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Question

^{(The puzzle statement now
identifies these numbers as equivalent.)}


                                                         O O
                                                       O O O O
                                                     O O O O
                                     O             O O O O
                                     O           O O O O
                                     O O O     O O O O
                                     O O O O O O O O
                                     O O O O O O O
                                     O O O O O O
                                     O O O O O O O O
                                     O O O O O O O O O O

#1. &hairsp; You are (now)&hairsp;here

                                                               ıııııı
                                                            Oıııııı
                                                         ıııııııı         ıııııı
                                                       Oıııııııı     Oıııııııı
                                                     ıııııııııı  ıııııııııı
                                                   OııııııııııOıııııııııı
                                                 ıııııııııOOııııııııııı
                                               OııııııOOıııııııııııııı
                           ıııııııııı        ıııııOOııııııııııııııııı
                        ııııııııııOOOOOO   ıııOOııııııııııııııııııı
                       ıııOıııııOOOOOOOOOOıOııııııııııııııııııııı
                       ıııııııOOOOOOOOOOOOOOııııııııııııııııııı
                       ııııOOOOOOOOOOOOOOOOOOOıııııııııııııııı
                       OO      OOOOOOOOOOOOOOOOOıııııııııııı
                      O          OOOOOOOOOOOOOOOOOıııııııı
                     O            OOOOOOOOOOOOOOOOOOıııı
                    O              OOOOOOOOOOOOOOOOOOııO
                                    OOOOOOOOOOOOOOOOOOOO
                                      OOOOOOOOOOOOOOOOOOO
                                         OOOOOOOOOOOOOOOOO
                                             OOOOOOOıOıOOıOı
                                                 ıOOıOıOOıOıOOı
                                                ıOOıOOıOıOOıOıOOOıı
                                                     ıOııOıOOıı
                                                         ıOııOıOıı

#2. &hairsp; Hummingbirds don’t know the words

             ıııııııııııııııııııııııııııııı
             ııı                        ııı
             ııı   ıııııııııııııııııı   ııııııııııııııııııııııııııı
             ııı   ııı            ııı   ııı                     ııı
             ııı   ııı   ıııııı   ııı   ııı   ııııııııııııııı   ııı
             ııı   ııı      ııı   ııı   ııı   ııı         ııı   ııı
             ııı   ıııııı   ııı   ııı   ııı   ııı   ııı   ııı   ııı
             ııı            ııı   ııı   ııı         ııı   ııı   ııı
             ıııııııııııııııııı   ııı   ııııııııııııııı   ııı   ııı
                                  ııı                     ııı   ııı
                      ııııııııııııııııııııııııııııııııııııııı   ııı
                      ııı                                       ııı
                      ııı   ııııııııııııııııııııııııııııııııııııııı
                      ııı   ııı
                      ııı   ııı   ııııııııııııııı
                      ııı   ııı   ııı         ııı
                      ııı   ııı   ııı   ııı   ııı
                      ııı   ııı         ııı   ııı
                      ııı   ııııııııııııııı   ııı
                      ııı                     ııı
                      ııııııııııııııııııııııııııı

#3. &hairsp; Through the labyrinth we unwind

                                                O
                                               O O
                                              OıOıO
                                             O O O O
                                    O       OıOıOıOıO       O
                                   O O     O O     O O     O O
                                  OıOıO   OıOıO   OıOıO   OıOıO
                                 O O O O O O O O O O O O O O O O
                                OıOıOıOıOıOıOıOıOıOıOıOıOıOıOıOıO
                               O O     O O             O O     O O
                              OıOıO   OıOıO           OıOıO   OıOıO
                             O O O O O O O O         O O O O O O O O
                            OıOıOıOıOıOıOıOıO       OıOıOıOıOıOıOıOıO
                           O O             O O     O O     O O
                          OıOıO           OıOıO   OıOıO   OıOıO
                         O O O O         O O O O O O O O O O O O
                        OıOıOıOıO       OıOıOıOıOıOıOıOıOıOıOıOıO
                       O O     O O     O O     O O
                      OıOıO   OıOıO   OıOıO   OıOıO
                     O O O O O O O O O O O O O O O O
                    OıOıOıOıOıOıOıOıOıOıOıOıOıOıOıOıO

#4. &hairsp; $\sf \small N \scriptsize ONE$ders of the ancient world

Obviously (?) the $\sf\scriptsize ONE$-and$\small/$or-$\sf\scriptsize NONE$ders here represent patterns of ~~mathematical constructs~~ equivalent numbers.

