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.
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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.
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.
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.
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.
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 it was at the start of the move, or to any of the eight neighbours of that grid position.
The winner is the first player to complete a lap (cross the finish line).
Combining the following rules in various ways, there are many variants of the game.
The track
The track need 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.
On track may be also some turbos with number and arrow in one of eight directions. When vehicle going through this area, his principal point is moved by number of squares wrote in area of that turbo in direction of arrow.
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 disallow them entirely.
A player running off the track may be allowed to continue in the following way: The car must brake and turn around, and then enter the track again crossing the boundary at a point behind that where it left. At high speeds this will take a considerable number of moves.
Another possibility is to add "damage points" to any car running off the track, 1 for each square of the last movement; the car brakes but can re-enter the track anywhere; when, say, 5 damage points are reached, the car cannot run anymore.
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 the 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.
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:
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.
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.
Triplanetary was a science fiction rocket ship racing game [1] 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.