Alice and the Fractal Hedge Maze

Question

This is an entry to the 12^th fortnightly challenge.

---

Alice: Would you tell me, please, which way I ought to go from here?
Cheshire Cat: That depends a good deal on where you want to get to.
Alice: I don't much care where...
Cheshire Cat: Then it doesn't much matter which way you go!
Alice: ... so long as I get somewhere.
Cheshire Cat: Oh, you're sure to do that, if only you walk long enough.

Alice is in the most puzzling part of Wonderland yet. Following the white rabbit, she emerged found herself in the middle of a hedge maze. The rabbit provided her with a map before scurrying off, but it only seemed to make her more confused. She needs your help to figure out how to escape the maze.

The maze has 12 potential exits, numbered on the map. Each of the squares labeled A, B, C and D represent smaller copies of the entire maze. These submazes each have their own submazes, like infinitely many nested Matryoshka dolls, except that every doll has four dolls nested inside it.

Below the map is a bird's eye view of the actual maze, where you can see how the passages become smaller and smaller in a fractal fashion (only three levels of recursion are actually pictured). Fortunately, Alice has an ample supply of cakes and elixirs to change her size as necessary.

One last note: the little orange curve between B and D is a bridge which can be crossed over and walked under, but jumping from the bridge to the path below is not allowed.

Map

Bird's Eye View

Sources

Though I created this particular puzzle, the concept of a fractal maze is nothing new. Here are some other notable examples of cool fractal mazes, which served as inspiration for this one.

• As far as I can tell, the concept of a fractal maze was created my Mark J. P. Wolf. He has made at least two mazes, taken from mathpuzzle.com.

  • His first maze and his second maze.

• These are from the blog Skeptic's Play:

  • Maze 1 (circle), maze 2 (Sierpinksi), and maze 3 (carpet).

• Two devious looking mazes which I found referenced in this forum, but couldn't find the original sources for.

  • One and two.

Answer 1

Here is a link to a prezi. If there is anyway to export it that would be great but I'm not exactly sure how it works.

Just keep clicking next through it.

https://prezi.com/oh2efo-ejbv9/untitled-prezi/?utm_campaign=share&utm_medium=copy

GIF

And the path in the format from-to (level). If the number includes a letter it comes from/goes to an internal maze box, if it does not it goes to the outside of the current level.

IN-A3 (1)
3-B1 (2)
1-12 (3)
B12-B11 (2)
11-A10 (3)
10-C4 (4)
4-B6 (5)
6-D8 (6)
8-10 (7)
D10-7 (6)
B7-7 (5)
D7-9 (4)
A9-8 (3)
B8-D3 (2)
3-B4 (3)
4-B6 (4)
6-D8 (5)
8-10 (6)
D10-7 (5)
B7-D10 (4)
10-8 (5)
D8-6 (4)
B6-4 (3)
D4-5 (2)
A5-1 (1)
OUT

Answer 2

The path you can take out is

Enter A3
Enter B1
Exit B12
Enter B11
Enter A10
Enter C4
Enter B6
Enter D8
Exit D10
Exit B7
Exit C7
Exit A9
Exit B8
Enter D3
Enter B1
Exit B12
Enter B11
(Repeat)
Exit B8
Enter D1
Exit D12
Enter D11
Enter A10
(Repeat)
Exit A9
Exit D10
Enter B7
Enter D10
Exit D8
Exit B6
Exit D4
Exit A5

How I reached this conclusion

A few rules that you can make

Enter 4, Exit 7
Enter 4
Enter B6
Enter D8
Exit D10
Exit B7
Exit 7

Enter 10, Exit 9
Enter 10
Enter C4
Enter 4, Exit 7
Exit C7
Exit 9

Enter 11, Exit 2|8|9|10
Enter 11
Enter A10
Enter 10, Exit 9
Exit A9
Exit 2|8|9|10

Please help me with formatting ._.

Answer 3

This puzzle definitely caught my fancy! The leap was when I realized that all moves could be expressed as compositions of "prime" moves, which don't enter any of the submazes: 1-12, 2-8, 2-10 and 8-10 (and their reverses). So I just drew the maze with blank submazes, added the prime moves to the submazes, saw which moves I could do now and added those to the submazes, and repeated until I had "built" the solution (3-5).

3-5: B 1-12*
     B 11-8
     D 3-5

11-8: A 10-9

10-9: C 4-7

4-7: B 6-7

6-7: D 8-10*

3-5: D 4-6

4-6: B 1-12*
     B 11-8
     D 1-12*
     D 11-8

I...haven't actually expanded the solution out all the way, but I'm mostly sure it's identical to gtwebb's.

Answer 4

Found the same path as @gtwebb. I used the following notation: 1 - node 1 on external square, A7 - node 7 on A, CB11 - node 11 on B inside C. So from CAB6 there are edges to CA4 and CABD8.

First I ran the classic BFS... which didn't stop (until my patience ran out). Priority queue came to rescue, causing nodes with minimal length, which are closest to the external border, to be handled first. This found solution in under 1000 iterations and under 1 second.

Here's the script. It uses a, b, c instead of 10, 11, 12 to make things little bit easier.

And here's the solution:

A3 - AB1 - AB12 - AB11 - ABA10 - ABAC4 - ABACB6 - ABACBD8 - ABACBD10 - ABACB7 - ABAC7 - ABA9 - AB8 - AD3 - ADB4 - ADBB6 - ADBBD8 - ADBBD10 - ADBB7 - ADBD10 - ADBD8 - ADB6 - AD4 - A5 - 1