# Racetrack designers, assemble!

Note: Turns out the intended answer to this doesn't work after all (see comments to Sconibulus's answer). The rules have been changed slightly after the fact to ensure there can be valid answers (see extra rule #1). Anyone who's tried to solve this, I apologize for the time wasted. I should have verified this properly before posting.

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, except those on the starting line,1 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.

1 (This part was added two weeks after the puzzle was posted.)

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.

• Just out of curiosity - have you solved the problem already? (Fine if you haven't, of course). – Brandon_J May 29 at 13:12
• @Brandon_J Yeah, I have one example prepared. I can share my solution (if it's different) once someone earns the checkmark. – jafe May 29 at 13:15
• Cool, thank you! – Brandon_J May 29 at 13:20
• Question about your second added rule: Suppose a player knows they can't win, but there are multiple paths that achieve the minimum number of moves. Are we allowed to cherry-pick from them or must we treat them all as possible? – Dark Thunder May 29 at 14:35
• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it. (I understand there was an issue with your intended solution actually not working; do any of the other solutions here work? If this is unsolvable, some of the commentary on Unanswered puzzle - leave, self-answer or write a postmortem? may be useful; also, this mega-answer, which I actually put together just so I could put it in this comment [yes, really!], may help.) – Rubio Jun 10 at 9:59

## 4 Answers

I drew it!

There could be a little mistake in there, but nothing that couldn't be fixed. So to recap:

The paths shown are surrounded by walls such that they are the only possible paths to take.

Typically paths are constructed such that they require certain levels of speed. This means we can prevent players from waiting (usually), and it also prevents players from changing paths at an intersection. The "home-stretch" of each path does not need this constraint because each player will try to finish at fast as possible anyway.

The # of moves each path requires was specifically chosen such that certain players in certain start locations would cause crashes. If player 1 chooses to take the short path, they will crash should either player 2 or 3 be on the medium path. By this logic, player 1 must choose the long path in order to not crash at the first intersection. The same reasoning almost works for player 2 not being able to take the short path but there is a problem.

Player 2's handicap allows them to choose to wait at the starting line. This means player 2 could wait 1 turn and then not crash into player 3 (who would be on the medium path in this scenario). This would result in a tie. This is why the 2nd intersection point was added. The path lengths make it so player 2 would end up crashing into player 1 at the second intersection should player 2 have waited 1 turn. This also means that player 1 would be in danger of crashing into someone else during the intended scenario, which is why the long path allows for a waiting point between the intersections (W1).

Most of the list of possible starting combinations and waiting options are easy to eliminate because a player wouldn't choose to kill themselves. There's a couple tricky ones left after that but there's only one scenario players would choose in the end:

Player 1 takes the longest path, player 2 takes the medium path, and player 3 takes the shortest path. Player 2 will not need to wait, and player 1 will choose to wait 1 turn (in between the intersection points) to not crash. Player 3 wins in 34 turns, player 2 in 35 turns, and finally player 1 in 37 turns.

Everything below is my original post:

@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.

Edit

I didn't realize that player 2 could stand on the starting point during his first turn, and that does mean he could survive taking the shortest path if he waited 1 turn, and then would tie with player 3 who would be on the medium path.

To correct that, there would have to be a second intersection point, between the longest path and the shortest path. Player 1 still has to take the longest path to live, so the 2nd intersection can be used to let player 1 block a theoretical player 2 on the shortest path (who waited one turn). This would mean player 2 would have to wait two turns to live, and would then lose to player 3 in that scenario.

• Sorry for taking so long to respond. Your design looks very promising. I'm just not able to verify it at the moment because I can't keep track of all those moves in my head while on mobile... I'd love to be able to award the checkmark for this, so I'll get back when I have the chance to sit down and go through all the variations. – jafe Jun 12 at 18:36
• Hmm, how about this variation: #1 takes the red path, #2 takes the green path but skips one turn. Now #3 cannot move forward because they'd crash with #2, so they have to circle back to the start line to pick up speed (losing 4 turns). Meanwhile, #1 gets to the P2 point long before #3 has time to get in the way. It ends up being a tie between #1 and #2. Since neither #1 nor #2 can do any better, and #3 going last has no say in the matter, I think this ends up being best play? – jafe Jun 12 at 20:42
• That said, I think I'll just remove the possibility of #2 starting with a 0-length move altogether. (See the comments under the question) – jafe Jun 12 at 20:45
• Hmm. Player 2 would end up taking one turn longer. However, killing off player 3 causes Player 2 to "tie" instead of lose outright, and you're saying that would be preferred. I'll admit I hadn't considered that one. – Dark Thunder Jun 13 at 1:35
• With the New And Improved Rules this is correct, of course. I'll give you the checkmark because you were the first to post a full track where #3 wins. – jafe Jun 13 at 12:24

A slightly different approach, drawn terribly

Player 1 is blocked by Player 2's opening move, but Player 2's momentum forces them to take a longer path, letting Player 3 win.

• This is almost exactly what I had in mind. – jafe May 30 at 19:41
• My version, with a couple of different starting positions. – jafe May 30 at 20:11
• If player 2 doesn't use his handicap, I have him crossing the line in the same number of moves as you have above... and player 3 does not win that game. – Dark Thunder May 31 at 12:19
• @jafe I see Dark Thunder's point, I do believe. – Brandon_J May 31 at 13:05
• Yeah, I see what you mean now. My logic was that not using the handicap loses 2 moves (1 for taking an extra move to reach the bottleneck, and 1 for having to pass a turn to wait for p1 to move out of the way), but I failed to take into account that using the handicap also loses 2 moves (1 extra move northward and 1 southward). Should have played out those games until the end as well. So looks like this was based on a flawed concept to begin with... – jafe May 31 at 13:28

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.

• I may have overlooked the possibility of car 2 using his handicap to stay still at move 1. Hmh. – Bass May 29 at 20:54
• I like your thinking! I have hard time visualizing how the 3-player track would look, though. Do you mean something like this? – jafe May 29 at 22:14
• I think p2 being able to stay still on the first move does make a difference. If p2 stays still, p1 still has to stop and circle back in order to avoid crashing with p3. And now p2 would come to the intersection last and it's p3 who has to give way to avoid crashing. – jafe May 29 at 22:38

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.

• Player 1 can pick any spot on the starting line as they're the first to move. I probably should have made that explicit in the rules... – jafe May 29 at 11:49
• @jafe aaah i seee – athin May 29 at 11:52
• Made an edit to rule #1 to clarify this. – jafe May 29 at 11:53