Part 1 of 3
2xN Optimal Domination Numbers


D(2,N) = the domination number for a board with 2 rows and N columns.

D(2,1) = 2
------------
D(2,2) = 4
D(2,3) = 4
D(2,4) = 4
D(2,5) = 4
D(2,6) = 4
D(2,7) = 6
------------
D(2,N+6) = D(2,N) + 4  for N >= 2



Part 2 of 3
2xN Sufficiency Proofs


Sufficiency is proved by showing a feasible solution
using the domination number of knights.

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D(2x1) <= 2

x
x

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Extendable solution by repeating the first six columns
D(2x2) <= 4    D(2,6k+2) <= 4k+4

x x            - - x x - - x x
x x            - - x x - - x x
               \---------/
                 repeats

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Extendable solution by repeating the first six columns
D(2,3) <= 4    D(2,6k+3) <= 4k+4

- x x          - - x x - - - x x
- x x          - - x x - - - x x
               \---------/
                 repeats

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Extendable solution by repeating the first six columns
D(2,4) <= 4    D(2,6k+4) <= 4k+4

- - x x        - - x x - - - - x x
- - x x        - - x x - - - - x x
               \---------/
                 repeats

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Extendable solution by repeating the first six columns
D(2,5) <= 4     D(2,6k+5) <= 4k+4

- - x x -       - - x x - - - - x x -
- - x x -       - - x x - - - - x x -
                \---------/
                  repeats

________________________________________________________
Extendable solution by repeating the first six columns
D(2,6) <= 4    D(2,6k+6) <= 4k+4

- - x x - -    - - x x - - - - x x - -
- - x x - -    - - x x - - - - x x - -
               \---------/
                 repeats

________________________________________________________
Extendable solution by repeating the first six columns
D(2,7) <= 6       D(2,6k+1) <= 4k+2

- - x x x - -     - - x x - - - - x x x - -
- - x x x - -     - - x x - - - - x x x - -
                  \---------/
                    repeats



Part 3 of 3
All 2xN Necessity Proofs


Necessity is proved by showing that
at least the domination number of knights is required
to cover a subset of the board.

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D(2,1) >= 2

o   Since no square can be attacked,
o   all squares must be covered by occupying them.

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D(2,2) >= 4

o o   No knight can cover more than one o-square,
o o   so it takes 4 knights to cover all 4 of them.

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D(2,3) >= 4

o o -   Same proof as D(2,2).
o o -

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D(2,4) >= 4

o o - -   Same proof as D(2,2).
o o - -

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D(2,5) >= 4

o o - - -   Same proof as D(2,2).
o o - - -

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Extendable optimality proof by repeating the first six columns.
D(2,6k+6) >= 4k+4

o o - - - -   Each repeatable 2x2 block requires a separate 4 knights.
o o - - - -
\---------/
  repeats

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Extendable optimality proof by repeating the first six columns.
D(2,6k+1) >= 4k+2

o o - - - - a   Each repeatable 2x2 block requires a separate 4 knights.
o o - - - - b   a-square and b-square each require a separate knight.
\---------/     Total is 4 times number of 2x4 blocks plus 2.
  repeats

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Extendable optimality proof by repeating the first six columns.
D(2,6k+2) >= 4k+4

o o - - - - a a  Each repeatable 2x2 block requires a separate 4 knights.
o o - - - - a a  The right-hand 2x2 block requires an extra 4 knights.
\---------/      Total is 4 times number of 2x4 blocks plus 4.
  repeats

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Extendable optimality proof by repeating the first six columns.
D(2,6k+3) >= 4k+4

o o - - - - a a -   Same proof as D(2,6k+2).
o o - - - - a a -
\---------/
  repeats

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Extendable optimality proof by repeating the first six columns.
D(2,6k+4) >= 4k+4

o o - - - - a a - -    Same proof as D(2,6k+2).
o o - - - - a a - -
\---------/
  repeats

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Extendable optimality proof by repeating the first six columns.
D(2,6k+5) >= 4k+4

o o - - - - a a - - -    Same proof as D(2,6k+2).
o o - - - - a a - - -
\---------/
  repeats

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