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Question


                                                    1 1
                                                  1 1 1 1
                                                1 1 1 1
                                1             1 1 1 1
                                1           1 1 1 1
                                1 1 1     1 1 1 1
                                1 1 1 1 1 1 1 1
                                1 1 1 1 1 1 1
                                1 1 1 1 1 1
                                1 1 1 1 1 1 1 1
                                1 1 1 1 1 1 1 1 1 1

#1. You are here

            1111111111111111111111111111111
            111                         111
            111   1111111111111111111   111111111111111111111111111
            111   111             111   111                     111
            111   111   1111111   111   111   111111111111111   111
            111   111       111   111   111   111         111   111
            111   1111111   111   111   111   111   111   111   111
            111             111   111   111         111   111   111
            1111111111111111111   111   111111111111111   111   111
                                  111                     111   111
                      111111111111111111111111111111111111111   111
                      111                                       111
                      111   111111111111111111111111111111111111111
                      111   111
                      111   111   111111111111111
                      111   111   111         111
                      111   111   111   111   111
                      111   111         111   111
                      111   111111111111111   111
                      111                     111
                      111111111111111111111111111

#2. Through the labyrinth we wind


                           1ooo11ooo1
                       111ooooo1   1oo111o
                       1 oo1111   1      1
                       o1   o1   1        1
                      o1    1   1          1
                       o1   1   1           1
                        1    1   1          1
                         1    1   11ooo1oo1o111oo
                          1ooo11o1     oooo1
                          o1 1111  ooo1111
                         1111    1111111
                         1          1111
                                     1111
                                       o1
                                       o1
                                o1111   o1
                             oo1     o1 o1
                            o1         1111
                           o1      ooo1oo111o
                           1   ooo1      o1
                           1111           1
                            1

#3. A rose is a rose is a rose

                                                1
                                               1 1
                                              1o1o1
                                             1 1 1 1
                                    1       1o1o1o1o1       1
                                   1 1     1 1     1 1     1 1
                                  1o1o1   1o1o1   1o1o1   1o1o1
                                 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
                                1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1
                               1 1     1 1             1 1     1 1
                              1o1o1   1o1o1           1o1o1   1o1o1
                             1 1 1 1 1 1 1 1         1 1 1 1 1 1 1 1
                            1o1o1o1o1o1o1o1o1       1o1o1o1o1o1o1o1o1
                           1 1             1 1     1 1     1 1
                          1o1o1           1o1o1   1o1o1   1o1o1
                         1 1 1 1         1 1 1 1 1 1 1 1 1 1 1 1
                        1o1o1o1o1       1o1o1o1o1o1o1o1o1o1o1o1o1
                       1 1     1 1     1 1     1 1
                      1o1o1   1o1o1   1o1o1   1o1o1
                     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
                    1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1

#4. $\sf \small O \scriptsize NE$ders of the ancient world

Obviously (?) the $\sf\scriptsize ZERO$-and$\small/$or-$\sf\scriptsize ONE$ders here represent a pattern of mathematical constructs.

#5. What picture could be fifth, but not at other #s here? Why?

                                                               1o111111ooooo
                                                         oooooo1o11111ooooo
                                                   oooooooooooo1o1111ooooo
                                             oooooooooooooooooo1o111ooooo
                                       oooooooooooooooooooooooo1o11ooooo
                                 oooooooooooooooooooooooooooooo1o1ooooo
                           ooooooooooooooooooooooooooooooooooo11o1oooo
                     oooooooooooooooooooooooooooooooooooooooo111o1ooo
               ooooooooooooooooooooooooooooooooooooooooooooo1111o1oo
         oooooooooooooooooooooooooooooooooooooooooooooooooo11111o1o
   ooooooooooooooooooooooooooooooooooooooooooooooooooooooo111111o1

#6. One for the road

The answer can be pictured in infinitely many ways. The ?-shaped placeholder presently at #5 is meant to be replaced. Only numbers composed of o zeros and$\small/$or 1 ones are pertinent. Two-dimensional shapes and surrounding words are just gratuitous embellishments. If you’re getting nowhere after considering all this, why not actually go nowhere, to $\sf \small N \scriptsize ONE$derland, for comparison?

