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3-colored adjacent algebraic algorithm arrangements balls biconnected components binary sequences binomial coefficients bipartite graph chosen color combinatorial consecutive corners counting problems cube digital sum digits directed graph disjoint distinct boxes distinct objects distribute edges equation equivalent Euler circuit Example Exercise expansion exponential generating function Find a recurrence five formal products formula four graph G graph in Figure Grundy function Grundy number guess Hamiltonian circuit I-Iow identical objects induction Instant Insanity integer integer solutions isomorphic kernel least letters lexicographic order Mastermind game matching mathematical mathematical induction maximal flow minimal multiplication n-digit Nim game number of different number of partitions obtain outcomes pair permutation pick pile planar graph player polynomial position possible probability progressively finite game recurrence relation secret code Section shortest path Show solve spanning tree subgraph subset Suppose symmetries Theorem undirected unoriented vertex