Suppose you have a triangular chessboard of size $n$, whose "squares" are ordered triples $(x,y,z)$ of nonnegative integers that add up to $n$. A rook can move to any other point that agrees with it in one coordinate -- for example, if you are on $(3,1,4)$ then you can move to $(2,2,4)$ or to $(6,1,1)$, but not to $(4,3,1)$.


What is the maximum number of mutually non-attacking rooks that can be placed on this chessboard?


More generally, is anything known about the graph whose vertices are these ordered triples and whose edges are rook moves?



Hexagonal rooks
Here is a paper about this problem: ["Non-attacking queens on a triangle"](http://www.cin.ufpe.br/~pcp/nonattacking_queens.pdf).


And here's another one ["Putting Dots in Triangles"](http://arxiv.org/abs/0910.4325)


