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Joe Joyce wrote on 2010-11-13 UTC
To answer your question about squares used for triangular movement:
You remove some of the squares in a particular pattern. Each removed square is at the center of a hexagonal arrangement of squares that are always present. Pick any square from that always-present hex arrangement of squares, and remove neighboring squares such that any is always surrounded by an alternating pattern of present and absent squares. 

This pattern allows triangular movement, some of which may pass through the absent squares to the present squares on the other side. This pattern leaves the board with these features:
1 - all absent squares are at the center of a hexagonal ring of present squares. 
2 - All present squares are members [junction points] of 3 rings of present squares. 
3 - All present squares have 3 absent neighbors, and 3 present neighbors.

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