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What's the 14-letter answer I'm looking for?


                                                               ((||| - (—— —— |||)), | —— |||||
—— —— —— —— ——  |  ——   —— —— ——  (||||| - (—— —— —— —— |||||))                                ||| —— —— ——, |||, —— ——   —— —— —— ——, | —— ||||| —— —— —— —— ——                                                         
          
    
          

          X X X X X X X X X X
          X X X X X X X X X X
          X X X X X X X X X X
          X X X X X X X X X X
          X X X X X X X X X X
 X X X X X                   X X X X X X X X X X X X X X X X X X X X
 X X X X X                   X X X X X X X X X X X X X X X X X X X X
 X X X X X                   X X X X X X X X X X X X X X X X X X X X
 X X X X X                   X X X X X X X X X X X X X X X X X X X X
 X X X X X                   X X X X X X X X X X X X X X X X X X X X
                 

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    $\begingroup$ From the title and the X's I know the answer. Not sure how the dashes and pipes work though... $\endgroup$
    – caPNCApn
    Jan 28, 2023 at 16:16
  • $\begingroup$ @cap Ah, should've used a more cryptic title. I was actually unsure if I should use that title but hoped it wouldn't make too much sense. The problem with titles sometimes... But keep trying to make sense how they work $\endgroup$ Jan 28, 2023 at 16:29
  • 2
    $\begingroup$ it was the X's I recognized first and the title merely confirmed it. Working on it... $\endgroup$
    – caPNCApn
    Jan 28, 2023 at 17:30

1 Answer 1

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The 14-letter answer is:

EULER'S IDENTITY - the name given to the equation $e^{i\pi}+1=0$ which is considered an example of 'mathematical beauty' (hence the title, which also emphasises 'ID' for 'identity').

The general shape of this can be seen in the arrangement of Xs:

If an individual character is 5 Xs high and wide then we have an arrangement consisting of 1 character, 2 characters at a higher level (superscript), and 4 characters at the same level as the first - exactly as the formula does.

And, in fact, this observation is key to decoding the remainder of the puzzle...

Let's split up the Xs into the seven 5x5 character blocks and assign each row and column within a character block a number from 1 to 5, beginning the numbering from the top-left corner:

Dividing the image into numbered cells with a coordinate system

Now count up the number of bar (|) and dash (—) symbols in any given comma-separated run of symbols, and interpret these as representing 'column number [x]' or 'row number [y]'. Shade these columns or rows. Occasionally, a term appears with a subtraction sign; in these instances, what follows is a combination of bars and dashes suggesting an individual cell that should not be shaded in that column/row.

This may be tricky to understand at first reading, so here's an explanation of each term in the bar/dash sequences:

—— —— —— —— ——  |  ——   —— —— ——  (||||| - (—— —— —— —— |||||))

row5, col1, row1, row3, col5 without r4c5

At the higher level:

((||| - (—— —— |||))

col3 without r2c3

| —— |||||

col1, row1, col5

Returning to 'regular level':

||| —— —— ——

col3, row3

|||

col3

—— ——   —— —— —— ——

row2, row4

| —— ||||| —— —— —— —— ——

col1, row1, col5, row5

Highlighting/shading just the cells indicated here results in the following image:

Euler's identity revealed in the image

In other words, Euler's identity, $e^{i\pi}+1=0$, is revealed for all to see!

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    $\begingroup$ Solved :) Nicely explained and great job! $\endgroup$ Feb 1, 2023 at 14:54

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