There are n players standing around in a circle. Carol, who is not one of the...

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There are n players standing around in a circle. Carol, who is not one of the n players, goes around the circle clockwise and taps the head of every second person still standing. As soon as a player has their head tapped, that person sits down. Carol keeps going around the circle until only one person is left standing. and that person is declared the winner. For example, if there are n=9 players, then Carol taps players 2, 4, 6, 8 on her first trip around the circle. Then in her next trip around the circle, she taps players 1, 5, 9. Finally, in her last trip around the circle she taps player 7. Thus, player 3 is the winner. 7 start 7 For each integer n, let f(n) denote the position number of the winning player when the game consists of n players. For example, /(9) = 3. (a) For each n satisfying 15 nS 15, determine the value of f(n). 1 2 3 4 5 6 7 8 9 10 3 11 12 13 14 15 10 11 12 13 15 16 16 f(n) No proof is necessary. (b) If n is even, represent f(n) as a function of f (2). Use induction to prove that your formula is correct. (c) If n is odd, represent f(n) as a function of f("). Use induction to prove that your formula is correct. (d) Let N be the set of positive integers. Determine whether the function f: N→ N is injective and/or surjective. Clearly justify your answers. (e) Suppose there are n = 2023 people around the circle, with you being player t. Determine the value of t that ensures you win this game. There are n players standing around in a circle. Carol, who is not one of the n players, goes around the circle clockwise and taps the head of every second person still standing. As soon as a player has their head tapped, that person sits down. Carol keeps going around the circle until only one person is left standing. and that person is declared the winner. For example, if there are n=9 players, then Carol taps players 2, 4, 6, 8 on her first trip around the circle. Then in her next trip around the circle, she taps players 1, 5, 9. Finally, in her last trip around the circle she taps player 7. Thus, player 3 is the winner. 7 start 7 For each integer n, let f(n) denote the position number of the winning player when the game consists of n players. For example, /(9) = 3. (a) For each n satisfying 15 nS 15, determine the value of f(n). 1 2 3 4 5 6 7 8 9 10 3 11 12 13 14 15 10 11 12 13 15 16 16 f(n) No proof is necessary. (b) If n is even, represent f(n) as a function of f (2). Use induction to prove that your formula is correct. (c) If n is odd, represent f(n) as a function of f("). Use induction to prove that your formula is correct. (d) Let N be the set of positive integers. Determine whether the function f: N→ N is injective and/or surjective. Clearly justify your answers. (e) Suppose there are n = 2023 people around the circle, with you being player t. Determine the value of t that ensures you win this game.

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Youve provided an image describing a game and several questions related to it The game involves a person named Carol who taps the heads of every second person in a circle and this person sits down the ... View the full answer