Racetrack (game)

Racetrack is a paper and pencil game of unknown origins, played by two or more players. It is also known under names such as Vector formula, Vector rally, Vector race, Graph racers, PolyRace, Paper and pencil racing, or the Graph paper race game. "Racetrack" is played on a squared sheet of paper ("quad pad", e.g. Letter preprinted with a 1/4" square grid, or A4 with a 5 mm square grid). The game simulates a car race. The rules for moving represent a car with a certain inertia and physical limits on traction. Hence, the cars will move on pleasing "racing lines" around the track, very reminiscent of how real racing cars move. As one must e.g. slow down before a dangerous bend in the track, the game requires some foresight and planning for successful play. The game is popular as an educational tool teaching vectors.

The basic game

The rules here are explained in simple terms. As will follow from a later section, if the mathematical concept of vectors is known, some of the rules may be stated more briefly. The rules may also be stated in terms of the physical concepts velocity and acceleration.

The track

On a squared sheet of paper, a freehand loop is drawn as the outer boundary of the racetrack. A large ellipse will do for a first game, but some irregularities make the game more interesting. Another freehand loop is drawn inside the first. It can be more or less parallel with the outer loop, or the track can have wider and narrower spots (pinch spots), with usually at least two squares between the loops. A straight line is drawn anywhere across the two loops. This is the starting and finishing line. Choose a direction for the race to be run, e.g., counter clockwise.

Preparing to play

The order of players is agreed upon. Each player chooses a color or mark (such as x and o) to represent the player's car. Each player marks a starting point for his or her car - a grid intersection at or behind the starting line.

The moves

All moves will be from one grid point to another grid point. Each grid point has eight neighbouring grid points: Up, down, left, right and the four diagonal directions. Players take turns to move their cars according to some simple rules. Each move is marked by drawing a line from the starting point to the new point, using whatever colour or mark that player has chosen.

* Each player's first move must be to one of the eight neighbours of their starting position. (The player can also choose to stand still.)
* On each turn after that, the player can choose to move the same number of squares in the same direction as on the previous turn; the grid point reached by this move is called the "principal point". The player also has the choice of any of the eight neighbours of the principal point.
* Cars must stay within the boundaries of the racetrack. This applies to the entire length of the move, not just the start and end. If a player is unable to move according to the rules, the player has crashed (see below).

Hence, if the player's previous move e.g. was two squares to the left and four squares upwards, then the next move will take the car either two more squares to the left and four upwards from where is was at the start of the move, or to any of the eight neighbours of that grid position.

Finding a winner

The winner is the first player to complete a lap (cross the finish line).

Since this game is turn based, it means that even though the first player crosses the finish line initially, he is actually not a winner if one of the following players finish in his or hers equivalent turn.

The winner must then be found otherwise. A common way of doing so is by saying that the player who crosses the finish line with the most squares is the winner.

For example:Player 1 finishes in his 20th turn of the race, and thus he crosses the finish line first. Player 2, however, has only played 19 turns, and if he or she manages to cross the finish line within the 20th turn, it could be considered a tie.

Additional and alternative rules

Combining the following rules in various ways, there are many variants of the game.

The track

The track needs not be a closed curve; the starting and finishing lines could be different.

Before starting to play, the players may go over the track, agreeing in advance about each grid point near the boundaries as to whether that point is inside or outside the track.

Alternatively, the track may be drawn with straight lines only, with corners at grid points only. This removes the need to decide dubious points. Players may be allowed to touch the walls, but not to cross them.

The moves

Instead of allowing moves to any of eight neighbours of the principal point, one may use the four neighbours rule, limiting moves to the principal point or any of its "four" nearest neighbours.

When drawing the track, slippery regions with oil spill may be marked, wherein the cars cannot change velocity at all, or only according to the four neighbours rule. The rule may e.g. apply to all moves "beginning" in the slippery region.

Collisions and crashes

Cars may be allowed to occupy the same point simultaneously. However, the most common and entertaining rule is that the line segments are allowed to intersect, but that a car cannot move to or through a grid point that is occupied by another car, as they would collide.

