Racetrack (game)
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- This page is about the paper and pencil game. For the facility used to conduct races, see 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 Formula, Vector rally or Vector race, 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.
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[edit] 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.
The moves
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.
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 player also has the option of staying at the starting point. 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.
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.
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).
[edit] 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 staring 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 90 and 45 degree corners at grid points only. This removes the need to decide dubious points. Players may or may not 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 (that may be reached by a Western chess king's move), one may use the four neighbours rule, limiting moves to the principal point or any of its four nearest neighbours (that may be reached by a Chinese chess general's move).
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 (staring 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.
[edit] 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:
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- (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 in all directions is not unrealistic.
[edit] 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.
