Pullen's Fractal Maze
Hello! On mathpuzzle.com, in "Material added 13 May 05", you have an item
for "Walter D. Pullen's Fractal Maze Generator", where you have a special
10x4 pin 7 chip Maze, and ask "Can anyone solve the following maze". It
doesn't look like anybody has submitted an answer yet, so I will solve a
Maze created by my own program. ;) It turns there's one unique shortest
solution, and it's of length 43, and requires entering a depth of 7 (this
verified by computer). That's a complicated Maze, with a long and deep
solution, a very good choice! :)
Below is the solution. Let's label the 40 pins around the outsides as
follows: Indicate the edge by its direction (NWSE) followed by a number
ranging from 110 left to right, or 14 top to bottom. The chips are
numbered 17, so e.g. "1S10" means the rightmost bottom pin on chip #1,
and "W1" means the topmost left pin on the outer rim. Notice how the
solution has two instances of a long subsequence of stack 65515, but
going in opposite directions, i.e. the way to cross chip #6 between 6E4
and 6S7. (Add a link between E4 and S7 on the outer rim and the solution
becomes much shorter!) Anyway:
From  Enter 5E4  Stack:
From E4 Enter 4S10  Stack: 5
From S10 Leave E2  Stack: 54
From 4E2 Enter 1E3  Stack: 5
From E3 Leave E1  Stack: 51
From 1E1 Enter 2W3  Stack: 5
From W3 Enter 6E4  Stack: 52
From E4 Enter 5E1  Stack: 526
From E1 Enter 5S9  Stack: 5265
From S9 Enter 4S10  Stack: 52655
From S10 Leave E2  Stack: 526554
From 4E2 Enter 1E3  Stack: 52655
From E3 Leave E1  Stack: 526551
From 1E1 Enter 1E2  Stack: 52655
From E2 Enter 5S3  Stack: 526551
From S3 Leave N2  Stack: 5265515
From 5N2 Leave N5  Stack: 526551
From 1N5 Leave S3  Stack: 52655
From 5S3 Leave S10  Stack: 5265
From 5S10 Leave S7  Stack: 526
From 6S7 Enter 5N2  Stack: 52
From N2 Leave S3  Stack: 525
From 5S3 Leave E2  Stack: 52
From 2E2 Leave W1  Stack: 5
From 5W1 Enter 5S10  Stack:
From S10 Enter 5S3  Stack: 5
From S3 Leave N2  Stack: 55
From 5N2 Enter 6S7  Stack: 5
From S7 Enter 5S10  Stack: 56
From S10 Enter 5S3  Stack: 565
From S3 Enter 1N5  Stack: 5655
From N5 Enter 5N2  Stack: 56551
From N2 Leave S3  Stack: 565515
From 5S3 Leave E2  Stack: 56551
From 1E2 Enter 1E1  Stack: 5655
From E1 Leave E3  Stack: 56551
From 1E3 Enter 4E2  Stack: 5655
From E2 Leave S10  Stack: 56554
From 4S10 Leave S9  Stack: 5655
From 5S9 Leave E1  Stack: 565
From 5E1 Leave E4  Stack: 56
From 6E4 Leave W3  Stack: 5
From 5W3 Goto ++++
Walter D. Pullen

I have to admit, these things are addictive.
I found a solution only (?) 6 levels deep.
The north and south exits are numbered 1 to 10 from left to right. The west
and east exits are numbered 1 to 4 from top to bottom. Parentheses represent
submazes. The submazes are labeled both before and after the parentheses,
to keep me from getting lost.
Let point A be the intersection just northwest of the northwest corner of
submaze 1.
Let point B be the intersection just north of gate S9.
A path from A to B is:
A  1(N55(N2S3)5E2)1  1(E1E3)1  4(E2S10)4  B
This path, and its reverse (from B to A), appear several times in this
solution to the maze.
For brevity, it will be written as "AB" or "BA".
[]
 2(S3 1(N5 5(N2 AB S9)5  2(E15(S9 BBA S3)5E2)2 W1)1 W4)2
 2(E2 5(S3N2)5  6(S7 5(S10 5(S3 AB S9)5 E1)5 E4)6 W3)2
 1(E1E3)1  4(E2S10)4
 5(E1 5(S9 BA N2)5  6(S7 5(S10 5(S3 AB S9)5 E1)5 E4)6 W3)5
 [+]
Eric von Laudermann

Hi Ed!
I'm glad to see the fractal maze making a return on MathPuzzle. If you
recall, I wrote a Javascript simulator for the maze and submitted
colorcoded solutions. This last puzzle is quite big so I will just
submit the solution in my simulator's notation:
The solution is a list of nodes. A node is either a , +, or a pin
designated as [chiplabel][side][pin#]. The chips labeled 17 are the
internal chip; chip X is the big chip. Sides are N/E/S/W for
north/east/south/west respectively. Pins are numbered 1..10 from
lefttoright or toptobottom.
These solution enters only 5/6 at the top most level respectively.
There are others, possible shorter solutions.
Solution#1:
Outline: 5(5(2(5(411(5))6(5(5(1(5)14))))1(5(411(5))))6(5(5(1(5)14)))))