Indeed, @ABcDexter, o and 1 are the only pieces in play. (To have the pictures behave rebusly would have been a good idea, but they are merely whimsical shapes allowed by the underlying rule in question. A valid answer doesn't actually have to resemble anything.) — humn, Jul 9 '16 at 19:38
The fourth looks a lot like the Sierpinski triangle. en.wikipedia.org/wiki/Sierpinski_triangle — Tony Ruth, Jul 10 '16 at 6:33
How did you do that in the title? I didn't think it was possible to use formatting there. — sumelic, Jul 10 '16 at 7:07
These are produced manually, @ABcDexter. #6 took more time to concoct because it is actually the most complex and meant to contain clues that avert some false leads. Very simple and similar mathematical diagrams, one for each #, are likely to help in the solution. — humn, Jul 11 '16 at 19:18

Deusovi · Accepted Answer · 2016-10-05 00:28:36Z

7

In the $n$th picture, the strings are all

representations of the number $1$ where the base is one of the primitive $n$th roots of unity. (That is, solutions to $z^n=1$ that don't work for any smaller $n$.)

So any picture containing

111111

would fit in slot 5, but not any other.

answered Oct 5 '16 at 0:28

Deusovi♦

116k11 gold badges392 silver badges497 bronze badges

add a comment |

humn · Accepted Answer · 2016-07-12 18:32:14Z

^{(Note from puzzle's poser:
When this answer's tables were first posted,
the puzzle statement and comments weren't clear that spaces do matter
to the extent that they separate contiguous sequences of os and/or 1s.
At the end of this post are similar tables that take this into account.)}

Not an answer just another way to look at each line since it was stated spacing doesn't matter I removed all line breaks and spaces. For the tables they can be read left to right then top to bottom. So for number 3 it is (one 1, threes 0s, two 1s, three 0s, four 1s, etc)

#1

111111111111111111111111111111111111111111111111111111111111111111

table

+======+=======+
| Ones | Zeros |
+======+=======+
| 66   | 0     |
+======+=======+

#2

 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

table

+======+=======+
| Ones | Zeros |
+======+=======+
| 476  | 0     |
+======+=======+

#3

`1ooo11ooo1111ooooo11oo111o1oo111111o1o111o1111o111111111111ooo1oo1o111oo1ooo11o1oooo1o11111ooo111111111111111111111111o1o1o1111o1oo1o1o1o11111o1ooo1oo111o1ooo1o1111111`

table

+======+=======+
| Ones | Zeros |
+======+=======+
| 1    | 3     |
| 2    | 3     |
| 4    | 5     |
| 2    | 2     |
| 3    | 1     |
| 1    | 2     |
| 6    | 1     |
| 1    | 1     |
| 3    | 1     |
| 4    | 1     |
| 12   | 3     |
| 1    | 2     |
| 1    | 1     |
| 3    | 2     |
| 1    | 3     |
| 2    | 1     |
| 1    | 4     |
| 1    | 1     |
| 5    | 3     |
| 24   | 1     |
| 1    | 1     |
| 1    | 1     |
| 4    | 1     |
| 1    | 2     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 5    | 1     |
| 1    | 3     |
| 1    | 2     |
| 3    | 1     |
| 1    | 3     |
| 1    | 1     |
| 7    |       |
+======+=======+

#4

1111o1o1111111o1o1o1o11111111111o1o11o1o11o1o11o1o111111111111111111o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1111111111o1o11o1o11o1o11o1o111111111111111111o1o1o1o1o1o1o1o11o1o1o1o1o1o1o1o1111111111o1o11o1o11o1o11o1o111111111111111111o1o1o1o11o1o1o1o1o1o1o1o1o1o1o1o1111111111o1o11o1o11o1o11o1o111111111111111111o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1

table

+======+=======+
| Ones | Zeros |
+======+=======+
| 4    | 1     |
| 1    | 1     |
| 7    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 11   | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 18   | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 10   | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 18   | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 10   | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 18   | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 10   | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 2    | 1     |
| 1    | 1     |
| 18   | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    | 1     |
| 1    |       |
+======+=======+

#5

o?1?o?1?1?o?1?o?1?o?1?

#6

1o111111ooooooooooo1o11111ooooooooooooooooo1o1111ooooooooooooooooooooooo1o111ooooooooooooooooooooooooooooo1o11ooooooooooooooooooooooooooooooooooo1o1oooooooooooooooooooooooooooooooooooooooo11o1oooooooooooooooooooooooooooooooooooooooooooo111o1oooooooooooooooooooooooooooooooooooooooooooooooo1111o1oooooooooooooooooooooooooooooooooooooooooooooooooooo11111o1oooooooooooooooooooooooooooooooooooooooooooooooooooooooo111111o1

table

+======+=======+
| Ones | Zeros |
+======+=======+
| 1    | 1     |
| 6    | 11    |
| 1    | 1     |
| 5    | 17    |
| 1    | 1     |
| 4    | 23    |
| 1    | 1     |
| 3    | 29    |
| 1    | 1     |
| 2    | 35    |
| 1    | 1     |
| 1    | 40    |
| 2    | 1     |
| 1    | 44    |
| 3    | 1     |
| 1    | 48    |
| 4    | 1     |
| 1    | 52    |
| 5    | 1     |
| 1    | 56    |
| 6    | 1     |
| 1    |       |
+======+=======+

Just some more views

#4

First line of each paired line repeats. Line 3 is the same as line 9. Lines 5 and 7 just have a double 1 that is added within the 0101 sequence.

1111o1o1111111o1o1o1o1

1111111111o1o11o1o11o1o11o1o
111111111111111111o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o

1111111111o1o11o1o11o1o11o1o
111111111111111111o1o1o1o1o1o1o1o11o1o1o1o1o1o1o1o

1111111111o1o11o1o11o1o11o1o
111111111111111111o1o1o1o11o1o1o1o1o1o1o1o1o1o1o1o

1111111111o1o11o1o11o1o11o1o
111111111111111111o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o

1

Note from puzzle's poser: The following tables are probably similar to what this answer's poster would have made if the puzzle statement and comments had been clear that spaces do matter to the extent that they separate contiguous sequences of os and/or 1s.

#1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

table

+=========+======+==========+
| repeats | ones | sequence |
+=========+======+==========+
|    66   |   1  | 1        |
+=========+======+==========+

#2

1111111111111111111111111111111 111 111 111 1111111111111111111 111111111111111111111111111 111 111 111 111 111 111 111 1111111 111 111 111111111111111 111 111 111 111 111 111 111 111 111 111 1111111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 1111111111111111111 111 111111111111111 111 111 111 111 111 111111111111111111111111111111111111111 111 111 111 111 111111111111111111111111111111111111111 111 111 111 111 111111111111111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111111111111111 111 111 111 111111111111111111111111111

table

+=========+======+=========================================+
| repeats | ones | sequence                                |
+=========+======+=========================================+
|    1    |  31  | 1111111111111111111111111111111         |
|    3    |   3  | 111                                     |
|    1    |  19  | 1111111111111111111                     |
|    1    |  27  | 111111111111111111111111111             |
|    7    |   3  | 111                                     |
|    1    |   7  | 1111111                                 |
|    2    |   3  | 111                                     |
|    1    |  15  | 111111111111111                         |
|   10    |   3  | 111                                     |
|    1    |   7  | 1111111                                 |
|   14    |   3  | 111                                     |
|    1    |  19  | 1111111111111111111                     |
|    1    |   3  | 111                                     |
|    1    |  15  | 111111111111111                         |
|    5    |   3  | 111                                     |
|    1    |  39  | 111111111111111111111111111111111111111 |
|    4    |   3  | 111                                     |
|    1    |  39  | 111111111111111111111111111111111111111 |
|    4    |   3  | 111                                     |
|    1    |  15  | 111111111111111                         |
|   14    |   3  | 111                                     |
|    1    |  15  | 111111111111111                         |
|    3    |   3  | 111                                     |
|    1    |  27  | 111111111111111111111111111             |
+=========+======+=========================================+

