Bulletin of Mathematics ISSN Printed: 2087-5126; Online: 2355-8202 Vol. 10, No. 01 (2018), pp. 1–11 http://talenta.usu.ac.id OPERATOR MEX TO WIN THE THREE-PILE FRAENKEL GAME Susi Parlini, Ihda Hasbiyati and M. D. H. Gamal Abstract. This paper discusses a formula to reconstruct the P -position of three- pile Fraenkel game by using mex operator. P -position is a dynamic position where a gamer can decide his victory. The result shows that P -position of three-pile Fraenkel game is a series (A n,j ,B n,j ,C n,j ), with the value of A n,j is the smallest positive integer which doesnot exist in (A n,j ,B n,j ,C n,j ) with 0 ≤ i<n, while the value of B n,j , and C n,j can be recursively obtained from A n,j . From this discussion, it can be concluded that there are (j + 1) series of P -position for each n integer, with the value of j is the integer started from zero to the lowest value of n 2 . 1. INTRODUCTION Generally, a combinatorial game is assented to have characteristics as fol- lows: (a) there are two players, (b) each player moves one after another, (c) there is a starting position which is settled in the beginning, (d) for each position (except for the loser position), and for each player, there are several choices to move, and every chosen move determines the next position, (e) if a player got a position that blocks him to move, he is stated in a defeated position, (f) both players have complete information about their position, (g) there is no lucky factor, the last result (win or lose) can be achieved by selecting accurate moves, (h) the game ends by the number of moves. Received 15-01-2018, Accepted 16-02-2018. 2010 Mathematics Subject Classification: 91A46, Key words and Phrases: Mex operator, P -position, Fraenkel game. 1