Alternative use: Race track

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 rally (in Scandinavia), or Graph racers, 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. As the cars have a certain inertia, one must e.g. slow down before a dangerous bent in the track. Thus, the game requires foresight and planning for successful play.

The basic game

The rules are here 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 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 (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., anticlockwise.

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.

Eight neighbours

All moves will be from one grid point to another grid point. Any grid point is said to have eight neighbours, being the eight grid points that can be reached by going one square up or down, and/or one square to the right or to the left.

The first round

Each player's first move must be one square up or down, and/or one square right or left. This will take the car to one of the eight neighbours of the starting point chosen by that player. The move is marked by drawing a line segment from the starting point to the new point, using the appropriate colour, and marking the new point by the appropriate mark.

Subsequent moves

Play proceeds with players taking turns and marking the track with their new positions. In both the horizontal and vertical direction, one permitted move is to repeat the previous move. E.g., if the player last time moved two squares to the right and four squares up, the player may do so again. The position resulting from this move is called the principal point. A move to any of the eight neighbours of that point is also allowed.

Crash rule

The cars must stay within the boundaries of the racetrack. In each move, this applies to the starting point, the end point, and the entire line segment connecting them.

Finding a winner

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

Additional rules and alternative rules

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

Collision

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

Reducing advantage of moving first

If the collision rule above is used, there is 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., with three players, 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 looser move first in the next game.

Modified rule for finding a winner

At the end of the game, one may complete a round. E.g., with the same three players as before, 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.

Modified crash rule

A player running off the track may be allowed to continue in the following way: The car must be breaked 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. Typically, this will take a considerable number of moves.

Some sets of rules for racetrack 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.

Four neighbours rule

Instead of alowing moves to any of eight neighbours of the principal point (that may be reached by a Western chess king's move), one may limit moves to the principal point or any of its four nearest neighbours (that may be reached by a Chinese chess general's move).

Deciding dubious points

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.

Oil spill

When drawing the track, slippery regions may be marked, wherein the cars cannot change velocity at all: All moves beginning in a slippery region must go to the principal point.

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 squareroot-of-two 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.

History and contemporary use

The origins of the game are unknown, but it certainly existed in 1980, 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.

External links

• Graph racers
• Paper and pencil racing
• On the speedway
• Racetrack
• Racetrack