#3

1ooo11ooo1 111ooooo1 1oo111o 1 oo1111 1 1 o1 o1 1 1 o1 1 1 1 o1 1 1 1 1 1 1 1 1 1 11ooo1oo1o111oo 1ooo11o1 oooo1 o1 1111 ooo1111 1111 1111111 1 1111 1111 o1 o1 o1111 o1 oo1 o1 o1 o1 1111 o1 ooo1oo111o 1 ooo1 o1 1111 1 1

table

+======+=======+======+=======+======+=======+======+=======+=================+
| ones | zeros | ones | zeros | ones | zeros | ones | zeros | sequence        |
+======+=======+======+=======+======+=======+======+=======+=================+
|   1  |   3   |   2  |   3   |   1  |       |      |       | 1ooo11ooo1      |
|   3  |   5   |   1  |       |      |       |      |       | 111ooooo1       |
|   1  |   2   |   3  |   1   |      |       |      |       | 1oo111o         |
|   1  |       |      |       |      |       |      |       | 1               |
|   0  |   2   |   4  |       |      |       |      |       | oo1111          |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
|   2  |   3   |   1  |   2   |   1  |   1   |   3  |   2   | 11ooo1oo1o111oo |
|   1  |   3   |   2  |   1   |   1  |       |      |       | 1ooo11o1        |
|   0  |   4   |   1  |       |      |       |      |       | oooo1           |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   4  |       |      |       |      |       |      |       | 1111            |
|   0  |   3   |   4  |       |      |       |      |       | ooo1111         |
|   4  |       |      |       |      |       |      |       | 1111            |
|   7  |       |      |       |      |       |      |       | 1111111         |
|   1  |       |      |       |      |       |      |       | 1               |
|   4  |       |      |       |      |       |      |       | 1111            |
|   4  |       |      |       |      |       |      |       | 1111            |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   1   |   4  |       |      |       |      |       | o1111           |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   2   |   1  |       |      |       |      |       | oo1             |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   4  |       |      |       |      |       |      |       | 1111            |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   0  |   3   |   1  |   2   |   3  |   1   |      |       | ooo1oo111o      |
|   1  |       |      |       |      |       |      |       | 1               |
|   0  |   3   |   1  |       |      |       |      |       | ooo1            |
|   0  |   1   |   1  |       |      |       |      |       | o1              |
|   4  |       |      |       |      |       |      |       | 1111            |
|   1  |       |      |       |      |       |      |       | 1               |
|   1  |       |      |       |      |       |      |       | 1               |
+======+=======+======+=======+======+=======+======+=======+=================+

#4

1 1 1 1o1o1 1 1 1 1 1 1o1o1o1o1 1 1 1 1 1 1 1 1 1 1o1o1 1o1o1 1o1o1 1o1o1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1 1 1 1 1 1 1 1 1 1o1o1 1o1o1 1o1o1 1o1o1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1o1o1o1o1o1o1o1o1 1o1o1o1o1o1o1o1o1 1 1 1 1 1 1 1 1 1o1o1 1o1o1 1o1o1 1o1o1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1o1o1o1o1 1o1o1o1o1o1o1o1o1o1o1o1o1 1 1 1 1 1 1 1 1 1o1o1 1o1o1 1o1o1 1o1o1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1

table

+=========+========+===================================+
| repeats | length | sequence                          |
+=========+========+===================================+
|     3   |    1   | 1                                 |
|     1   |    5   | 1o1o1                             |
|     5   |    1   | 1                                 |
|     1   |    9   | 1o1o1o1o1                         |
|     9   |    1   | 1                                 |
|     4   |    5   | 1o1o1                             |
|    16   |    1   | 1                                 |
|     1   |   33   | 1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1 |
|     8   |    1   | 1                                 |
|     4   |    5   | 1o1o1                             |
|    16   |    1   | 1                                 |
|     2   |   17   | 1o1o1o1o1o1o1o1o1                 |
|     8   |    1   | 1                                 |
|     4   |    5   | 1o1o1                             |
|    16   |    1   | 1                                 |
|     1   |    9   | 1o1o1o1o1                         |
|     1   |   25   | 1o1o1o1o1o1o1o1o1o1o1o1o1         |
|     8   |    1   | 1                                 |
|     4   |    5   | 1o1o1                             |
|    16   |    1   | 1                                 |
|     1   |   33   | 1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1 |
+=========+========+===================================+

