# Solve Recursive Series for Winning Game: A vs B

• 8daysAweek
In summary, there is a game with two players, A and B, where they take turns shooting at each other simultaneously. The objective is for one player to have a higher life point value than 1, while the other player's life points are below 1. To determine the minimum starting life points for player B to win, a series of equations can be used but it may be helpful to write them in matrix form and examine the pattern of convergence.
8daysAweek
There is a game with two players: A and B.
Each turn the players shoot at each other simultaneously.
Player A has 100 life points and the damage he inflicts is 50% of his remaining life points. Player B deals 25% respectively. Life points are rational numbers.
A player wins the game when his life points are higher than 1, while his opponent's life points are smaller than 1.
Find the minimum, natural starting life points that player B should have in order to win the game.
I decided to start by representing the life points of each player as a series. I got this:

$$a_0 = 100$$
$$b_0 = X$$
$$a_n = a_{n-1}-{0.25}b_{n-1}$$
$$b_n = b_{n-1}-{0.5}a_{n-1}$$

But I got stuck here unable to solve the equations.

Any help or ideas will be appreciated.*This is not homework

Last edited:
it may help to write it in matrix form
$$\begin{pmatrix} a_{n+1} \\ b_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & -\frac{1}{4} \\ -\frac{1}{2} & 1\\ \end{pmatrix} \begin{pmatrix} a_{n} \\ b_{n} \end{pmatrix}$$

then maybe examine the form of the matrix for several rounds, starting at ao, bo

## 1. How do you solve a recursive series for a winning game between A and B?

The process for solving a recursive series for a winning game between A and B involves breaking down the problem into smaller, solvable parts. This can be done by identifying the base case, or the simplest version of the problem, and then using this to create a recursive formula that can be applied to larger versions of the problem.

## 2. What is the base case for a recursive series for a winning game between A and B?

The base case for a recursive series for a winning game between A and B is the simplest version of the game, where there is only one possible outcome. For example, in a game between A and B where the first player to reach 10 points wins, the base case would be when one player reaches 9 points and the other has 8 points or less.

## 3. How do you create a recursive formula for a winning game between A and B?

To create a recursive formula for a winning game between A and B, you first need to identify the base case and then determine how to get from the base case to the next step. This can be done by looking at the rules of the game and determining how each player can score points or move closer to winning. The recursive formula should then be applied to larger versions of the problem until the final outcome is reached.

## 4. Can a recursive series for a winning game between A and B have multiple base cases?

Yes, a recursive series for a winning game between A and B can have multiple base cases. This may occur if there are different ways for the game to end or if there are multiple possible outcomes that lead to a win for one player. In this case, each base case would need to be considered when creating the recursive formula.

## 5. Are there any limitations to using recursive series to solve a winning game between A and B?

While recursive series can be an effective way to solve a winning game between A and B, there are some limitations to this approach. For example, the game must have a clear set of rules and a defined end point in order for a recursive formula to be created. Additionally, the complexity of the game may increase with each step, making it more difficult to calculate the final outcome. In these cases, other methods may be more suitable for solving the game.

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