Racecar Maniac #1

(If all goes well this will become a series.)

Attribution: Inspired by Racetrack in Ben Orlin's Math Games with Bad Drawings

You are a racecar maniac. You have decided that antarctica is the nicest place to accelerate your car, so of course you go to antarctica. Your goal is to lead the car to the flag. The rules are intentionally not precise, since you will have to figure out how the car works yourself.

The squares on the top right (if present) indicates the length of the common path: a number in green is the time required (in frames) to reach the flag. A red box indicates that the puzzle is not possible to be solved. The optimal path is not necessarily unique.

A complete answer should solve all 14 puzzles. For solvable puzzles, give the solution. Otherwise, demonstrate that the puzzle is not solvable. Partial answers are allowed and encouraged.

Have fun racing!

Hint: Hopefully this post gains momentum.

14

It must be somewhere...

• I assume the difference between 12 and 13 is in the number of blank spaces to the flag, but it's not very clear. Maybe make the background grid visible in some way.
– fljx
Commented Feb 14 at 9:05

I think the rule is

• The car has the initial velocity of (0, 0).
• At every step, you can accelerate by +1, 0, or -1 unit, independently for X and Y directions, and then change position according to the new velocity.

The paths for levels 1 to 11 are the same as in Jujustum's answer, but some require an alternative explanation.

To ease explanation, let's define

right is the positive X direction, and down is the positive Y direction.

Level 6

Accelerate +1 in both X and Y directions, twice.

Level 7

First, accelerate +1 in both X and Y directions. Then accelerate +1 in X and -1 in Y to get the velocity of (2, 0) and reach the goal.

Level 8

Same as Level 7 for the first two steps. Then, accelerate -1 in X and -1 in Y to get the velocity of (1, -1).

Level 10

Accelerate -1 in X to get the velocity of (-1, 0) and go back 1 square. Then accelerating +1 in X gives the velocity of (0, 0), keeping the car in place. Now accelerate +1 twice to reach the goal.

Level 13

I can't upload an image at the moment, so I'll share a penpa link instead. This path takes 7 steps.

• The square labeled 1 (the one right below the start) cannot be avoided since the first acceleration in Y will put you somewhere on that row.
• Then the path tries to accelerate in X as much as possible. I think velocity of 3 in X is not reachable, so the path goes on with the constant X velocity of 2. Y velocity is then adjusted to keep the car on the land and reach the goal with the Y velocity of 2.

Level 14

Again, here is a penpa link in place of an image. This path takes 8 steps.

• This time, the focus is on maximizing the X acceleration. Accleration of +1 twice in X from the start is impossible, so the car accelerates +1 and then +0, and then keeps accelerating until it builds up X velocity up to 5.
• On the process, the car happens to nicely jump onto the second line at step 4, and step on the middle of I at velocity 5.
• The optimality of this path can be proven by $$1 + 1 + 2 + 3 + 4 + 5 + 6 = 22 < 25$$, which is the theoretical maximum X displacement in 7 steps under the constraint that accelerating twice in the first two steps is impossible.

Possible answer (with missing rules, according to the author of the puzzle)

Level 1

Let's drive the car!

Level 2

The car is able to move at a constant speed.

Level 3

The car can build up speed to cross more than one tile in a single frame.

Level 4

The car must land on a tile.

Level 5

By building up speed, the car can go over empty spaces as long as it lands on a tile.

Level 6

The car can move diagonally.

Level 7

The car keeps its momentum after a 45 degree turn.

Level 8

The car can decelerate after building up speed.

Level 9

Not all tiles need to be crossed.

Level 10

The car can drive in reverse. It takes a frame for the car to switch from reverse mode to drive mode.

Level 11

Your acceleration is 1 tile / frame squared (if you're driving at a speed of 2 tiles / frame, you'll be able to drive at 3 tiles / frame on the next frame).

Level 12

Here, there isn't enough space to drive up to 4 tiles/frame, which makes the 4-tile gap impossible to cross.

Level 13

Let's use previous knowledge to solve this harder level! Here's a solution in 6 frames, that I believe to be optimal.

Level 14

The 14th puzzle is hidden in the title! Here's a solution in 9 frames, that I believe to be optimal.

• You are missing a significant portion of the puzzle. I also miscalculated level 11; it should be 7 moves, not 6, which I'll edit. Here's a hint: the car can move in all directions.
– Sny
Commented Feb 14 at 9:33
• Level 11 is correct now, but can you extend the rule more generally? In paticular both your Level 13 and Level 14 routes are illegal. Here's the hint again: the car can move in all directions, not just straight and diagonal. How would you formalize this notion of "gaining momentum", and generalize it?
– Sny
Commented Feb 14 at 10:15