#6

1o111111ooooo oooooo1o11111ooooo oooooooooooo1o1111ooooo oooooooooooooooooo1o111ooooo oooooooooooooooooooooooo1o11ooooo oooooooooooooooooooooooooooooo1o1ooooo ooooooooooooooooooooooooooooooooooo11o1oooo oooooooooooooooooooooooooooooooooooooooo111o1ooo ooooooooooooooooooooooooooooooooooooooooooooo1111o1oo oooooooooooooooooooooooooooooooooooooooooooooooooo11111o1o ooooooooooooooooooooooooooooooooooooooooooooooooooooooo111111o1

table

+=======+======+=======+======+=======+=================================================================+
| Zeros | Ones | Zeros | Ones | Zeros | sequence                                                        |
+=======+======+=======+======+=======+=================================================================+
|       |   1  |   1   |   6  |   5   | 1o111111ooooo                                                   |
|   6   |   1  |   1   |   5  |   5   | oooooo1o11111ooooo                                              |
|  12   |   1  |   1   |   4  |   5   | oooooooooooo1o1111ooooo                                         |
|  18   |   1  |   1   |   3  |   5   | oooooooooooooooooo1o111ooooo                                    |
|  24   |   1  |   1   |   2  |   5   | oooooooooooooooooooooooo1o11ooooo                               |
|  30   |   1  |   1   |   1  |   5   | oooooooooooooooooooooooooooooo1o1ooooo                          |
|  35   |   2  |   1   |   1  |   4   | ooooooooooooooooooooooooooooooooooo11o1oooo                     |
|  40   |   3  |   1   |   1  |   3   | oooooooooooooooooooooooooooooooooooooooo111o1ooo                |
|  45   |   4  |   1   |   1  |   2   | ooooooooooooooooooooooooooooooooooooooooooooo1111o1oo           |
|  50   |   5  |   1   |   1  |   1   | oooooooooooooooooooooooooooooooooooooooooooooooooo11111o1o      |
|  55   |   6  |   1   |   1  |       | ooooooooooooooooooooooooooooooooooooooooooooooooooooooo111111o1 |
+=======+======+=======+======+=======+=================================================================+

6 revs, 2 users 99% · Accepted Answer · 2017-04-13 12:50:51Z

(Wikified worksheet - feel free to correct or add.)

Comparison to $\sf \small N \scriptsize ONE$derland

The arrows at #1 differ only in that their digits are opposites. Likewise for the pyramids at #4.
Incremental building blocks for pyramids at #4 are the same 1010.
To highlight patterns, this puzzle minimizes the digit o zero. The other puzzle minimizes ı one.
Labyrinths are at different #s. The only difference in their constructions is that the top left spiral here has one more column of digits.

Some more-or-less obvious tabulations and observations made when test-solving this puzzle

  #1.               number                  length

                      1                        1       1  is at #1, #3, #4


  #2.               number                  length

                     111                       3
                   1111111                     7       1111111  is at #2, #3
               111111111111111                15
             1111111111111111111              19
         111111111111111111111111111          27       lengths at #2 are odd
       1111111111111111111111111111111        31
   111111111111111111111111111111111111111    39


  #3.         number         difference     length

                   1                           1       1  is at #1, #3, #4
                1111               +++.        4
             1oo111o            +.....-        7
             1111111             ++...+        7       1111111  is at #2, #3
            1ooo11o1           +---..-.        8
           111ooooo1          +.+..--..        9
          1ooo11ooo1         +---++....       10
     11ooo1oo1o111oo    ++......+-.++.-       15


  #4.               number                  length

                      1                        1       1  is at #1, #3, #4
                    1o1o1                      5
                  1o1o1o1o1                    9
              1o1o1o1o1o1o1o1o1               17       building block at #4 is  1o1o
          1o1o1o1o1o1o1o1o1o1o1o1o1           25        same as at #4 in Nonederland
      1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1o1       33


  #6.         number         difference     length

            1o1ooooo                           8       almost all differences
            11o1oooo            +-+....        8            at #6 are  +-+...
            111o1ooo             +-+...        8
            1111o1oo              +-+..        8
            11111o1o               +-+.        8       many differences are
            111111o1                +-+        8            similar at #6
           1o11ooooo          +-..---+-        9            in Nonederland
          1o111ooooo         +-+.......       10
         1o1111ooooo        +-+........       11
        1o11111ooooo       +-+.........       12
       1o111111ooooo      +-+..........       13

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