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For the square path, one solid placement (0,10,13)1 features 103871 solutions, another (3,10,13) features 69789. Together with just three others, (0,10,19)1, (0,10,20)1, (0,5,10)1, they encompass 205736 solutions or about 2/3 of the total. Note that all of these include the 3rd or 5th column of the middle row, positions 10 and 13, with the two "big boys" using both of them. Altogether, just 79 solid placements (of 684 possibilities) are used.
While positions 10 and 13 show up prominently for placement of a solid piece, a few nearby positions never show up at all: 8,9,14, and 15. (The two spots nearest the border of the middle row.)
For the diamond path, with far fewer solutions, there are also fewer solid placements, just 37. There are 8 of them with more than 400 solutions each, (0,17,22),(0,3,6)1,(0,17,20)1,(1,3,6)1,(0,7,17)1, (0,6,17)1,(1,4,17), and (0,6,16)1 (in descending order of number of solutions). All but one of them includes at least one corner (0 or 17 or both). It is interesting that no solutions have a solid in the middle row!
1 These placements refer to a pattern reflected left-right.
These patterns have another distinctive feature:
Some solutions fold.
That is, they can be divided into two 3x4 halves, which can also be fitted together to form a 4x6 solution. These 4x6's also feature a "Z" path of squares or diamonds.
The solution shown here is the ONLY Diamond Z path one
which folds. For the Square Z path, there are many.