One may have a rule requiring players to try to avoid collisions, but such a rule requires some interpretation. Another possibility is to penalize collisions in some way, but not disallowing them entirely.

A player running off the track may be allowed to continue in the following way: The car must be braked down and turned around, and then it must enter the track again, crossing the boundary at a point "behind" the point where it left it. At high speeds, this will take a considerable number of moves; other forms of penalty may be considered.

Some sets of rules allow the line segment representing a move to cross the boundary twice, with the start and end points inside the track. However, with heavily convoluted racetracks, this may allow some unreasonable shortcuts.

Finding a winner

At the end of the game, one may complete a round. E.g., with three players A, B and C (starting on that order), if B is the first to cross the finish line, C is allowed one more move to complete an A-B-C cycle. The winner is the player whose car is the greatest distance beyond the finish line.

If the common collision rule mentioned above is used, there is still a considerable advantage in moving first. This may be partially counterbalanced by having the players choose their individual starting points in reverse order. E.g., first C chooses a start point, then B, then A. Then, A makes the first move, followed by B, then C.

Another possible rule is to let the loser move first in the next game.

Mathematics and physics

Each move may be represented by a vector. E.g., a move two squares to the right and four up may be represented by the vector (2,4).

The eight neighbour rule allows changing each coordinate of the vector by ±1. E.g., if the previous move was (2,4), the next one may be any of the following nine:::(1,5) (2,5) (3,5)::(1,4) "(2,4)" (3,4)::(1,3) (2,3) (3,3)

If each round represents 1 second and each square represents 1 metre, the vector representing each move is a velocity vector in metres per second. The four neighbour rule allows accelerations up to 1 metre per second squared, and the eight neighbours rule allows accelerations up to √2 metres per second squared. (If each square represents 10 metre instead, the size of the track and the maximum acceleration will be more realistic.)

The speed built up by acceleration can only be reduced at the same rate. This restriction reflects the inertia or momentum of the car. Note that in physics, speeding, braking, and turning right or left all are forms of "acceleration", represented by one vector. For a sports car, having the same maximum acceleration without loss of traction in all directions is not unrealistic; see Circle of forces. Note, however, that the circle of forces strictly applies to an individual tyre rather than an entire vehicle, that a slightly elongated ellipsis would be more realistic than a circle, and that the theory of traction involving this circle or ellipsis is quite simplified.

History and contemporary use

The origins of the game are unknown, but it certainly existed in the 1960s, and it is reported to have been invented by engineers. Considering the close links to physics, this is quite plausible. Today, the game is used by math and physics teachers around the world when teaching vectors and kinematics. However, the game has a certain charm of its own, and may be played as a pure recreation.

Related game: Triplanetary

Triplanetary was a science fiction rocket ship racing game [http://www.sjgames.com/triplan/] that was sold commercially between 1973 and 1981. It used similar rules to Racetrack but on a hexagonal grid and with the spaceships being placed in the center of the grid cells rather than at the vertices. The game used a laminated board which could be written on with a grease pencil.

In Triplanetary, the spaceship moved the same amount as the previous turn - and could be accelerated to one of the neighbouring hexagons by firing the engines and using up a unit of fuel (which was in very limited supply). There were planets marked on the map - each had an arrow facing towards the planet marked in each of the surrounding hexagons that forced the ship to move one additional step in the indicated direction in order to simulate gravity. Remarkably, one could use these simple rules to produce stable orbits around planets - or to 'slingshot' around them to change direction and speed without consuming any fuel. Several scenarios were introduced in the game - one of which was a race requiring each player to visit every planet in the solar system and then return to earth.

External links

* [http://www.sjbaker.org/paper_and_pencil_games/graph_racers/ Graph racers]
* [http://ideaexplore.net/racetrack.pdf Racetrack]
* [http://www.boardgamegeek.com/image/114849 "Racetrack, as it was played in the 1960s."]
* [http://www.vectorracer.net Vector Racer (aka Racetrack) online] (needs Java)
* [http://home.mnet-online.de/flome/index.html#PolyRace PolyRace] Program for Windows to play online and locally
* [http://www.karopapier.de karopapier.de] Online game to race against others (German only, translations available)