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2008d.00459 Huddy, S.Ryan Jones, Michael A. Shelton, Brittany C. Redundancy in nimber sequences for three-element subtraction sets. Pi Mu Epsilon J. 12, No. 7, 393-403 (2007). 2007 Department of Mathematics, Worcester Polytechnic Institute, Worcester, MA EN M40 strategic games periods, preperiods number sequences game theory congruence redundant values redundant elements subtraction games games of no chance cyclic structures cyclic redundancy Summary: In single-pile Nim, two players alternate taking turns removing a restricted number of coins (which make up the subtraction set) from a single pile until a player has no move and loses the game. Optimal play is described by the nimber sequence. Using the nimber sequence for the subtraction set $\{ a, b\}$, we determine when $c$ is never used in optimal play, or redundant, for the game under $\{ a, b, c\} $. Further, we explicitly determine the nimber sequence for the game under $\{ a, b, c\} $ when $b$ is redundant in $\{ a, b\} $. Besides a special property, these nimber sequences have period length $b + c$ and no preperiod.