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Polylines can be regarded as unit lines confined to the square grid. I use the convention that polylines are allowed to cross each other, whilst polysticks cannot. Physical pieces can be constructed for the polysticks in a number of ways.

  Monostick - 1 (1 line)

  Disticks - 2 (4 lines)

1	2

  Tristicks - 5 (15 lines)

1	2	3	4	5

  Tetrasticks - 16 (64 lines)

1	2	3	4	5
I	L	Y	N	V
6	7	8	9	10
W	T	Z	R	X
11	12	13	14	15
U	F	H	P	J
16
O

  For higher orders of polylines, the number of pieces is as follows: