Racetrack

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Example route with the first nine moves by two players, start at the bottom counterclockwise

Racetrack is a pencil and paper strategy game of unknown origin for two or more players. It is also known under names such as car racing , vector formula , vector races , box races , Vector Rally or Vector Race , Graph Racers , PolyRace , paper-and-pencil race or graph paper racing game . Racetrack is played on a squared sheet of paper (5 mm grid). The game simulates a car race . Since the cars have a certain inertia , z. B. be braked before a dangerous curve. For a successful game, the game therefore requires foresight and planning.

Base game

Here the rules are explained in simple sentences. In a later section, when the mathematical concept of the vector is known, some of the rules can be shortened.

route

A closed line is drawn freehand on a squared sheet of paper as the outer boundary of the racetrack. A large ellipse is enough to get you started, but some irregularities make the game more interesting. Another closed line is drawn freehand within the first. It can run more or less parallel to the outer line, or the route can have wider or narrower areas, with usually at least two boxes between the lines. A straight line is drawn somewhere between the two lines. This is the start and finish line. Choose a direction of travel for the race, e.g. B. counterclockwise.

Game preparation

The order of the players is decided. Everyone chooses a color or marker (like x or o) that represents the player's car. Each player draws a starting point for his or her car - a grid crossing point on or behind the starting line.

Trains

It is said that every grid point has eight neighbors , which means the eight grid points that are reached by moving one box up or down and / or one box to the left or right. Each move leads from the current grid point (starting point) to another grid point (end point). The move is entered by drawing a route from the starting point to the end point in the respective player's color. The new point receives the marker chosen by the player (e.g. x or o). The players take turns drawing.

The following rules apply when drawing:

  • The first turn of each player must lead to one of the eight neighbors of the starting point. The player also has the option to stop at his starting point.
  • In every further move the so-called main point for this move is determined. The main point is obtained by repeating the previous move, both horizontally and vertically. If the player last dragged two boxes to the right and four boxes up, the main point is now two boxes to the right and four above the current starting point. The player now has the option to move directly to the main point or to one of his eight neighbors.
  • The cars must stay within the track boundaries. This applies to the starting point, the end point and the entire route that connects the two of each train
End of game and winner

The winner is the first player to complete a round (cross the finish line). The game is then over.

Further and alternative rules

By combining the following rules in various ways, countless variants of the game are created.

  • route
    • The route does not have to be a closed course; The start and finish lines may be different.
    • Before the start of the game, the players check the route and decide ahead of time whether a point near the edge of the route is inside or outside the route.
    • Alternatively, the route may only consist of straight lines that contain only 90 ° or 45 ° curves precisely on grid points. Decisions about unclear route points become superfluous. Players can be allowed or prohibited from touching the edges but not crossing them.
  • Trains
    • Instead of allowing moves to each of the eight neighbors of the main point (which can be reached on a king move in western chess ), one can use the four-neighbor rule and only allow moves to the main point and its four nearest neighbors (those on a general move in Chinese Chess ).
    • When drawing the route, slippery areas are drawn in as an oil slick, where the cars cannot change their speed at all or only according to the four-neighbor rule . This rule is used for trains that are e.g. As in the slippery region begin .
    • One possibility to compensate for the disadvantage of the last person to draw is to give each player the opportunity to skip one or a limited number of moves and make up for them later. If the move is made up, the player moves forward in the turn order. This makes the race fairer, since the likelihood of a disadvantage due to the sequence of moves in the course of the race is distributed among all players, and it has a tactical instrument comparable to a "standing attempt".
  • Collisions and accidents
    • The cars can be allowed to occupy the same point at the same time. However, the most commonly used and most fun rule is that while the routes are allowed to overlap, a car cannot drag onto or cross a grid point when it is occupied by another car as if the cars were colliding.
    • A rule can be introduced that requires players to try to avoid collisions. However, such a rule requires a certain interpretation.
    • Another option is to somehow punish collisions, but not completely prohibit them:
      • In the case of collisions, it can be defined that the driver with the higher speed is “punished” more or differently than the slower one; the faster one could have to start again from zero, while the slower one may continue to drive at the current speed.
      • The slower driver could take over the speed of the faster one and the faster one has to keep going at the speed of the slower one.
      • The one coming from behind “transfers” its speed to the one in front; the latter may (or must - be careful of curves!) drive faster, whereas the rear has to start again from zero.
  • Leaving the route
    • A player who leaves the track can be allowed to continue in various ways, provided that he does not have to be eliminated or loses the opportunity to win:
      • The car must be braked to zero (skip 1 lap and then start again at the edge of the track) or re-enter the track before leaving the point and start at zero there.
      • The car is not braked, but has to re-enter the track and cross the edge of the track at a point that is immediately in front of the point at which it left the track (usually this requires an arc of several trains).
      • As a variant, “entrances” can be designated in advance along the route, and the route can only be re-entered via these entrances, so the driver must return to the next entrance.
      • Furthermore, only walking pace would be permitted outside the course (per train not faster than to a neighboring point), or the rule for changing the speed would be retained (which could require a large arc to enter).
      • More physically oriented is the variant of restricting the train options after leaving the route in such a way that the car has to slow down each speed component to a maximum of 1 and damage is proportional to the length of the “slide out”. Too much damage is the end.
    • Some rule sets allow the leg of a train to cross the edge twice if the start and end point are within the course. However, this could allow unwanted shortcuts on very intricate racetracks.

At high speed, penalties require a significant number of moves; other forms of punishment can be considered - there are no limits to the range of rules.

  • Determine the winner
    • At the end of the game, the current round can be completed. So if z. B. of the three players A, B and C (who start in this order), B reaches the goal first, C is still allowed to make a move to complete a complete ABC cycle. The winner is then the player whose car drives the furthest over the finish line. Because - to stay with the above example - B can be the first to reach the goal, but C might make it even further across the finish line.
    • When the above rule of collision is applied, there is no small advantage in pulling first. This can be partially offset if the players choose their starting point in reverse order. For example C chooses the starting point first, then B, last A. Then A makes the first move, followed by B and C.
    • Alternating orders are also possible
    • Another option is to let the loser go first in the next game.

mathematics and physics

Each move can be represented by a vector . For example, a move two boxes to the right and four boxes up can be represented by the vector (2,4). The eight-neighbor rule allows any coordinate of the vector to be changed by ± 1. If the previous move was (2,4), the next move may be one 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 of the game represents one second and each box represents a meter, the vector for a move means a speed in meters per second. The four-neighbor rule allows an acceleration of up to 1 meter per square second, the eight-neighbor rule an acceleration of up to √2 meters per square second. (If you agree instead to 10 meters per box, the size of the route and the acceleration will be more realistic.)

The speed created by the acceleration can only be reduced by the same rate. This constraint reflects the inertia or momentum of the cars. Note that in physics picking up speed, braking, or turning left or right are all forms of "acceleration" represented by a vector. It is not unrealistic for a sports car to allow the same maximum acceleration in all directions.

History and present

The game's origins are unknown, it certainly existed as early as the 1960s, and it is reported that it was invented by engineers. Eugen Oker writes in his book on pencil and paper games that it appeared in America " for the first time in a scientific journal" . If you consider the close connection to physics, that sounds believable. Nowadays the game is used by math and physics teachers all over the world to teach vectors and kinematics . However, the game has a certain charm of its own and can be played for pure pastime.

Web links

Commons : Racetrack  - collection of images, videos and audio files

Individual evidence

  1. ^ Eugen Oker: The most beautiful games with pencil and paper , Droemersche Verlagsanstalt, 1980, ISBN 3-426-07612-0 , page 98/99