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Fix the image before completing the sequence. What do I want you to draw your attention to?


103, 91, 79, 0, x, x, 28, x, x, x, 0, 78, x, x, x, x, x, x, 5

enter image description here


Hints:

You need to place 11 new matches into the image to make it correct. The image represents something well-known that's needed to understand the sequence.

16x8, overlapped

Some weird patterns, huh?... The center matchstick of each 1x2 rectangle seems to always repeat in the pattern 01010101. Isn't it peculiar that halfway down the image, there's always a straight horizontal line of matchsticks across the entire width? What are the dimensions of the grid again? You can also look at the columns for potential hints, it's not like they're patternless. Take a close look at the raw patterns found in Lypyrhythm's partial answer. Anything weird you notice there?
Always keep in mind the 1x2 rectangles have overlapping edges...

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3
  • $\begingroup$ Does the image itself represent a number or sequence? $\endgroup$ Nov 25 at 13:59
  • $\begingroup$ @sarsaparilla Number? No. Sequence? Yes, a very simple one. $\endgroup$ Nov 25 at 14:47
  • $\begingroup$ @sarsaparilla To clarify, the image contains a sequence which is using all of the matches. What it's supposed to get across is more along the lines of "the method used to represent the sequence with the matches" $\endgroup$ Nov 25 at 16:13
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+50
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Lukas wants to draw our attention to …

… the PSE Advent Calendar.

The completed sequence is:

103, 91, 79, 0, 119, 61, 28, 79, 21, 15, 0, 78, 119, 14, 79, 21, 61, 119, 5

The matches represent …

… the numbers from 0 to 127 in binary form. Each 1×2 “box” has seven possible positions for matches. These positions are bits in a binary representation of a seven-bit number. Bit 0 is the centre match, bits 1 through 6 are anticlockwise starting from the top left upright match.

These numbers are arranged in a straightforward fashion: Row-wise starting from 0 in the top left corner to 127 in the bottom right corner. The boxes overlap, so that a match on the border could be a bit that’s set on either of the neighbours or on both.

     -- 6 --
    |       |             (1 << 0) == 1
    1       5             (1 << 1) == 2
    |       |             (1 << 2) == 4
     -- 0 --              (1 << 3) == 8
    |       |             (1 << 4) == 16
    2       4             (1 << 5) == 32
    |       |             (1 << 6) == 64
     -- 3 --

And the sequence?

The seven-match boxes are, of course, seven-segment displays in disguise. You can assemble the bits of the numbers or look up the index in the match pattern. (But be careful with overlapping matches.) The numbers read:

 103  91  79   0   x   x  28   x   x   x   0  78   x   x   x   x   x   x   5
  _   _   _       _   _       _   _   _       _   _   _   _   _   _   _
 |_| |_  |_      |_| |_|     |_| |_| |_|     |   |_| |_| |_| |_| |_| |_|  _
 |    _| |_      |_| |_| |_| |_| |_| |_|     |_  |_| |_| |_| |_| |_| |_| |

The missing matches …

… had been identified by Lypyrhythm based on the regular pattern of the binary boxes. Now that we know what the boxes are, we can verify that finding.

Eleven machsticks are missing and eleven numbers in the sequence are missing. If we look at the missing matches, we can see that they all belong to letter shapes. (There are some minor ambiguities: The stem of the d could belong to a U and the stem of the A could be part of an N. These disambiguities are resolved by the fact that u/v and n are already represented unambiguously by lower-case seven-stick symbols.)

Below are the letters drawn out, i.e. no longer overlapped, and with the missing sticks highlighted with ugly DOS-blocks. (Sorry for that.)

    _ |  |_     _ |  |_     _ |  |_     _ |  ▐_
            |  |  |  |   _  _  _  _ |_ |_ ▐_ |_

    _ |  |_     ▄ |  |_     _ |  |_     _ |  |_
  |  |  |  || |▐ || || | _| _| _| _||_||_||_||_|

  | _|| ||_|  | _|| ||_|  | _|| ||_|  | _|| ||_|
            |  |  |  |   _  _  _  _ |_ |_ |_ |_

  | _|| ||_|  | _|| ||_|  | _|| ||_|  | ▄▌| ||_|
  |  |  |  || || || || | _| _| _| _||_||_||_||_|
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
    _ |  |_     _ |  |_     _ |  |_     _ |  ▐▄
            |  |  |  |   _  _  _  _ |_ |_ |_ |_
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
    _ |  |_     _ |  |_     _ |  |_     _ |  |_
  |  |  |  || || || || | _| _| _| _||_||_||_||_|
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
  | _|| ||_|  | _|| ||_|  | _|| ||_|  | _|| ||_|
            |  |  |  |   _  _  _  _ |_ |_ |_ |_
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
  | _|| ||_|  | _|| |▐▄|  | _|| ||_|  | _|| ||_|
  |  |  |  || || || || ▌ _| _| _| _||_||_||_||_|

We can fill in the blanks:

The x's are:

    3 × A   2 × E   2 × d   2 × n   1 × L   1 × t

and we get:

 103  91  79   0 119  61  28  79  21  15   0  78 119  14  79  21  61 119   5
  _   _   _       _           _               _   _       _           _
 |_| |_  |_      |_|  _|     |_   _  |_      |   |_| |   |_   _   _| |_|  _
 |    _| |_      | | |_| |_| |_  | | |_      |_  | | |_  |_  | | |_| | | |

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12
  • $\begingroup$ The 11 missing matches also come into play for the final sequence. Once you've included that the green checkmark will come :) $\endgroup$ Nov 26 at 21:36
  • $\begingroup$ Do you mean “clockwise starting from the top right upright match”? Following your instructions, I get a “ꟼ” instead of a “P”. $\endgroup$
    – Lypyrhythm
    Nov 27 at 10:01
  • $\begingroup$ @LukasRotter: You mean it's no coincidence that there are 11 missing matches and 11 x's? I've updated the answer. $\endgroup$
    – M Oehm
    Nov 27 at 10:26
  • $\begingroup$ @Lypyrhythm: No, I mean aniclockwise from the top left. 103 + 8 (bottom) + 16 (se) are the full 127. The 8 and 16 are missing from the "8" to get a "P". I've included a sketch. $\endgroup$
    – M Oehm
    Nov 27 at 10:28
  • $\begingroup$ (I apologize for reducing the matchstick aesthetic of the puzzle to rather rough text sketches. My only excuse is that I might have developed a mild case of spirtophobia while editing tha answer.) $\endgroup$
    – M Oehm
    Nov 27 at 10:30
4
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Partial answer

Look at the matchsticks one by one from left to right – you should realize they are placed in a particular order. As you can see in the image below, the pattern in the first row is:

  • blank, blank, stick, stick, blank, blank, stick, stick, etc.

The second row consists of horizontal matchsticks placed in the following sequence:

  • blank, stick, blank, stick, blank, stick, etc.

Do this for the remaining 30 rows of matchsticks. If you notice there is a gap where there is supposed to be a matchstick, insert one to correct the pattern. They should add up to exactly 11 new matchsticks (1 = matchstick, 0 = gap):

  1. 001100110011001(1)0
  2. 0101010101010101
  3. 00001111000011(1)10
  4. 0000000011111111
  5. 00110011001100110
  6. 01010(1)0101010101
  7. 1111(1)11111111111
  8. 0000000011111111
  9. 1111111111111111
  10. 0101010101010101
  11. 00001111000011110
  12. 0000000011111111
  13. 1111111111111(1)11
  14. 0101010101010(1)01
  15. 1111111111111111
  16. 1111111111111111
  17. 001100110011001(1)0
  18. 010101010101010(1)
  19. 00001111000011110
  20. 1111111111111111
  21. 00110011001100110
  22. 0101010101010101
  23. 1111111111111111
  24. 1111111111111111
  25. 1111111111111111
  26. 0101010101010101
  27. 00001111000011110
  28. 1111111111111111
  29. 111111(1)111111111
  30. 0101010(1)01010101
  31. 1111111(1)11111111
  32. 0000000011111111

The 11 matchsticks to add are highlighted green:

matchsticks solution

I am still trying to figure out what the corrected image is supposed to represent. It somehow reminds me of a rot13(sybbe cyna bs n (jryy-xabja?) ohvyqvat).

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5
  • $\begingroup$ rot13(Gung'f gur pbeerpg vzntr, tbbq wbo! Guvax nobhg jung gur frpbaq uvag pbhyq zrna (16k8 tevq jvgu bireyncf) naq ybbx ng gur cnggreaf lbh sbhaq ntnva. Vg unf abguvat gb qb jvgu ohvyqvatf.) $\endgroup$ Nov 25 at 19:45
  • $\begingroup$ (Reworded comment) Which parts of the diagram have you overlaid on each other to get a 16x8 image? (You haven't said...) Also, I'm finding it quite hard to follow your first spoiler as it stands - what do each of the four separate comma-separated runs of binary in each 'row' represent in themselves? (i.e. which matches are you counting for each?) Thanks! $\endgroup$
    – Stiv
    Nov 26 at 10:38
  • $\begingroup$ @Stiv FWIW, I assume they didn't overlay anything in the image on each other. The could have used the image I've provided in my second hint, which gives a 16x8 grid of 1x2 rectangles. For each of the 8 rows there, the four binary sequences would be "vertical sticks on upper part of the rects", "horizontal sticks in middle of the rects", "vertical sticks on lower part of the rects", "horizontal sticks on lower part of the rects" $\endgroup$ Nov 26 at 11:13
  • $\begingroup$ @LukasRotter Ah, okay - so here each 'row' actually refers to each double row in the original diagram. It would probably still be useful for Lypyrhythm to mention this explicitly (and what the counts represent), because as far as a reader is aware at a glance all we have to go on is the original image. Right, now to make sense of the rest of the puzzle, people! $\endgroup$
    – Stiv
    Nov 26 at 11:43
  • $\begingroup$ I just clarified the explanation of my answer. Is it better now? $\endgroup$
    – Lypyrhythm
    Nov 26 at 19:25

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