Who is missing here?

It seems my last few puzzles have all been far too easy and quickly solved. Let's see how long this one stands against the genius of the combined PSE community.

$$[56\;68\;76\;80\;80\;76\;68\;56]^2 \;\;\;\sum=560$$ $$[26\;38\;52\;52\;52\;52\;38\;26]^2 \;\;\;\sum=336$$ $$[36\;58\;58\;58\;58\;58\;58\;36]^2 \;\;\;\sum=420$$ $$[168\;180\;188\;192\;192\;188\;180\;168]^2 \;\;\;\sum=1456$$ $$[14\forall64] \;\;\;\sum=896$$

If you think you know the answer for the above, don't be too hasty. Before writing your answer, consider this:

$$\between 1\ \&\ 3$$ $$\sec$$ $$\between 2\ \&\ 4$$ $$3 \oplus 4$$ $$\underrightarrow{1}$$ $$\nexists\in\Game\quad\underrightarrow{3}$$

And just to clarify: You are seeking the unique and unambiguously defined answer (one word) to the question

Who is missing here?

• This is what happens when we allow one or two rascals to solve puzzles too quickly. Everyone gets punished. – humn Apr 12 '17 at 19:11
• Punished, you call it? – Gareth McCaughan Apr 12 '17 at 21:12
• Is the change from 5 to 1 in the second block of formulae (or whatever they are) a correction? Or does it just make things easier somehow while leading to the same answer? – Gareth McCaughan Apr 18 '17 at 14:20
• @GarethMcCaughan I think it could be seen both ways. I realized after making the puzzle that I used a correct, but not so common assumption. The puzzle leads to the identical answer with 1 and the "common" assumption or with 5 and the alternative assumption. The second part, however, only makes sense after a break-through from the first. – BmyGuest Apr 18 '17 at 14:30

A “piece” of the puzzle, having only inconclusively considered the second part...

If you think you know the answer for the above, don't be too hasty. Before writing your answer, consider this:   . . . $\require{begingroup} \begingroup \let \BS = \boldsymbol \let \S = \small \let \T = \textsf \def \B { \color{black} } \def \V #1#2#3#4#5#6#7#8{ [ #1 ~ #2 ~ #3 ~ #4 ~ #5 ~ #6 ~ #7 ~ #8 ] }$

Sorry to have missed you,  $\S\BS{ \V {7}{14}{14}{14}{14}{14}{14}{7} \times \V{0}{14}{14}{14}{28}{14}{14}{0} }\raise-1ex\strut$,  perhaps known as  $\S\BS{ \V {7}{ 7}{7}{7}{7}{7}{7}{7} \times \V{0}{16}{8}{8}{8}{8}{8}{0} }$,  but also known as...

...Pawn, the chess piece.

Fortunately the tag helped us find you!   Signed,

$\S\kern32mu \B{ \V{ 56}{ 68}{ 76}{ 80}{ 80}{ 76}{ 68}{ 56}^2 } \kern32mu$ – Bishop, the chess piece
$\S\kern32mu \B{ \V{ 26}{ 38}{ 52}{ 52}{ 52}{ 52}{ 38}{ 26}^2 } \kern32mu$ – Knight,   〃
$\S\kern32mu \B{ \V{ 36}{ 58}{ 58}{ 58}{ 58}{ 58}{ 58}{ 36}^2 } \kern32mu$ – King,     〃
$\S \B{ \V{168}{180}{188}{192}{192}{188}{180}{168}^2 } \kern03mu$ – Queen,   〃
$\S\kern88mu \B{ [ 14 \, \forall \, 64 ] } \kern92mu$ – Rook,   〃

What clicked was that chess is a board game with 6 different pieces, while 5 not-missing players are presented in the puzzle, and that an unobstructed rook can range over some or another 14 squares from ll 64 squares of the board.   To be sure, this puzzle oozes 8×8-ness, so the tag probably would have been unnecessary in the long run.

These formulations represent our...

...attack ability (perhaps move ability) summaries from each rank or file, and ultimately square, of a chess board.

For example, $\S\BS{ \V{56}{68}{76}{80}{80}{76}{68}{56}^2 }$ summarizes the number of squares vulnerable to attack, or that may be moved to, by an unobstructed bishop standing in each of the 64 squares:

 Attack ability of a bishop            Rank         Example of a "9" square
__________________________  totals       __________________________
|                          |      ]      |                          |
|  7  7  7  7  7  7  7  7  |   56        |  .  .  .  .  .  .  .  .  |
|  7  9  9  9  9  9  9  7  |   68        |  1  .  .  .  .  .  .  .  |
|  7  9 11 11 11 11  9  7  |   76        |  .  2  .  .  .  .  .  .  |
|  7  9 11 13 13 11  9  7  |   80        |  .  .  3  .  .  .  .  .  |
|  7  9 11 13 13 11  9  7  |   80        |  .  .  .  4  .  .  .  9  |
|  7  9 11 11 11 11  9  7  |   76        |  .  .  .  .  5  .  8  .  |
|  7  9  9  9  9 [9] 9  7  |   68        |  .  .  .  .  . [B] .  .  |
|  7  7  7  7  7  7  7  7  |   56        |  .  .  .  .  7  .  6  .  |
|__________________________| [           |__________________________|
File
totals  [ 56 68 76 80 80 76 68 56  ]  

As such, you could be formulated  $\S\BS{ \V {7}{14}{14}{14}{14}{14}{14}{7} \times \V{0}{14}{14}{14}{28}{14}{14}{0} }$.

 Attack ability of                                  Examples of "4" (with en
a bottom-homed pawn                   Rank         passant) and "2" squares
__________________________  totals       __________________________
|                          |      ]      |                          |
|  0  0  0  0  0  0  0  0  |    0        |  .  .  .  .  .  .  .  .  |
|  1  2  2  2  2  2  2  1  |   14        |  .  .  .  .  .  .  .  .  |
|  1  2  2  2  2  2  2  1  |   14        |  1  .  2  .  .  .  .  .  |
|  2 [4] 4  4  4  4  4  2  |   28        |  3 [P] 4  .  .  .  .  .  |
|  1  2  2  2  2  2  2  1  |   14        |  .  .  .  .  .  .  .  .  |
|  1  2  2  2  2  2  2  1  |   14        |  .  .  .  .  1  .  2  .  |
|  1  2  2  2  2 [2] 2  1  |   14        |  .  .  .  .  . [P] .  .  |
|  0  0  0  0  0  0  0  0  |    0        |  .  .  .  .  .  .  .  .  |
|__________________________|  [          |__________________________|
File
totals  [  7 14 14 14 14 14 14  7  ]  

Or you could be formulated  $\S\BS{ \V {7}{ 7}{7}{7}{7}{7}{7}{7} \times \V{0}{16}{8}{8}{8}{8}{8}{0} }$.

 Move ability of
a bottom-homed pawn                   Rank         Example of a "2" square
__________________________  totals       __________________________
|                          |      ]      |                          |
|  0  0  0  0  0  0  0  0  |    0        |  .  .  .  .  .  .  .  .  |
|  1  1  1  1  1  1  1  1  |    8        |  .  .  .  .  .  .  .  .  |
|  1  1  1  1  1  1  1  1  |    8        |  .  .  .  .  .  .  .  .  |
|  1  1  1  1  1  1  1  1  |    8        |  .  .  .  .  .  .  .  .  |
|  1  1  1  1  1  1  1  1  |    8        |  .  .  .  .  .  1  .  .  |
|  1  1  1  1  1  1  1  1  |    8        |  .  .  .  .  .  2  .  .  |
|  2  2  2  2  2 [2] 2  2  |   16        |  .  .  .  .  . [P] .  .  |
|  0  0  0  0  0  0  0  0  |    0        |  .  .  .  .  .  .  .  .  |
|__________________________|  [          |__________________________|
File
totals  [  7  7  7  7  7  7  7  7  ]  

The case for  $\S\BS{ \V {7}{14}{14}{14}{14}{14}{14}{7} \times \V{0}{14}{14}{14}{28}{14}{14}{0} \,}$ as your formulation

A pawn’s formulation of  $\S \V{7}{14}{14}{14}{14}{14}{14}{7} \times \V{0}{14}{14}{14}{28}{14}{14}{0} \,$ is based on attacking, which fits the strictest interpretation of king’s  $\S \V{36}{58}{58}{58}{58}{58}{58}{36}^2$.   A king’s formulation based on moving, however, would be $\S \V{36}{58}{58}{58}{\BS{60}}{58}{58}{36} \times \V{\BS{38}}{58}{58}{58}{58}{58}{58}{36} \,$, because castling is an additional move, without a parallel attack posture.