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

.
.               ııııııııııOııııııııııOııııııııııOııııııııııOııııııııııOıııııııııı
.
.                      ıııııııııOıııııııııOıııııııııOıııııııııOıııııııııOııııııııı
.
.                             ııııııııOııııııııOııııııııOııııııııOııııııııOıııııııı
.
.                                    ıııııııOıııııııOıııııııOıııııııOıııııııOııııııı
.
.                                           ııııııOııııııOııııııOııııııOııııııOıııııı
.
.                                                  ııOııııııııOııııııııOıııııOııOııııı
.
.                                                         ııııOııııOııııOııııOııııOıııı
.
.                                                                ıııOıııOıııOıııOıııOııı
.
.                                                                       ııOııOııOııOııOıı
.
.                                                                              ıOıOıOıOıOı
.
.                                                                                         O
.                                                                             ıııııOOOOOOı
.                                                                       OOOOOOıııOıOOOOOı
.                                                                 OOOOOOOOOOOOııOOıOOOOı
.                                                           OOOOOOOOOOOOOOOOOOıOOOıOOOı
.                                                     OOOOOOOOOOOOOOOOOOOOOıOOıOOOıOOı
.                                               OOOOOOOOOOOOOOOOOOOOOOOOOOııOOıOOOıOı
.                                         OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıııOOıOOOıı
.                                   OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıOııOOıOOOı
.                             OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıOOııOOıOOO
.                       OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıOıOııOOıOOO
.                 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıOııOııOOıOOO
.           OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıOıııOııOOıOOO
.     OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOıOııııOııOOıOOO

#6. &hairsp; Destination unknown

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 ı ones are pertinent. Two-dimensional shapes and surrounding words are just gratuitous embellishments. If you are still nowhere after considering all this, why not visit $\sf \small O \scriptsize NE$derland for comparison?

score 7 · Accepted Answer · answered Oct 05 '16 at 00:26

7

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

representations of the number $0$ 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

ııııı

would fit in slot 5, but not any other.

answered Oct 05 '16 at 00:26

Deusovi

146,248
16
519
609

$\color{#0c0}{\LARGE\raise-.1ex\checkmark}\kern-.5em$ with a $\large\bf\raise-.2ex?\small{:}~$ How did you realize the answer$\large\raise-.2ex? ~~$ Also with a $\small\bf(~),{:}$ $\small($The numbers have a purely geometric explanation as well, naturally related to your explanation.$\small)$ – humn Oct 06 '16 at 07:17
(Well, actually, the "purely geometric" explanation includes vectors.) – humn Oct 06 '16 at 08:25
1

@humn: You probably remember I had the core idea of numbers in different bases a while back - I only realized yesterday that the bases might be changing. Then I saw that picture 3's lengths were always divisible by 3. That, combined with the name, gave it to me. (And I actually checked my answers using the vector interpretation and a piece of paper.) – Deusovi Oct 06 '16 at 13:33

score 3 · Answer 2 · edited Apr 13 '17 at 12:50

(Wikified worksheet - feel free to correct or add.)
(Also a gradually unfolding solution by the poser.)

Comparison to $\sf \small O \scriptsize NE$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 ı one. The other puzzle minimizes o zero.
Labyrinths are at different #s. The only difference in their constructions is that the top left spiral here has one fewer column of digits.

