...urface with N(f) = 0 but MF(f) = 2. Maps on surfaces which satisfy N(f) = MF(f) are called Wecken maps, and this paper continues the recent work of Brimley, Griisser, Miller, and the second author in =-=[1]-=- investigating the abundance (or lack thereof) of the Wecken maps among the set of all selfmaps on surfaces with boundary. For maps on surfaces with boundary, the numbers N(f) and MF(f) depend only on...