Question 1: You Are Given A Deck Containing N Cards

Question 1you Are Given A Deck Containing N Cards While Holding The

You are given a deck containing N cards. While holding the deck facedown: 1. Deal all the cards facedown onto a table into Y piles as one would do when playing with a group of people (i.e., card 1 to P1, card 2 to P2, ..., card Y to PY, then continue in a round-robin fashion). 2. Combine all the piles into a deck by stacking P1 onto P2, then the combined P1+P2 onto P3, and so forth. This constitutes one round. 3. Pick up the deck from the table and repeat steps 1-2 until the deck returns to its original order. 4. For each round, vary the number of piles according to a repeating pattern: start with 3, then 4, then 5, then loop back to 3, and so on. Write a C program to determine the number of rounds needed for the deck to return to its original order. The program should not use arrays to represent the deck, but instead should utilize appropriate data structures (such as linked lists). The program should take the number of cards as a command line argument and output the total number of rounds required to restore the original order. Ensure the code compiles and runs correctly, and include a comment block explaining your methodology.

Paper For Above instruction

Introduction

The problem involves determining the number of rounds required to return a deck of N cards to its original order after performing a series of shuffling steps, with varying numbers of piles in each round. Each round comprises two steps: dealing the deck into Y piles and then stacking those piles to form a new deck. The process repeats until the deck's order matches its initial arrangement. The challenge emphasizes using data structures like linked lists instead of arrays to simulate the deck operations, reflecting a realistic card shuffling scenario and ensuring memory efficiency and dynamic flexibility.

Methodology

The methodology involves modeling the deck of cards using a linked list, which inherently supports dynamic insertion and extraction operations crucial for simulating shuffles without arrays. Initially, the deck is represented as a singly linked list, with each node encapsulating a card's identity (e.g., its original position). The primary operations for each shuffle round include dealing and stacking, implemented by traversing and reorganizing the linked list accordingly.

The key steps are as follows:

1. Initialization: Create a linked list representing the deck in its original order, with nodes numbered from 1 to N.

2. Dealing into Y piles: Traverse the linked list, distributing nodes into Y separate linked lists (piles). This is analogous to dealing cards into multiple stacks, respecting the order.

3. Stacking piles: Concatenate the Y lists into a single list, stacking one pile onto the next, simulating the recombination of the dealing process.

4. Cycle detection: After each round, compare the current order of the nodes against the original configuration to determine if the deck has returned to its initial order.

5. Repeating with varying Y: The number of piles Y cycles through 3, 4, 5, then repeats. The process continues until the deck is restored, and the count of rounds is recorded.

This iterative process leverages linked list operations for dealing and stacking, which accurately emulate the physical process of shuffling cards in a manner that arrays cannot efficiently replicate due to their fixed size and memory management constraints.

Implementation Details

- Use a singly linked list to represent the deck, with each node storing the card's position label.

- For dealing, create temporary linked lists for each pile by sequentially distributing nodes.

- For stacking, connect the tail of each pile to the head of the next to form a single linked list.

- Use a loop to perform rounds, incrementing a counter each time.

- After each shuffle, verify whether the current order matches the initial order by traversing and comparing node positions.

- Continue until the deck restores to the initial order, then output the total number of rounds.

The goal of this modular approach is to emulate real card shuffling, handle dynamic pile sizes efficiently with linked lists, and accurately determine the shuffle cycle count.

