Incremental Span Stepwedges

An incremental span stepwedge is a stepwedge that spans its complete interval, and that has the same integer increment between all pairs of neighboring steps. There are only certain combinations that satisfy these constraints. All possible cases are summarized in the table below.

A common example would be a stepwedge representing 8-bit pixel values. In this case, an 8-bit value can represent 256 unique levels, spanning the range [0, 255]. What integer increments exist that will exactly span this range? An obvious value would be an increment of one. This would make a 256 level stepwedge (0, 1, 2, … 253, 254, 255). This is an incremental span stepwedge because it has the same integer increment between all pairs of steps (1) and it spans the full 8-bit range (0 through 255). Another obvious value would be an increment of 255 which would make a two-level stepwedge (0, 255). Both of these stepwedges span the entire range (i.e. they contain both extreme values 0 and 255) and both have integer increments (i.e. 1 and 255 in these examples).

Other possibilities exist that are less obvious. An increment of 2 will not work, because that produces a stepwedge that does not span the full range (0, 2, 4, … 252, 254). Notice that this stepwedge does not contain the value 255; therefore it is disqualified. An increment of 3 will work, since it produces a spanning stepwedge (0, 3, 6, 9, … 249, 252, 255). If you exhaustively examine all possible increments, you will find that the 8-bit case has only eight incremental span stepwedges:

Here is a table that presents all possible incremental span stepwedges for bit sizes from 1 through 16.