For X ≤ 1/8 all 1's is best. It always has an average weight of 1.
...111...
...111...
...111...
The following pattern has an average weight of (8X+8)/9 which is tied for the best only when X=1/8 weight of 1 (it is never the sole maximum)
...111111...
...1X11X1...
...111111...
...111111...
...1X11X1...
...111111...
From 1/8 to 1/4 the following pattern is the highest with a weight of (3+8X)/4
...111111...
...1X1X1X...
...111111...
...1X1X1X...
Above 1/4 this pattern of alternating strips of 1's and X's is the best with an average weight of (1+6X)/2:
...111111...
...XXXXXX...
...111111...
...XXXXXX...
At X=1/8 precisely the first 3 are tied, at X=1/4 precisely the last two are tied. |