The game MSP(m) discussed in this paper, is played alternately by two players, removing at least one stone and at most m stones from the single pile at each turn. Each player is prohibited from removing the same number of stones as the number of stones removed by the opponent at the immediately previous turn. A player who makes the pile empty wins the game. Furthermore, when a player cannot remove any number of stones due to the prohibition rule, the opponent becomes the winner. This game is denoted by MSP(m). In our previous report 8 we formulated the initial numbers of stones, called unsafe number of stones, at the pile such that the second player of MSP(m) is promised to become the winner. However, we discovered that there were some serious errors in the previous report. In this paper we remove the errors in the previous one, and add some new results. We categorize MSP(m)s into types 1,2, 3,..., k, k + 1,... according to certain properties of m's. Then for each type of MSP(m)'s we investigate the periodicity of unsafe numbers. For vast majority of MSP(m)'s, from the periodicity we can efficiently decide which player is promised to become the winner of the game with a given initial number of stones.
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