Marvin is playing a solitaire game with marbles. There are n bowls (for some positive integer n), and initially each bowl contains one marble. Each turn, Marvin may eitherremove a marble from a bowl, orchoose a bowl A with at least one marble and a different bowl B with at least as many marbles as bowl A, and move one marble from bowl A to bowl B.The game ends when there are no marbles left, but Marvin wants to make it last as long as possible.Prove that the game must end after at most n(n+1)/2 turns.Prove that for every positive integer k, there is an n such that Marvin can make the n-bowl game last for at least kn turns.Marisa comes along and asks to join the game. Marvin and Marisa revise the rules: they will alternate taking turns, starting with Marisa. When the game ends, whoever took the last turn is the winner.Prove that among any three consecutive values of n, there is at least one value for which Marvin has a winning strategy.

There is always at least one consecutive value of n for which Marvin has a successful approach.

By induction, we can demonstrate that the game must conclude after a maximum of n(n+1)/2 turns for the first half of the issue, Here Induction Means a method for demonstrating that a statement P(n) is true for every natural number n, and that the infinitely numerous examples P(0), P(1), P(2), P(3), etc. all hold, is mathematical induction. Informal analogies, such a ladder or a falling domino, are used to explain this technique .

Base case: If there is just one bowl, Marvin can get rid of it in one turn, ending the game after one.

Step one of induction: Assume that there are k bowls and that the game terminates after no more than k(k+1)/2 turns. When there are k+1 bowls, we shall demonstrate that the game terminates after at most (k+1)(k+2)/2 turns. The number of bowls lowers by one if Marvin takes a marble out of one, and the game must terminate after no more than k(k+1)/2 turns through induction hypothesis

When Marvin transfers a marble from bowl A to bowl B, where bowl A contains at least one marble and bowl B contains an equal number of marbles, bowl A loses one marble and bowl B gains one marble. The number of bowls stays the same because bowl B has at least as many marbles as bowl A. As a result, the game must still terminate after k(k+1)/2 turns at the latest.

In any scenario, the game ends after at most k(k+1)/2 turns, thus when there are k+1 bowls, the game must end after at most (k+1)(k+2)/2 turns. For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.

Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.

For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.

Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.

Marvin can select a bowl containing at least k marbles on each turn and transfer one marble to a bowl containing fewer than k marbles .Marvin can transfer one marble from the first bowl to the second bowl, for instance, if there are k+1 marbles in the first bowl and k marbles in the second bowl. This guarantees that there are at least k marbles in each bowl. The game will go for at least kn turns because Marvin can take at least one turn every bowl.

For the third component of the issue, we can demonstrate that there is always at least one value of n for which Marvin has a successful strategy.

Assume that n is a positive integer. We shall demonstrate that either Marvin or Marisa has a winning strategy for the game of n bowls or the game of (n+1) bowls.

If Marvin has a winning strategy for the N-bowl game, he can use that method to win the N-bowl game.

If Marisa has a winning strategy for the n-bowl game and Marvin does not, then Marisa does as well. In this scenario, Marvin can use Marisa's successful approach to the n-bowl game as his own successful approach to the (n+1)-bowl game.

To learn more about induction at: brainly.com/question/8419798

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