# The Mystery of the MathJax Lines

Observed from an old-timey spaceship1 and strewn across the Puzcle Plateau 2 lie 14 mysterious lines.

$\rlap{\kern 0 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 1 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 2 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} % \rlap{\kern 3 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 4 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 5 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 6 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} % \rlap{\kern 7 pc\raise 0 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 1 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 2 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 3 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 4 pc\Rule{3.1pc}{0.1pc}{0pc}} % \rlap{\kern 6 pc\raise 5 pc\Rule{4.1pc}{0.1pc}{0pc}} \rlap{\kern 6 pc\raise 6 pc\Rule{4.1pc}{0.1pc}{0pc}}$

On-scene computerological assays determined that these lines were made with 100% pure MathJax.

$\rlap{\kern 0 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 1 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 2 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} % \rlap{\kern 3 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 4 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 5 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 6 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} % \rlap{\kern 7 pc\raise 0 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 1 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 2 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 3 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 7 pc\raise 4 pc\Rule{3.1pc}{0.1pc}{0pc}} % \rlap{\kern 6 pc\raise 5 pc\Rule{4.1pc}{0.1pc}{0pc}} \rlap{\kern 6 pc\raise 6 pc\Rule{4.1pc}{0.1pc}{0pc}}$
|            |
shift right     shift up


Seems the ancients (and you) could translate these 14 lines (shift each of them left/right/up/down, no rotation) by making single-digit substitutions in the two space-separated number columns. The left column specifies sideways shifts while the right column specifies upward shifts.

Sacrosanct $\small\texttt{\Rule}$ portions within these lines define their fixed orientations and lengths, and thus are never altered. A 4-unit-long horizontal line, for instance, is “ruled” by $\small\texttt{\Rule\{}$4$\small\texttt{.1pc\}{0.1pc}}\,$ while a 3-unit-long vertical line is ruled by $\small\texttt{\Rule{0.1pc}\{}$3$\small\texttt{.1pc\}}\,$.

A celebrity television documentarian breathlessly breaks the news of speculation that these lines were meant to assemble into an ancient type of puzzle, with a unique solution. This was suggested by a find on the nearby Game Plain, where four similar lines...
$\kern1pc\rlap{\kern 4 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 5 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} % \rlap{\kern 0 pc\raise 0 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 0 pc\raise 1 pc\Rule{3.1pc}{0.1pc}{0pc}}$ $\kern6pc\small \raise2pc\matrix{ \texttt{\$}\hfil \\[-.5ex] \texttt{\rlap{\kern 4 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}}}\\[-.5ex] \texttt{\rlap{\kern 5 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}}}\\[-.5ex] \texttt{%}\hfil \\[-.5ex] \texttt{\rlap{\kern 0 pc\raise 0 pc\Rule{3.1pc}{0.1pc}{0pc}}}\\[-.5ex] \texttt{\rlap{\kern 0 pc\raise 1 pc\Rule{3.1pc}{0.1pc}{0pc}}}\\[-.5ex] \texttt{\$}\hfil \\[-.5ex] }$

...were one morning discovered in a tic-tac-toe reconfiguration. (Not specifically a puzzle there.)
$\kern1pc\rlap{\kern 1 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 2 pc\raise 0 pc\Rule{0.1pc}{3.1pc}{0pc}} % \rlap{\kern 0 pc\raise 2 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 0 pc\raise 1 pc\Rule{3.1pc}{0.1pc}{0pc}}$ $\kern6pc\small \raise2.3pc\matrix{ \texttt{\$}\hfil \\[-.5ex] \texttt{\rlap\{\kern } \;\boxed{ \tt 1 }\; \texttt{ pc\raise }\;~ { \tt 0 }\;~ \texttt{ pc\Rule{0.1pc}{3.1pc}{0pc}\}}\\[-.5ex] \texttt{\rlap\{\kern } \;\boxed{ \tt 2 }\; \texttt{ pc\raise }\;~ { \tt 0 }\;~ \texttt{ pc\Rule{0.1pc}{3.1pc}{0pc}\}}\\[-.5ex] \texttt{%}\hfil \\[-.5ex] \texttt{\rlap\{\kern } \;~ { \tt 0 }\;~ \texttt{ pc\raise }\;\boxed{ \tt 2 }\; \texttt{ pc\Rule{3.1pc}{0.1pc}{0pc}\}}\\[-.5ex] \texttt{\rlap\{\kern } \;~ { \tt 0 }\;~ \texttt{ pc\raise }\;~ { \tt 1 }\;~ \texttt{ pc\Rule{3.1pc}{0.1pc}{0pc}\}}\\[-.5ex] \texttt{\$}\hfil \\[-.5ex] }$

Here is your chance to lend a shred of credibility to the story:

Translate the original 14 lines by making single-digit substitutions within the two space-separated number columns in order to form a familiar type of puzzle with a unique solution.

