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Racetrack is a paper-and-pencil game where two or more players race around a track drawn on grid paper. The game uses vectors to represent each player's movement.

General rules:

1. The track has a start line which also works as the finish line. Every player starts in the same direction. The first player picks their spot on the starting line first, then the second player and so on. Whoever first crosses the finish line from the other side wins. Touching the finish line is not enough – you need to be able to cross over the line with a legal move. If multiple players cross the finish line with the same number of moves, the game is a draw.


2. Players take turns moving. The first move is one square forward from the start line. On following turns, the player can proceed the same number of squares in the same direction as their previous turn – this point is called the principal point. Alternatively, they can choose any of the 8 neighbouring points next to the principal point (see image below).


3. A player may stop completely if the car's current location is a neighbour to the principal point. In this case, the player simply passes their turn. On the next move, the new principal point is the car's current location.


4. Driving through a wall or another car is not allowed. Crossing an opponent's line is fine, but a player cannot drive through (or into) the exact point currently occupied by another car. If a player has no legal moves, they crash and lose the game.

An example of possible moves. The black arrow represents the player's previous turn. The dark blue point is the principal point; it and its eight neighbours are all legal choices for the next move.

_{Image credit: Nø, CC BY-SA 4.0}

Extra rules specific to this puzzle:

1. There are three players. The second player gets a handicap – instead of starting with a move of length one, they can start with a move to any of the eight neighbours of a possible starting point, as long as the move is otherwise legal. However, they can't start from the same spot on the start line as another player. Players 1 and 3 get no handicap and start normally.


2. Assume all players play optimally, with winning as their first priority and minimizing the number of moves needed to cross the finish line as the second.

Some examples:

Example 1. A trivial win for player 1 (yellow).

Example 2. A draw between players 1 and 2. Player 1 crosses the finish line first, but both take the same number of turns to cross the finish line so the game is a draw.

Example 3. A win for player 2 (red). There are multiple ways to win; two are shown here.

The puzzle:

Design a track where, with best play, player 3 wins.

OK, so here's my track design, or at least the functional part of it.

I call it The Candle of Chicken.

Unlike the examples above, I've marked the spots that are not walls, that is, there's only a very narrow path available from the start at the bottom.

The starting points are symmetrical, and there's only one choice to make: Either stop after the first move, or keep going. The track is designed so that there are zero choices to make after that. In the map above, I've marked the only possible moves with red dots.

The player that goes first is at a disadvantage: the second player will always choose to go, so the first player will unavoidably crash if he also chooses to go. On the other hand, if the first player "flinches" and stops, thus losing the game of chicken, it takes four extra moves to back up to the starting line and re-accelerate before he can make it through the first curve, so he's going to be hopelessly behind.

Now, this design only beats one opponent that started earlier, but that's no problem, we can just chain another one right after it; just make sure that the other path takes the same number of moves to reach the bottom of the second chicken candle.

After the second candle, car 3 is certainly in the lead (possibly even the only car in the running), so we'll probably add some sweet jumps to the remaining part of the lap so that he can celebrate.

Will this work?

Because

- Player 1 moves forward.

- Player 2 is forced to move forward.

- Player 3 moves forward.

- Player 1 and 2 pass their turn as they can't move.

- Player 3 moves to the right.

- And so on, trivially Player 3 will pass the finish line first without a draw.

@Bass had a very similar idea to mine. Funny how that works. I was going to wait and try to actually draw it, but I feel like it isn't worth the effort, now.

All three starting locations can be constrained such that each follows it's own path. Player 2's handicap is useless and because the corridor requires certain levels of "speed" they cannot afford to wait before starting. It would look something like this:




I know, my Paint skills are of legend. Anyway, each path does what it has to and they all meet at a common point eventually. The point is shared, but because of the player speed, the paths remain individual.





After this each path leads back around to the finish line. Why does this work? We have to construct the paths such that their lengths are different. Green is the longest, red in the middle, and blue the shortest. The caveat is that the distance each path takes to the intersection point is, for example, 10 moves for green, 11 moves for red, and 12 moves for blue. This means the player on the longest overall path gets to the intersection first (by one turn). If player 3 is on this path, both players 1 and 2 will die in a horrible crash because player 3 will be sitting at the intersection when they get there. Player 3, conversely, can take the shortest path without fear of being killed at the intersection. The only outcome then is: Player 1 takes the longest path to not die, player 2 takes the medium path to not die, and player 3 takes the shortest path and wins.