 Move ability of
a bottom-homed king                   Rank     The "7" square includes castling
__________________________  totals       __________________________
|                          |      ]      |                          |
|  3  5  5  5  5  5  5  3  |   36        |  .  .  .  .  .  .  .  .  |
|  5  8  8  8  8  8  8  5  |   58        |  .  .  .  .  .  .  .  .  |
|  5  8  8  8  8  8  8  5  |   58        |  .  .  .  .  .  .  .  .  |
|  5  8  8  8  8  8  8  5  |   58        |  .  .  .  .  .  .  .  .  |
|  5  8  8  8  8  8  8  5  |   58        |  .  .  .  .  .  .  .  .  |
|  5  8  8  8  8  8  8  5  |   58        |  .  .  .  .  .  .  .  .  |
|  5  8  8  8  8  8  8  5  |   58        |  .  .  .  1  2  3  .  .  |
|  3  5  5  5 [7] 5  5  3  |   38        |  .  .  4  5 [K] 6  7  .  |
|__________________________| [           |__________________________|
File
totals  [ 36 58 58 58 60 58 58 36  ]  

Considering, without solving, the second set of clues

What if the first set of clues were enumerated?

$\S \def \L #1#2#3#4{ \B{#1} & \B{#2} & \sf #3 & \T{#4} \\ }{} \begin{array}{c}{} \L { \T{(missing)} \phantom{^2} }{0}{P}{pawn}{} \L { \V{56}{68}{76}{80}{80}{76}{68}{56}^2 }{1}{B}{bishop}{} \L { \V{26}{38}{52}{52}{52}{52}{38}{26}^2 }{2}{N}{knight}{} \L { \V{36}{58}{58}{58}{58}{58}{58}{36}^2 }{3}{K}{king}{} \L { \V{168}{180}{188}{192}{192}{188}{180}{168}^2 }{4}{Q}{queen}{} \L{ [14 \forall 64] \phantom {^2} }{5}{R}{rook}{}\end{array}$
$\S\B{\T{New interpretations of the second set of clues}}$ $\S\B{\T{would be suggested,}}$ $\S\B{\T{without apparent benefit . . .}}$

$\S\begin{array}{c|cc}{} \B { \between 1 ~ \& ~ 3 } &{} \T{between B/bishop & K/king} & \T{CDEFGHIJ / N / queen} ~?{} \\[1ex]{} \B { \sec } &{} \T{secant of nothing / half of "second"("knight")}{} & ~~~~~~~~~~~~~~~ 1 \T{ / "kni"} ~~?{} \\[1ex]{} \B { \between 2 ~ \& ~ 4 } &{} \T{between N/knight & Q/queen)} & ~~ \T{OP / K / bishop} ~~?{} \\[1ex]{} \B { 3 \oplus 4 } &{} \T{K(11)/"king" xor Q(17)/"queen"} & ~~ \T{Z(26) / "egikqu"} ~?{} \\[1ex]{} \B { \underrightarrow{1} } &{} \T{ bishop moves right} ~???{} \\{} \B { \nexists\in\Game\quad \underrightarrow{3} } &{} \T{not an element of the game , king moves right} ~?{}\end{array}$

Incidental interpretations of  $\S\BS{ \V{56}{68}{76}{80}{80}{76}{68}{56}^2 }$ dealt out of play along the way

• Dot product. Too obvious, and more likely written as  $\S | 56 ~ 68 ~ 76 ~ 80 ~ 80 ~ 76 ~ 68 ~ 56 |^2$ anyway.

• Borders of a square (before seeing the clue).   An interesting brush with the true solution’s square formation.

                    56 56 68                                     56 68 76
68    76  ?                                  80    80  ?
76 80 80                                     76 68 56

• Outer product, like a multiplication table. Eerily suggestive in binary.

                    56  |  3136  3808  4256  4480  4480  4256  3808  3136
68  |  3808  4624  5168  5440  5440  5168  4624  3808
76  |  4256  5168  5776  6080  6080  5776  5168  4256
80  |  4480  5440  6080  6400  6400  6080  5440  4480
80  |  4480  5440  6080  6400  6400  6080  5440  4480
76  |  4256  5168  5776  6080  6080  5776  5168  4256
68  |  3808  4624  5168  5440  5440  5168  4624  3808
56  |  3136  3808  4256  4480  4480  4256  3808  3136
|_________________________________________________
56    68    76    80    80    76    68    56

First 4 columns as binary:  ghostly lettering?
0110001000000     0111011100000     1000010100000     1000110000000
0111011100000     1001000010000     1010000110000     1010101000000
1000010100000     1010000110000     1011010010000     1011111000000
1000110000000     1010101000000     1011111000000     1100100000000
1000110000000     1010101000000     1011111000000     1100100000000
1000010100000     1010000110000     1011010010000     1011111000000
0111011100000     1001000010000     1010000110000     1010101000000
0110001000000     0111011100000     1000010100000     1000110000000

Rotated and accentuated:  game pieces?
........          .@@..@@.          .@@..@@.          ........
.@@..@@.          @.@..@.@          @@....@@          ........
@@....@@          @..@@..@          ...@@...          .@@..@@.
.@@@@@@.          @......@          @.@@@@.@          @.@..@.@
...@@...          ...@@...          ...@@...          @@@@@@@@
.@....@.          @@....@@          ..@@@@..          ..@..@..
@@....@@          @.@@@@.@          .@@@@@@.          .@@..@@.
@@....@@          @......@          ........          ...@@...
..@@@@..          .@@@@@@.          @@@@@@@@          @@@@@@@@


$\endgroup$

• Very good! I knew the 'board game' would be a strong clue, but I guess it was needed. Seeing the connections though is still a remarkable feat. But as the puzzle states... PAWN is not the Who who is missing here ;-) Still holding back my check. (Pun indented) – BmyGuest Apr 23 '17 at 23:11
• (But to be clear: Your answer so far is 'exactly' as intended.) – BmyGuest Apr 23 '17 at 23:14
• Found some interesting notation to consider – humn Apr 24 '17 at 3:51
• Wow, love the ASCI art! Seems indeed like a self-referencing hint. If only I had known... great find. – BmyGuest Apr 24 '17 at 6:34
• In case you're interested (in your own puzzle), @BmyGuest, I'm working on a new angle for the second set of clues, based on the title "Who is missing here?" The second set of clues represents many missing pieces indeed – humn May 2 '17 at 6:50

$$\small\begin{array}{clr} \textsf{Clue} & \textsf{Direct interpretation} \\[1ex] \hline [56~68~76~80~80~76~68~56]^2 & \textsf{dot product}~=~ 39872 & 1001101111000000_2 \\ [26~38~52~52~52~52~38~26]^2 & \textsf{dot product}~=~ 15056 & 11101011010000_2 \\ [36~58~58~58~58~58~58~36]^2 & \textsf{dot product}~=~ 22776 & 101100011111000_2 \\ [168~180~188~192~192~188~180~168]^2 & \textsf{dot product} = 265664 & 1000000110111000000_2 \\ [14\forall64] & \scriptsize \left[ \matrix{ 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 \\[-.5ex] 14~14~14~14~14~14~14~14 } \right] \end{array}$$
$$\small\begin{array}{ccr} \textsf{"...consider this"} & \textsf{Direct interpretation} & = ~ ? \\[1ex] \hline \between 1\ \&\ 3 & \textsf{between 1 & 3} & 2 \\ \sec & \textsf{secant of nothing} & 1 \\ \between 2\ \&\ 4 & \textsf{between 2 & 4} & 3 \\ 3 \oplus 4 & \textsf{3 xor 4} & 7 \\ \underrightarrow{1} & ? & \\ \nexists\in\Game\quad \underrightarrow{3} & ? & \end{array}$$