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

  #2.                            alternate      alternate
                number              digits      ı  counts

                     O                   O      0       O  is at #1, #2, #4, #6

                   ııO                 ı O      1
                                        ı       1

                  ıııı                ı ı       2
                                       ı ı      2

                ıııııı              ı ı ı       3       ıııııı  is at #2, #3
                                     ı ı ı      3

              ıOıOOıOı            ı ı O O       2
                                   O O ı ı      2

              ıııııııı            ı ı ı ı       4
                                   ı ı ı ı      4

             ıOııOıOıı           ı ı O O ı      3
                                  O ı ı ı       3

            ıOııOıOOıı          ı ı O O ı       3
                                 O ı ı O ı      3

            ıııııııııı          ı ı ı ı ı       5
                                 ı ı ı ı ı      5

          ıııııııııııı        ı ı ı ı ı ı       6
                               ı ı ı ı ı ı      6

        ıOOıOıOOıOıOOı      ı O O O ı ı O       3
                             O ı ı O O O ı      3

      ııııııııııOOOOOO    ı ı ı ı ı O O O       5
                           ı ı ı ı ı O O O      5

                       ı O O ı ı O O O O ı      4
                        O ı O O O ı ı O ı       4

                     ı ı ı ı ı O ı ı ı ı ı     10
                      ı ı ı ı ı ı ı ı ı ı      10

                    ı ı ı O ı ı ı ı ı ı ı      10
                     ı ı ı O ı ı ı ı ı ı ı     10

                    ı ı ı ı ı O ı ı ı ı ı      10
                     ı ı ı ı O ı ı ı ı ı ı     10

                  ı ı O ı ı ı ı ı ı ı ı ı      11
                   ı O ı ı ı ı ı ı ı ı ı ı     11

                  ı ı ı O ı ı ı ı ı ı ı ı      11
                   ı ı O ı ı ı ı ı ı ı ı ı     11

   ı ı O O O O O O O O O O ı ı ı ı ı ı ı ı     10
    ı ı O O O O O O O O O ı ı ı ı ı ı ı ı      10

  ı ı ı ı O O O O O O O ı ı ı ı ı ı ı ı ı      13
   ı ı ı O O O O O O O ı ı ı ı ı ı ı ı ı ı     13

ı ı ı ı ı O O O O O O ı ı ı ı ı ı ı ı ı ı      15
 ı O ı ı O O O O O ı ı ı ı ı ı ı ı ı ı ı ı     15

  #3.                 number                 length
                                                        O  would fit in at #3
                       ııı                      3
                     ıııııı                     6       ıııııı  is at #2, #3
                ııııııııııııııı                15
              ıııııııııııııııııı               18       building block at #3 is  ııı
         ııııııııııııııııııııııııııı           27
       ıııııııııııııııııııııııııııııı          30
  ııııııııııııııııııııııııııııııııııııııı      39


  #4.                 number                 length

                        O                       1       O  is at #1, #2, #4, #6
                      ıOıO                      4
                    ıOıOıOıO                    8
                ıOıOıOıOıOıOıOıO               16       building block at #4 is  ıOıO
            ıOıOıOıOıOıOıOıOıOıOıOıO           24        same as at #4 in Onederland
        ıOıOıOıOıOıOıOıOıOıOıOıOıOıOıOıO       32

  #6.                                                      number   length                                                            difference

