How did Gammon solve the puzzle in episode 2?

Question

While I was watching Phi-Brain Episode 2, I tried solving the puzzles on my own. But on that sliding block game, I couldn't figure out how Gammon solved that puzzle without moving the red car until the path was cleared. Here's how it looks like:

enter image description here

The black car on the farthest left and the white car near the exit are 3 blocks long which can be confirmed from this picture below.

enter image description here

So when you plot it out, the puzzle would look like this:

enter image description here

I'm kinda wondering how and I don't think it's possible.

I'm assuming that the red car has to be moved through the exit, and that the cars can only move forward/backward? — Maroon, Apr 29 '15 at 02:59
It seems to be solvable, but my solution isn't complete yet, and I'm willing to bet that I have some unnecessary steps — Maroon, Apr 29 '15 at 03:28
Update: no success. Being a bored, mathy person, I'm now trying to come up with a possible proof showing that it's not solvable, but I can't say I'm have much success with that either, and I rather not do something by brute-force. — Maroon, Apr 29 '15 at 06:17
it is explained in the episode, he cheated I think, by using cars to push other cars, which you aren't supposed to do. I'll try formulate an answer when I get to the train :p — Peter Raeves, Apr 29 '15 at 08:08
For what it's worth, this prompted me to ask a question at [Math.SE](http://math.stackexchange.com/q/1257238/126378), where it's been suggested that the puzzle can be solved. Unfortunately, I'm really too tired right now to make a fresh attempt of it. — Maroon, Apr 29 '15 at 09:29
Update: yeah, someone included a solution when answering my question, even though the focus of my question was something else. — Maroon, Apr 29 '15 at 18:31
@Maroon Read the replies, but didn't really understand the solution though. — Peter Raeves, Apr 29 '15 at 21:47
@PeterRaeves: I'll probably try solving again and see if I get something. — Maroon, Apr 29 '15 at 21:49
@Ezui I tried to find a solution, but I had to move the red car twice before the path was cleared. Was it mentioned in the episode that the red car shouldn't move? — Peter Raeves, Apr 29 '15 at 22:43
@PeterRaeves Well he did find a solution without moving it first. — , Apr 29 '15 at 22:54
@Ezui I'm quite sure that he moved it up and down offscreen and that in the last move it was at the back again (same as in my solution btw). You have to move it up once to move past the lower right horizontal car (car number 10) to the left. Then the red car is moved back down until the end of the game. It is impossible to solve otherwise. — Peter Raeves, Apr 29 '15 at 22:58
@PeterRaeves From what I remember, he just moved the other cars until he was able to clear the path. I guess I'll have to rewatch it then. — , Apr 29 '15 at 23:16
@Ezui At [18:34](https://youtu.be/gwwcs1KG8JE?t=18m34s), he moves it [up for the first time](http://i.stack.imgur.com/vo7tX.png) — Peter Raeves, Apr 29 '15 at 23:22

score 6 · Accepted Answer · edited Oct 05 '15 at 01:03

6

I ended up writing a descriptive model for it in IDP, letting the solvability prover from our univeristy prove whether a solution can be found. The fastest solution it could come up with was one finishing the game in 48 steps (see below). Therefore the problem is indeed solvable. My first answer though, saying Gammon cheated, was indeed incorrect. It was only after he had solved the puzzle, that the system was sabotaged and made Kaito cheat to save their lives.

I numbered the cars from top to bottom and left to right as in the following picture.

enter image description here

The solution is written in the form Move(t,cid,d) with t being the step number in the solution, cid being the car identifier and d being the distance the car travels during that time step. d is positive when driving up or to the right and d is negative when driving down or to the left.

Move = {
    1,9,1; 2,4,2; 3,2,1; 4,1,-1; 5,6,-3;
    6,7,1; 7,9,1; 8,3,3; 9,7,-2; 10,6,1;
    11,1,1; 12,2,-1; 13,5,3; 14,2,1; 15,1,-1;
    16,6,-1; 17,7,2; 18,8,2; 19,10,-4; 20,8,-2;
    21,7,-1; 22,6,1; 23,1,1; 24,2,-1; 25,5,-3;
    26,2,2; 27,1,-1; 28,6,-1; 29,7,1; 30,3,-3;
    31,7,-1; 32,6,1; 33,1,1; 34,2,-2; 35,4,-2;
    36,9,-4; 37,4,2; 38,2,1; 39,1,-1; 40,6,-1;
    41,7,1; 42,3,3; 43,7,-1; 44,6,3; 45,1,1;
    46,2,-1; 47,5,4;
}

edited Oct 05 '15 at 01:03

Maroon

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answered Apr 29 '15 at 08:11

Peter Raeves

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But Gammon did not knew this at first. He played by the rules. He only knew that it was possible after Kaito, with the help of the Armband of Orpheus, realized the trick behind the game. – Apr 29 '15 at 08:52
And if Gammon knew, he wouldn't have been kicking the doors of the car just to escape. – Apr 29 '15 at 09:22
@ezui yeah i rewatched the scene and there was indeed a solution without cheating. Ill change the answer when I have it calculated. My model has some error somewhere – Peter Raeves Apr 29 '15 at 14:51
Am I missing something. It can be solved by only 6 moves. [2:down, 4: left, 4: down, 3:down, 3: left, 5: up] – Thellimist Apr 30 '15 at 23:37
1

@Furkan The blocks represent cars, as you can see in the screenshot of OP's question, and cars can not move sideways (yet?). Therefore car number 2 cannot move downwards as you suggested. – Peter Raeves May 01 '15 at 00:42

How did Gammon solve the puzzle in episode 2?

1 Answers1