5E4 => XE45
5E1 => XE155
2E1 => XE1255
5S9 => XS95255
4S10 => XS1045255
XE2 => 4E25255
1E3 => XE315255
XE1 => 1E15255
1E2 => XE215255
5S3 => XS3515255
XN2 => 5N215255
XN5 => 1N55255
XN2 => 5N2255
6S7 => XS76255
5S10 => XS1056255
5S3 => XS3556255
1N5 => XN51556255
5N2 => XN251556255
XS3 => 5S31556255
XE2 => 1E2556255
1E1 => XE11556255
XE3 => 1E3556255
4E2 => XE24556255
XS10 => 4S10556255
XS9 => 5S956255
XE1 => 5E16255
XE4 => 6E4255
XW3 => 2W355
1E1 => XE1155
5S9 => XS95155
4S10 => XS1045155
XE2 => 4E25155
1E3 => XE315155
XE1 => 1E15155
1E2 => XE215155
5S3 => XS3515155
XN2 => 5N215155
XN5 => 1N55155
XN2 => 5N2155
XN5 => 1N555
XN2 => 5N25
6S7 => XS765
5S10 => XS10565
5S3 => XS35565
1N5 => XN515565
5N2 => XN2515565
XS3 => 5S315565
XE2 => 1E25565
1E1 => XE115565
XE3 => 1E35565
4E2 => XE245565
XS10 => 4S105565
XS9 => 5S9565
XE1 => 5E165
XE4 => 6E45
XW3 => 5W3
+
Solution #2:
Outline:
6(3(2(1(2(5(1(5)14)))4(5(1(5)14)2(5(411(5))))3(5(5(411(5)))))7(41(5(411(5)))7(5(5(1(5)14)))))))

6S1 => XS16
3W1 => XW136
2W4 => XW4236
1W1 => XW11236
2E2 => XE221236
5S3 => XS3521236
1N5 => XN51521236
5N2 => XN251521236
XS3 => 5S31521236
XE2 => 1E2521236
1E1 => XE11521236
XE3 => 1E3521236
4E2 => XE24521236
XS10 => 4S10521236
XS9 => 5S921236
XE1 => 2E11236
XE3 => 1E3236
4E2 => XE24236
5S3 => XS354236
1N5 => XN5154236
5N2 => XN25154236
XS3 => 5S3154236
XE2 => 1E254236
1E1 => XE1154236
XE3 => 1E354236
4E2 => XE2454236
XS10 => 4S1054236
XS9 => 5S94236
2E1 => XE124236
5S9 => XS9524236
4S10 => XS104524236
XE2 => 4E2524236
1E3 => XE31524236
XE1 => 1E1524236
1E2 => XE21524236
5S3 => XS351524236
XN2 => 5N21524236
XN5 => 1N5524236
XS3 => 5S324236
XE2 => 2E24236
XW1 => 4W1236
3E4 => XE43236
5E1 => XE153236
5S9 => XS9553236
4S10 => XS104553236
XE2 => 4E2553236
1E3 => XE31553236
XE1 => 1E1553236
1E2 => XE21553236
5S3 => XS351553236
XN2 => 5N21553236
XN5 => 1N5553236
XS3 => 5S353236
XS10 => 5S103236
XN4 => 3N4236
XN6 => 2N636
7E4 => XE4736
4S10 => XS104736
XE2 => 4E2736
1E3 => XE31736
5S9 => XS951736
4S10 => XS10451736
XE2 => 4E251736
1E3 => XE3151736
XE1 => 1E151736
1E2 => XE2151736
5S3 => XS35151736
XN2 => 5N2151736
XN5 => 1N551736
XN2 => 5N21736
XN5 => 1N5736
7N3 => XN37736
5E2 => XE257736
5S3 => XS3557736
1N5 => XN51557736
5N2 => XN251557736
XS3 => 5S31557736
XE2 => 1E2557736
1E1 => XE11557736
XE3 => 1E3557736
4E2 => XE24557736
XS10 => 4S10557736
XS9 => 5S957736
XE1 => 5E17736
XE4 => 7E4736
XN8 => 7N836
XS6 => 3S66
XN10 => 6N10
+
Regards,
Yogy Namara