                                                                O      1       O  is at #1, #2, #4, #6
                                                        ıOOOıOOOı      9                                                               +...+...+
                                                       ııOOıOOOOı     10                                                              +...+-....
                                                      ıOııOOıOOOı     11                                                             +-.+.-+....
                                                      ıOOıOOOıOOı     11                                                               -...-+...
                                                      ııOOıOOOıOı     11                                                              +.-+..-+..
                                                      ıııOOıOOOıı     11                                                               +.-+..-+.
                                                      ıııOıOOOOOı     11                                                                 +-...+.
                                                     ıııııOOOOOOı     12                                                            +...+-......
                                                      ıOıOıOıOıOı     11                                                            -.-.-+.+.+..
                                                      ıOOııOOıOOO     11                                                               -+..-+-.-
                                                     ıOıOııOOıOOO     12       similar  +-+...                                      +-+.........
                                                    ıOııOııOOıOOO     13       differences                                         +-+..........
                                                   ıOıııOııOOıOOO     14       pervade #6                                         +-+...........
                                                  ıOııııOııOOıOOO     15       in Onederland                                     +-+............
                                                ııOııOııOııOııOıı     17                                                       ++-+.-......+..++
                                          ıııOıııOıııOıııOıııOııı     23                                                 +++.++.-+.....+-.+.-+..
                                    ııııOııııOııııOııııOııııOıııı     29                                           ++++.+.......+-..+.-.+..-+...
                              ııOııııııııOııııııııOıııııOııOııııı     35       pattern anomaly               ++.+++....+-...+.........+-..-+....
                        ııııııOııııııOııııııOııııııOııııııOıııııı     41                               ++++++-.+....-...+..-.....+-....+.-+.....
                  ıııııııOıııııııOıııııııOıııııııOıııııııOııııııı     47                         ++++++.-....+..-...+...-..+....-.+.....-+......
            ııııııııOııııııııOııııııııOııııııııOııııııııOıııııııı     53                   ++++++..-....+...-...+....-..+.....-.+......-+.......
      ıııııııııOıııııııııOıııııııııOıııııııııOıııııııııOııııııııı     59             ++++++...-....+....-...+.....-..+......-.+.......-+........
ııııııııııOııııııııııOııııııııııOııııııııııOııııııııııOıııııııııı     65       ++++++....-....+.....-...+......-..+.......-.+........-+.........

    All wrap-by-6            NUMBERS WRAPPED INTO 6-DIGIT COLUMNS AND TOTALED
    totals reduce
   to sums of five            OOOOıO-
   building blocks            ıııOıı-   OıOııO-   ıOıOıı-             Oııııı-
  -----------------           OOıOOO    OıOOOı    OOıOOO              ıııııO-
   010101   001001            ------    ------    ------              ıııııı-
                              112021    020111    102011              ııııOı-
   101010   010010                                          Oııııı-   ıııııı-
                                                            ıOıııı-   ıııOıı-
            100100                                OııııO-   ııOııı-   ıııııı-
                                                  ııııOı-   ıııOıı-   ııOııı-
                                                  ıııOıı-   ııııOı-   ıııııı-
                              ıııııO-   OıııOO-   ııOııı-   ıııııO-   ıOıııı-
                              OOOOOı    ıOOOıı    ıOıııı    ıııııı    ıııııı
                              ------    ------    ------    ------    ------
                              111111    111111    444444    666666    aaaaaa


                                                                      Oııııı-
                              OııOOı-   OıııOı-                       ııııOı-
                              OOOıOı    OOOOOı              Oııııı-   ıııııı-
                              ------    ------              ııOııı-   ııOııı-
                              011102    011102              ııııOı-   ıııııı-
                                                            ıııııı-   Oııııı-
                                                  OıııOı-   Oııııı-   ııııOı-
                                                  ııOııı-   ııOııı-   ıııııı-
                              OOOıOO-   OıOıOı-   OıııOı-   ııııOı-   ııOııı-
                              OıOOOı    OıOıOı    ııOııı    ıııııı    ıııııı
                              ------    ------    ------    ------    ------
                              010101    020202    242424    686868    8a8a8a

                    OOOOOı-
                    OııOıı-   OOııOO-                                 Oııııı-
                    OOıOOO    ıOOOOı                                  ıııOıı-
                    ------    ------                                  ıııııı-
                    012012    101101                        OııOıı-   Oııııı-
                                                            ıııııı-   ıııOıı-
                                                            Oııııı-   ıııııı-
                                        OııOıı-   OOOıOı-   ıııOıı-   Oııııı-
                    OıOOıO-   OıOOıı-   OııOıı-   ıııOıı-   ıııOıı-   ıııOıı-
                    OOıOOı    OOıOOO    OııOıı    OOıOOO    Oııııı    ıııııı
                    ------    ------    ------    ------    ------    ------
                    011011    011011    033033    112112    366366    688688

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