Lines should touch only at intersections and end points. A complete answer will actively render the solution with 14 MathJax lines, correspondingly edited, but incomplete answers are welcome too. These answers cannot be hidden in spoilers due to the present mechanics of answer processing. No contrived clues this time, just some gratuitous liberties:  1Old-timey spaceship. 2Puzcle Plateau.

• Additional notes: If you work this out on graph paper, remember that the coordinates of left and bottom ends of lines can only range from 0 through 9. I tried to make things easy for experimenting with columns' numbers while pretending to edit a question (or an answer, but answers sometimes get submitted by accident before they're ready), with no need to really understand the MathJax commands.
– humn
Sep 22, 2016 at 10:07

The lines can be rearranged to form a

maze.

The coordinates required for the lines (in the order provided in the MathJax) are:

Vertical lines:
1, 2
2, 3
3, 3
0, 1
0, 5
4, 0
4, 4

Horizontal lines:
0, 1
1, 2
0, 6
1, 7
0, 8
0, 0
0, 9

Which generates the following (sorry, can't spoilerize it):

$\rlap{\kern 1 pc\raise 2 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 2 pc\raise 3 pc\Rule{0.1pc}{3.1pc}{0pc}} \rlap{\kern 3 pc\raise 3 pc\Rule{0.1pc}{3.1pc}{0pc}} % \rlap{\kern 0 pc\raise 1 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 0 pc\raise 5 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 4 pc\raise 0 pc\Rule{0.1pc}{4.1pc}{0pc}} \rlap{\kern 4 pc\raise 4 pc\Rule{0.1pc}{4.1pc}{0pc}} % \rlap{\kern 0 pc\raise 1 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 1 pc\raise 2 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 0 pc\raise 6 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 1 pc\raise 7 pc\Rule{3.1pc}{0.1pc}{0pc}} \rlap{\kern 0 pc\raise 8 pc\Rule{3.1pc}{0.1pc}{0pc}} % \rlap{\kern 0 pc\raise 0 pc\Rule{4.1pc}{0.1pc}{0pc}} \rlap{\kern 0 pc\raise 9 pc\Rule{4.1pc}{0.1pc}{0pc}}$

• Let's call the film crew so they can use this in their documentary. Good point about spoilers, which will be added to the puzzle statement. If you'd like another moment in the playpen, what if lines weren't allowed to go end to end? That spec was taken out of the statement for the sake of (at least some) brevity, but it did seem to force a silly solution. You could add it to this solution as a side note and recapture already-deserved attention after the statement is revised. (I'm also curious to know if you used anything but MathJax to figure this out.)
– humn
Sep 22, 2016 at 18:44
• @humn I used only MathJax to figure this out. I just kind of randomly changed the numbers around until I happened to hit something that vaguely resembled this type of puzzle. Once I knew what kind of puzzle I was trying to construct, I went at it a bit more systematically. It was difficult to make everything fit correctly, but eventually I got it. I'm sure there are multiple correct solutions. Sep 22, 2016 at 19:14
• @humn If you're suggesting trying the puzzle again, with the added constraint, I don't think I'm up for it just now. This one took a lot of fiddling to figure out. I would suggest posting that variant as a separate question, and see if someone else is able to do it. (Did your original solution meet this additional constraint?) Sep 22, 2016 at 19:16
• Nice work, thanks for the backstory, no push to get silly. Yes, I came up with a pathologically silly solution first, because long lines made the form almost leap out with no mystery. Didn't fool you, though. Thanks for the encouragement to tag on a follow-up puzzle, though it'll need some compulsive testing to see if it really does force a different-enough solution.
– humn
Sep 22, 2016 at 19:21
• @humn No worries! :) Sep 22, 2016 at 23:09