Here is a solution to the fractal maze, that may not be minimal.
First, notation.
I labeled the pins this way:
The top (North) pins from left to right: N0 ... N9
The bottom (South) pins from left to right: S0 ... S9
The left (West) pins from top to bottom: W0 ... W3
The right (East) pins from top to bottom: E0 ... E3
There are a number of equivalence classes of pins.
These are pins that are joined together within the maze,
without having to go up or down a level in the maze.
These are:
(A) N1 == s2
(B) N3 == S6
(C) E0 == E2
(D) E1 == S9
(E) E3 == S8
Let's call these axioms. They are obvious.
I solved the maze by building more equivalence classes that
usually depend on going only deeper into the maze, with a few
exceptions.
Notation:
B:Px  Inside box B at pin Px.
ABC:Px  Several levels down the maze, the deepest being C at pin Px.
I start with the simplest, and build up.
Theorem 1: E1 == N4.
Start at E1.
Go down into 5 at S2.
Use Axiom A to get to N1.
Go up out of 5 to N4.
QED
The next theorem goes above the starting level,
so it depends on the box as well as the pin.
Theorem 2: 1:E0 == 4:E1
Start at 1:E0.
Use Axiom C to get to 1:E2.
Go up out of 1, and into 4:E1.
QED
Theorem 2.1: 1:E0 == 4:S9
Start at 1:E0.
Use Theorem 2 to get to 4:E1.
Use Axiom D to get to 4:S9.
QED
Theorem 3: 1:N4 == 4:S9
Start at 1:N4.
Use Theorem 1 to get to 1:E1.
Go out of 1, and into 1:E0.
Use Theorem 2.1 to get to 4:S9.
QED
Theorem 4: S2 == S8
Start at S2.
Go down into 1:N4.
Go to 4:S9. (3)
Go up to S8.
QED
Theorem 4.1: S2 == E3
Start at S2.
Go to S8. (4)
Go to E3. (E)
QED
Theorem 5: E2 == W0
Start at E2.
Go down to 2:E0.
Go down to 25:S8.
Go to 25:S2. (4)
Go up to 2:E1.
Go up to W0.
QED
Theorem 6: S6 == S8
Start at S6.
Go down to 5:S9.
Go down to 55:S2.
Go to 55:S8. (4)
Go up to 5:E0.
Go up to S8.
QED
Theorem 6.1: S6 == E3
Start at S6.
Go to S8. (6)
Go to E3. (E)
QED
Theorem 7: N4 == E2
Start at N4.
Go down to 5:N1.
Go down to 51:N4.
Go to 54:S9. (3)
Go up to 5:S8.
Go up to E2.
QED
Theorem 7.1: N4 == E0
Start at N4.
Go to E2. (7)
Go to E0. (C)
QED
Now that these little transfomations have been found,
we can solve the maze.
Theorem 8: + == 
Start at +.
Go to 1:S1.
Go down to 16:W3.
Go down to 161:W0.
Go to 161:E2. (5)
Go up and down to 164:E1.
Go to 164:S9. (D)
Go up to 16:S8.
Go to 16:S6. (6)
Go up to 1:N4.
Go to 1:E1. (1)
Go up and down to 2:W2.
Go down to 26:E3.
Go down to 264:S9.
Go to 261:E0. (2.1)
Go up and down to 262:W2.
Go down to 2626:E3.
Go to 2626:S6. (6.1)
Go up to 262:N4.
Go to 262:E0. (7.1)
Go up to 26:E2.
Go to 26:N4. (7)
Go up and down to 23:S2.
Go to 23:E3. (4.1)
Go up and down to 24:W0.
Go to 24:E2. (5)
Go down to 245:S8.
Go to 245:S2. (4)
Go up to 24:E1.
Go to 21:E0. (2)
Go up and down to 21:E1.
Go to 21:N4. (1)
Go up to 2:S2.
Go up and down to .
QED
Rolan Christofferson

I've solved your newest fractal maze; my solution is below. This is a
solution of minimal depth (I think), and among all such solutions its
length is minimal (ditto).
I've had an idea for a fractal maze variant, but not managed to
implement it: the Tumbolian fractal maze (after Goedel, Escher, Bach).
Somewhere within the maze would be a Tumbolia node, and entering it
would manipulate the stack of nodes entered so far. For instance, the
Tumbolia node might perform a swap on the stack, so if you enter A,
enter B, enter C, and touch this node, you'd have to exit B, exit C, and
exit A to finish. Such a maze wouldn't be susceptible to the bottomup
approach I use to solve standard fractal mazes.
Perhaps some mathpuzzler out there can make something of this?
Alex Fink
Number the pins clockwise, with 1 at the upper left.

5.14
5.16
2.11
2.5.16
2.5.4.15
2.5.4.12
2.5.1.13
2.5.1.11
2.5.1.12
2.5.1.5.22
2.5.1.5.2
2.5.1.5
2.5.2
2.6.18
2.6.5.15
2.6.5.5.22
2.6.5.5.1.5
2.6.5.5.1.5.2
2.6.5.5.1.5.22
2.6.5.5.1.12
2.6.5.5.1.11
2.6.5.5.1.13
2.6.5.5.4.12
2.6.5.5.4.15
2.6.5.5.16
2.6.5.11
2.6.14
2.26
1.11
1.13
4.12
4.15
5.11
5.5.16
5.5.4.15
5.5.4.12
5.5.1.13
5.5.1.11
5.5.1.12
5.5.1.5.22
5.5.1.5.2
5.5.1.5
5.5.2
5.6.18
5.6.5.15
5.6.5.5.22
5.6.5.5.1.5
5.6.5.5.1.5.2
5.6.5.5.1.5.22
5.6.5.5.1.12
5.6.5.5.1.11
5.6.5.5.1.13
5.6.5.5.4.12
5.6.5.5.4.15
5.6.5.5.16
5.6.5.11
5.6.14
5.26
+
Alex Fink