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Unformatted text preview: CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Soln 4 1. (10 pts.) Recurrence relations Let f ( n ) be defined by the recurrence relation f ( n ) = 7 f ( n- 1 )- 10 f ( n- 2 ) (for all n ≥ 2) and f ( ) = 1, f ( 1 ) = 2. Prove that f ( n ) = 2 n for every n ∈ N . Answer: Proof : The proof is by strong induction over n . Let P ( k ) be the proposition that f ( k ) = 2 k . Base cases: If k = 0, then f ( ) = 1 = 2 . If k = 1, then f ( 1 ) = 2 = 2 1 . Thus the base cases P ( ) and P ( 1 ) are correct. Inductive hypothesis: Assume that for some k ∈ N satisfying k ≥ 1, P ( ) ∧ P ( 1 ) ∧···∧ P ( k ) is true, i.e., f ( j ) = 2 j for j = , 1 , 2 ,..., k . Inductive step: By the problem statement, we know that f ( k + 1 ) = 7 f ( k )- 10 f ( k- 1 ) . Since k ≤ k and k- 1 ≤ k , and both k and k- 1 are positive, the inductive hypothesis tells us that f ( k + 1 ) = 7 · ( 2 k )- 10 · ( 2 k- 1 ) = 7 · ( 2 k )- 5 · ( 2 k ) = ( 7- 5 ) · ( 2 k ) = 2 k + 1 . Thus, ( ∀ k ∈ N ) (( P ( ) ∧ P ( 1 ) ∧···∧ P ( k )) ⇒ P ( k + 1 )) , and we are done. 2 2. (20 pts.) Marbles In a game that’s about to sweep the nation, there is a bucket that contains some number of gold-colored marbles, silver-colored marbles, and bronze-colored marbles. When it is a player’s turn, the player may either: (i) remove one gold marble from the bucket, and add up to 3 silver marbles into the bucket; (ii) remove two silver marbles from the bucket, and add up to 7 bronze marbles into the bucket; or, (iii) remove a bronze marble from the bucket. These are the only legal moves. The last player that can make a legal move wins. Prove by induction that, if the bucket initially contains a finite number of marbles at the start of the game, then the game will end after a finite number of moves. Answer 1: An easy way to prove this is to think of the marbles as having values. For example, say that a gold marble is worth 100 dollars, a silver marble is worth 10 dollars, and a bronze marble is worth 1 dollar. This implicitly assigns a value to every possible bucket of marbles. Moreover, with this choice of values, every legal move will decrease the value by at least 1 dollar. Since the bucket can never have a negative value (and it starts with a finite value), the game cannot go forever. This intuition is formalized in the following proof. Lemma 1 : Let v be the value of the initial configuration and v n be the value of the configuration after n moves where n is a natural number. Then v n ≤ v- n. CS 70, Fall 2010, Soln 4 1 Proof : Proof by simple induction over n , the number of moves. Let P ( k ) be the proposition that the value after k moves is at most v- k ....
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This note was uploaded on 12/08/2010 for the course CS 70 taught by Professor Papadimitrou during the Fall '08 term at Berkeley.
- Fall '08