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A161531 The number of reachable states in a simple two-player counting game, in which each player starts with the pair (1,1) and one move is to add one of the opponent's numbers to one of your own numbers, but no number can grow above a pre-defined maximum n. The game continues until one of the players has no legal moves left. The winner is the one having the higher sum of his numbers. 0

%I

%S 1,4,13,34,75,152,301,482,794,1197,1842,2522,3582,4772,6408,8229,

%T 10728,13285,16851,20569,25361,30380,36927,43251,51692,60199,70668,

%U 81452,95083,107664,124360,140539,160222,179821,204026,226607,255551,283320

%N The number of reachable states in a simple two-player counting game, in which each player starts with the pair (1,1) and one move is to add one of the opponent's numbers to one of your own numbers, but no number can grow above a pre-defined maximum n. The game continues until one of the players has no legal moves left. The winner is the one having the higher sum of his numbers.

%C For many n (1 <= n <= 100), optimal play leads to a draw.

%e For n = 2 there are 4 valid states of the game: (A=1,1 B=1,1) starts the game. There is only one valid move, leading to (A=1,1 B=1,2). Now A also has only one choice (A=1,2 B=1,2), and B ends the game in the state (A=1,2 B=2,2) and wins. A has no valid move, since then one of his numbers would be larger than 2.

%o (Other) #include <climits> #include <iostream> #include <map> class State; typedef unsigned char byte; typedef std::map<State, int> ScoreMap; static inline int min(int a, int b) { return a < b ? a : b; } static inline int max(int a, int b) { return a > b ? a : b; } static inline int compare(int a, int b) { return a - b; }

%o static int MAX = 0; static ScoreMap score; struct State { byte a, b, c, d; State() : a(0), b(0), c(0), d(0) {} State(byte a, byte b, byte c, byte d) : a(min(a,b)), b(max(a,b)), c(min(c,d)), d(max(c,d)) {} int getChildren(State *children) { int n = 0; if (a + c <= MAX) children[n++ ] = State(c, d, a + c, b); if (a + d <= MAX && c != d) children[n++ ] = State(c, d, a + d, b);

%o if (a != b && b + c <= MAX) children[n++ ] = State(c, d, a, b + c); if (a != b && c != d && b + d <= MAX) children[n++ ] = State(c, d, a, b + d); return n; }; int getScore() { ScoreMap::iterator it = score.find(*this); if (it != score.end()) return it->second; State children[4]; int nchildren = getChildren(&children[0]); int iscore; if (nchildren == 0) { iscore = compare(a + b, c + d); } else { iscore = INT_MIN; for (int i = 0; i < nchildren; i++) iscore = max(iscore, -children[i].getScore()); } score.insert(ScoreMap::value_type(*this, iscore)); return iscore; }; };

%o static bool operator< (State a, State b) { if (a.a < b.a) return true; if (a.a > b.a) return false; if (a.b < b.b) return true; if (a.b > b.b) return false; if (a.c < b.c) return true; if (a.c > b.c) return false; if (a.d < b.d) return true; if (a.d > b.d) return false; return false; } int main() { std::cout << "max\tscore\tnstates" << std::endl; for (int max = 0; max < 1024; max++) { MAX = max; score.clear(); int iscore = State(1,1,1,1).getScore(); std::cout << max << "\t" << iscore << "\t" << score.size() << std::endl; } return 0; }

%K nonn

%O 1,2

%A Roland Illig (roland.illig(AT)gmx.de), Jun 12 2009

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