Question

In this assignment, you will model the game of Bulgarian Solitaire. The game starts with 45 cards. (They need not be playing cards. Unmarked index cards work just as well.) Randomly divide them into some number of piles of random size. For example, you might start with piles of size \(20,5,1,9\), and 10 . In each round, you take one card from cach pile, forming a new pile with these cards. For example, the sample starting configuration would be transformed into piles of size \(19,4,8,9\), and 5 . The solitaire is over when the piles have size \(1,2,3,4,5,6,7,8\), and 9 , in some order. (It can be shown that you always cnd up with such a configuration.) In your program, produce a random starting configuration and print it. Then keep applying the solitaire step and print the result. Stop when the solitaire final configuration is reached.

Short answer

Randomly start with piles summing to 45, perform steps removing one card per pile, and stop at configuration 1 to 9.

Step-by-step solution

Understand the Rules

The game of Bulgarian Solitaire involves starting with 45 cards and dividing them into random piles. In each step, take one card from each pile to form a new pile. The game ends when you reach a configuration of piles with sizes 1 to 9.

Generate Initial Configuration

Randomly distribute the 45 cards into any number of piles such that the sum of the cards in all piles equals 45. For example, you might randomly choose piles with sizes 20, 5, 1, 9, and 10.

Perform Solitaire Step

For each pile in your configuration, remove one card, and use these removed cards to form a new pile. If a pile is reduced to zero cards, it is eliminated from the configuration. For example, from the initial configuration of 20, 5, 1, 9, and 10, the next configuration becomes 19, 4, 0, 8, and 9, plus the new pile of 5. After removing empty piles, the next configuration is 19, 4, 8, 9, and 5.

Check for Final Configuration

Repeat Step 3 until the configuration of piles becomes exactly sizes 1, 2, 3, 4, 5, 6, 7, 8, and 9, regardless of the order. This is the condition that signifies the end of the solitaire game.

Iterate and Print Results

Continue performing the solitaire steps until the final configuration is reached. After each iteration, print the current configuration of piles to monitor progress until reaching the goal.

Key concepts

Algorithm Design

The design of an algorithm involves creating a step-by-step plan to solve a problem efficiently, and Bulgarian Solitaire provides an excellent example of this. The goal is to reach a specific card configuration from a randomized setup, demonstrating how algorithms can achieve well-defined outcomes. In this task, our algorithm must account for:

• Initial conditions: starting with 45 cards divided into various piles.
• Rules: in each iteration, remove one card from each pile to form a new pile.
• Stopping criteria: achieving the specific sequence
  1. Piles of size 1 to 9.
  2. Completion of goal leading to the end of the game.

This problem is a showcase for algorithm design, balancing randomness and structure to solve the card sorting challenge efficiently.

Random Number Generation

Random number generation is crucial for creating the initial setup in Bulgarian Solitaire. Your program needs to simulate the initial distribution of 45 cards across different piles at random, which requires generating random numbers. Various methods can achieve this, such as:

• Using programming libraries or functions that provide random number generation.
• Ensuring the sum of generated numbers always equals 45 to maintain game integrity.
• Creating a natural unpredictability in the pile sizes, which affects the game's progression.

Introducing randomness helps ensure that each game starts uniquely, adding to the challenge and learning experience while showing how randomness can play a role in simulations and programming.

Iteration in Programming

Iteration is an essential concept in programming that involves repeating a set of instructions until a specific condition is met. In Bulgarian Solitaire, iteration manifests as repeatedly applying the solitaire step to transform the card piles. The iterative process includes:

• Performing the card removal and new pile creation over multiple rounds.
• Checking conditions for each iteration to see if the game has reached its end state.
• Monitoring the progress by printing pile configurations after each step.

This repetitive process exemplifies robust programming practices, where loops are used to ensure ongoing checks and processes that adapt to the game's state until reaching the final configuration.

Card Game Simulation

Card game simulation involves creating a digital model of a card game to study its behavior and outcomes. Bulgarian Solitaire serves as a perfect example of card game simulation, requiring the program to model real-world card actions and rules. Key aspects include:

• Emulating the physical actions of organizing and managing card piles.
• Simulating realistic game processes in a virtual environment.
• Enabling the observation and analysis of the game as it progresses towards the end goal.

Through simulating Bulgarian Solitaire, we can test theories about game completion and configurations, allowing us to gain insights into both the game's nature and the effectiveness of our simulation.