  Given an impartial combinatorial game G, we create a class of related games
(CIS-G) by specifying a finite set of positions in G and forbidding players
from moving to those positions (leaving all other game rules unchanged). Such
modifications amount to taking cofinite induced subgraphs (CIS) of the original
game graph. Some recent numerical/heuristic work has suggested that the
underlying structure and behavior of such "CIS-games" can shed new light on,
and bears interesting relationships with, the original games from which they
are derived. In this paper we present an analytical treatment of the cofinite
induced subgraphs associated with the game of (three-heap) Nim. This
constitutes one of the simplest nontrivial cases of a CIS game. Our main
finding is that although the structure of the winning strategies in games of
CIS-Nim can differ greatly from that of Nim, CIS-Nim games inherit a type of
period-two scale invariance from the original game of Nim.
