  A, B, C, and D play dice games where the loser doubles the amount of money of all other players. Four games are played in all, and A, B, C, and D lose these games in this sequence only. Each player has Rs. 96 at the end of the fourth game. What is the amount with C at the end of the third game?

Let's analyze the situation systematically, step-by-step. 

### Problem Breakdown:

1. **Initial Setup**: Each player has some amount of money, which we need to determine.
2. **Game Rules**: In each game, the loser doubles the amount of money of the other three players.
3. **Sequence of Losses**: A loses the first game, B loses the second, C loses the third, and D loses the fourth.
4. **Final Outcome**: Each player has Rs. 96 at the end of the fourth game.
5. **Goal**: Find the amount with C at the end of the third game.

### Solution Strategy:

Let \( x \) be the initial amount each player starts with.

We work backward from the fourth game to find \( x \) and C's amount at the end of the third game.

**1. Fourth Game (D loses):**
- After D loses, each of A, B, and C has Rs. 96.
- D's amount is reduced by doubling the amounts of A, B, and C.
- Since D ends with Rs. 96 after losing, we calculate D's amount before losing:
  
  Let the amounts before the fourth game be \( A_3 \), \( B_3 \), \( C_3 \), and \( D_3 \).

  After D loses, the equation is:

  \[
  D_3 - 3 \times (A_3 + B_3 + C_3) = 96
  \]

  Since \( A_3 = B_3 = C_3 = 96 \) (after D loses):

  \[
  D_3 - 3 \times 96 = 96 \implies D_3 - 288 = 96 \implies D_3 = 384
  \]

  Therefore, the amounts before the fourth game were:

  \[
  A_3 = 96, \quad B_3 = 96, \quad C_3 = 96, \quad D_3 = 384
  \]

**2. Third Game (C loses):**
- Before C loses, \( A_2 = 96 \), \( B_2 = 96 \), and \( D_2 = 384 \).
- C's amount will change due to doubling the other players' amounts when losing:

  \[
  C_2 - 3 \times (A_2 + B_2 + D_2) = 96
  \]

  Plugging in values:

  \[
  C_2 - 3 \times (96 + 96 + 384) = 96 \implies C_2 - 3 \times 576 = 96 \implies C_2 - 1728 = 96
  \]

  \[
  C_2 = 1824
  \]

Therefore, **C has Rs. 1824 at the end of the third game.**

Would you like further details or have any questions?

### Related Questions:

1. What is the initial amount each player starts with?
2. How much does each player have after each game in detail?
3. How does the sequence of losses affect the game outcomes?
4. What would happen if one of the players had lost a different game?
5. How can we determine the impact on total amounts after each game loss?

**Tip:** When dealing with sequence-dependent problems, working backward can simplify finding unknown values.

Explore the solution to a dice game sequence problem where players lose in a specified order, affecting each other's amounts. Learn how to calculate the amount with C after the third game using arithmetic and game theory principles. Suitable for students in Grades 10-12.

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