are you over complicating?
log cabin's (usually) only need to have strips sorted into light and dark- lights go on 2 sides, darks the other 2...the log-cabins work...i'm a bit confused about your (half square triangles???) a log cabin block usually starts with a center square (traditionally red or yellow-the home hearth fire) then *logs* are added in a counter clockwise direction around the center square- for how ever rounds are required to make the size block you want.
and make blocks and put them aside- on a design wall if you have one- when you stand back and look you may find the ones you don't think work- work just fine- it is the values (lights and darks) that make log cabin blocks work