I've been thinking about Vault's defensive formations. More specifically I've been searching for the best way to put as few pieces as possible in my homerow so all these cells are under friendly attack (any enemy landing there will be killed immediately).
It turns out that, for all 3 main variants of the game (6x6, 8x8, 10x10) the best defensive formation is essentially the same: Put 4 consecutive pieces in the central cells of your homerow:
The 4-in-a-row formation is the smallest formation able to defend itself (no proof, but quite sure), so it is necessary even for the smallest version (6x6). But this formation can defend 3 extra cells to either side so it is also enough to defend up to 10 cells in a row.
In the smaller boards we can shift the formation a little bit while maintaining a perfect defense:
Indeed, any 2N-in-a-row formation can "crawl" quite well (last piece on one side jumps to be the first piece on the other side) so you can easily displace the 4-in-a-row formation along your homerow.
Now the questions:
Is the 4-in-a-row the most efficient defensive formation for Vault? Are there other 4-piece formations as efficient as the 4-in-a-row? How the increasing efficiency of this formation (from the 66% of the pieces in the 6x6 board to just the 40% of the pieces in the 10x10 board) affects the gameplay? Does it get compensated by the fact that the adversary's homerow is farther? Are there defensive formations that ensure that each homerow cell is under double friendly attack (so you can resists 2 consecutive enemy pieces landing there)? What about defensive formations above the homerow like this one?
Is less efficient and more difficult to repair, but it is also way more offensive!
All the previous considerations lead to a really conservative opening that I call the crab:
The resulting defensive formation is able to defend the homerow efficiently (with the 4-in-a-row formation) while bringing to the center the rest of the pieces.
Note that the two second-row pieces allow you to reconstruct the 4-in-a-row formation in case of attack. They can also reach the opponent's homerow (in order to jump to the opponent's homerow a piece should be placed in an even row: 2nd, 4th, 6th...)
Finally, The Crab can be easily adapted to the 6x6: