argument we just gave: if you color each of the 7 points of the Fano plane red or blue, you are guaranteed a red line or a blue line, but not both. Now apply the same argument to the 7 islands. Since the islands are joined in the same way as the points in an individual Fano plane, exactly one player will control 3 islands “in a line”.

Can one player force a win? This question was answered in Mary J. Riegel’s PhD thesis at the University of Montana in 2012, Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe. Mary proved that the first player is guaranteed a win using a "strategy stealing" argument; unfortunately, her proof is an existence (rather than constructive) proof, and we don’t know of any algorithms that guarantee a win. It seems this game is ready for someone to take the initiative to find such a strategy.

The instructions for the game include some strategies. If you control an island, you needn't place any additional pegs on that island; those pegs can go somewhere else to try to claim more islands. However, once you control an island, you may not want to control that island forever. If you have control over three non-collinear islands, you may want to disrupt one of them in order to try to get an island that makes a line with two of the others.

Our final post in this series (unless we change our minds) will connect projective planes to the game Spot It! Stay tuned!


Liz McMahon enjoys working in algebra, combinatorics, and finite geometry, plus reading, cycling, hiking, and traveling.

Gary Gordon is a combinatorialist who ran Lafayette College's REU program for 11 years; he enjoys baseball, climbing things, and the piano.

Liz and Gary both retired from Lafayette College in 2022; their favorite joint collaboration is the book The Joy of SET, which they coauthored with their two daughters Rebecca Gordon and Hannah Gordon.