Code Implementation

// Detailed comment block explaining the methodology
/* 
This program models a deck of N cards using a linked list, with each node representing a card.
It simulates the shuffling process described:
- Dealing into Y piles in a round-robin manner.
- Stacking the piles to reform the deck.
- Repeating until the original order is restored.
The pile count Y cycles through 3, 4, and 5.
Key components:
- Initialization: Create a linked list from 1 to N.
- Deal function: Distributes the deck into Y piles as linked lists.
- Merge function: Concatenates piles into a single linked list.
- Comparison function: Checks if the deck matches its original order.
- Main loop: Repeats the shuffle until the original order reappears, counting rounds.
This approach avoids arrays, using linked lists for dynamic memory efficiency.
*/
include 
include 
include 
typedef struct CardNode {
int value;
struct CardNode* next;
} CardNode;
// Function to create a new node
CardNode* create_node(int value) {
CardNode node = (CardNode)malloc(sizeof(CardNode));
node->value = value;
node->next = NULL;
return node;
}
// Initialize the deck in original order
CardNode* initialize_deck(int n) {
CardNode* head = create_node(1);
CardNode* current = head;
for (int i = 2; i 
current->next = create_node(i);
current = current->next;
}
return head;
}
// Copy the deck for comparison
CardNode copy_deck(CardNode head) {
if (head == NULL) return NULL;
CardNode* new_head = create_node(head->value);
CardNode* current_new = new_head;
CardNode* current_orig = head->next;
while (current_orig != NULL) {
current_new->next = create_node(current_orig->value);
current_new = current_new->next;
current_orig = current_orig->next;
}
return new_head;
}
// Compare two decks for equality
bool is_same_deck(CardNode d1, CardNode d2) {
while (d1 != NULL && d2 != NULL) {
if (d1->value != d2->value) return false;
d1 = d1->next;
d2 = d2->next;
}
return (d1 == NULL && d2 == NULL);
}
// Free the linked list
void free_deck(CardNode* head) {
while (head != NULL) {
CardNode* temp = head;
head = head->next;
free(temp);
}
}
// Deal into Y piles (represented as array of linked list heads)
CardNode* deal_into_piles(CardNode deck, int y, int* pile_sizes) {
// Allocate array for piles
CardNode piles = (CardNode) malloc(y  sizeof(CardNode));
CardNode tails = (CardNode) malloc(y  sizeof(CardNode));
for (int i = 0; i 
piles[i] = NULL;
tails[i] = NULL;
pile_sizes[i] = 0;
}
int index = 0;
CardNode* current = deck;
while (current != NULL) {
int pile_index = index % y;
// Detach node
CardNode* next_node = current->next;
current->next = NULL;
if (piles[pile_index] == NULL) {
piles[pile_index] = current;
tails[pile_index] = current;
} else {
tails[pile_index]->next = current;
tails[pile_index] = current;
}
pile_sizes[pile_index]++;
current = next_node;
index++;
}
free(piles);  // We keep only tail pointers to concatenate later
return piles;
}
// Concatenate piles to form a new deck
CardNode concatenate_piles(CardNode piles, int y, int pile_sizes) {
CardNode* new_deck = NULL;
CardNode* current_tail = NULL;
for (int i = 0; i 
if (piles[i] != NULL) {
if (new_deck == NULL) {
new_deck = piles[i];
// Move to the end of this pile
current_tail = piles[i];
while (current_tail->next != NULL) current_tail = current_tail->next;
} else {
current_tail->next = piles[i];
while (current_tail->next != NULL) current_tail = current_tail->next;
}
}
}
return new_deck;
}
int main(int argc, char* argv[]) {
if (argc != 2) {
printf("Usage: %s \n", argv[0]);
return 1;
}
int N = atoi(argv[1]);
if (N 
printf("Number of cards must be a positive integer.\n");
return 1;
}
// Initialize original deck
CardNode* original_deck = initialize_deck(N);
CardNode* current_deck = copy_deck(original_deck);
// Variables for cycling pattern: 3, 4, 5
int pile_pattern[] = {3, 4, 5};
int pattern_index = 0;
int rounds = 0;
int cycle_length = sizeof(pile_pattern) / sizeof(pile_pattern[0]);
int y;
do {
y = pile_pattern[pattern_index];
pattern_index = (pattern_index + 1) % cycle_length;
int pile_sizes = (int) malloc(y * sizeof(int));
CardNode** piles = deal_into_piles(current_deck, y, pile_sizes);
CardNode* new_deck = concatenate_piles(piles, y, pile_sizes);
// Free temporary structures
free(pile_sizes);
free(piles);
// Check if deck has returned to original order
bool same = is_same_deck(new_deck, original_deck);
// Prepare for next iteration
free_deck(current_deck);
current_deck = new_deck;
rounds++;
if (same) break;
} while (true);
printf("Number of rounds to restore original order: %d\n", rounds);
free_deck(current_deck);
free_deck(original_deck);
return 0;
}