FRQ 2022 Q1

by ADMIN 12 views

===========================================================

Introduction

The Hailstone sequence, also known as the Collatz Conjecture, is a mathematical concept where a number is repeatedly transformed according to a simple rule. In this article, we will delve into the implementation of Hailstone sequences and explore code reuse techniques. We will examine three problems related to Hailstone sequences and provide step-by-step solutions, along with explanations and notes on time and space complexity.

Problem a: Hailstone Sequence Length

Problem Statement

Given a positive integer n, find the length of the Hailstone sequence starting with n.

Solution

public static int hailstoneLength(int n) {
    int length = 1; // Start with the first term (n itself)
    
    // Continue the sequence until we reach 1
    while (n != 1) {
        if (n % 2 == 0) { // If n is even
            n = n / 2;
        } else { // If n is odd
            n = 3 * n + 1;
        }
        length++;
    }
    
    return length;
}

Notes

  • The hailstone sequence follows the Collatz conjecture pattern: if n is even, the next term is n/2; if n is odd, the next term is 3n+1.
  • We start with length = 1 to account for the initial value n.
  • We use a while loop to continue generating terms until we reach 1.
  • For each new term generated, we increment the length counter.
  • Time complexity: O(L) where L is the length of the sequence (unpredictable based on input).
  • Space complexity: O(1) as we only store the current value and length.

Problem b: Long Hailstone Sequences

Problem Statement

Given a positive integer n, determine if the Hailstone sequence starting with n is long.

Solution

public static boolean isLongSeq(int n) {
    // A sequence is long if its length is greater than its starting value
    int length = hailstoneLength(n);
    return length > n;
}

Notes

  • A hailstone sequence is considered "long" if its length is greater than its starting value.
  • We leverage the previously implemented hailstoneLength method.
  • Time complexity: Same as hailstoneLength, O(L).
  • Space complexity: O(1).
  • This method demonstrates good code reuse by calling the existing hailstoneLength method.

Problem c: Proportion of Long Hailstone Sequences

Problem Statement

Given a positive integer n, calculate the proportion of Hailstone sequences with starting values from 1 to n that are considered long.

Solution

public static double propLong(int n) {
    int longCount = 0;
    
    // Check each starting value from 1 to n inclusive
    for (int i = 1; i <= n; i++) {
        if (isLongSeq(i)) {
            longCount++;
        }
    }
    
    // Return the proportion doubled
    return (double) longCount / n;
}

Notes

  • This method calculates what fraction of sequences with starting values 1 through n are considered "long".
  • We count how many starting values from 1 to n (inclusive) produce long sequences.
  • We use the isLongSeq method to determine if each sequence is long.
  • The cast to double ensures we get a decimal result rather than integer division.
  • Time complexity: O(n × L), where L is the average length of sequences.
  • Space complexity: O(1).
  • This method demonstrates nested method calls, using previously implemented functionality.

Conclusion

In this article, we explored the implementation of Hailstone sequences and code reuse techniques. We examined three problems related to Hailstone sequences and provided step-by-step solutions, along with explanations and notes on time and space complexity. The solutions demonstrated good code reuse, leveraging previously implemented methods to solve new problems. The proportion of long Hailstone sequences method showcased nested method calls, using previously implemented functionality to calculate the desired proportion.

=====================================

Introduction

In our previous article, we explored the implementation of Hailstone sequences and code reuse techniques. We examined three problems related to Hailstone sequences and provided step-by-step solutions, along with explanations and notes on time and space complexity. In this article, we will address some frequently asked questions (FAQs) related to Hailstone sequences and provide additional insights.

Q&A

Q: What is the Collatz Conjecture?

A: The Collatz Conjecture, also known as the Hailstone sequence, is a mathematical concept where a number is repeatedly transformed according to a simple rule. If the number is even, it is divided by 2, and if it is odd, it is multiplied by 3 and added 1.

Q: Why is the Collatz Conjecture important?

A: The Collatz Conjecture is important because it has been extensively studied and remains one of the most famous unsolved problems in mathematics. It has been tested with millions of numbers, and the conjecture has been verified for all numbers up to a certain point, but a formal proof has not been found.

Q: What is the significance of the Hailstone sequence length?

A: The Hailstone sequence length is significant because it determines whether a sequence is long or not. A sequence is considered long if its length is greater than its starting value.

Q: How does the hailstoneLength method work?

A: The hailstoneLength method works by repeatedly applying the Collatz Conjecture rule to the input number n until it reaches 1. The method keeps track of the number of steps taken to reach 1 and returns the total count as the length of the Hailstone sequence.

Q: What is the time complexity of the hailstoneLength method?

A: The time complexity of the hailstoneLength method is O(L), where L is the length of the sequence. This is because the method needs to iterate through the sequence until it reaches 1, and the number of iterations is proportional to the length of the sequence.

Q: How does the isLongSeq method work?

A: The isLongSeq method works by calling the hailstoneLength method to get the length of the Hailstone sequence starting with the input number n. It then checks if the length is greater than n and returns true if it is, indicating that the sequence is long.

Q: What is the time complexity of the isLongSeq method?

A: The time complexity of the isLongSeq method is the same as the hailstoneLength method, which is O(L).

Q: How does the propLong method work?

A: The propLong method works by iterating through all numbers from 1 to n and checking if each sequence is long using the isLongSeq method. It then counts the number of long sequences and returns the proportion of long sequences.

Q: What is the time complexity of the propLong method?

A: The time complexity of the propLong method is O(n × L), where L is the average length of sequences. This is because the method needs to iterate through all numbers from 1 to n and check each sequence, and the number of iterations is proportional to the number of sequences and their average length.

Conclusion

In this article, we addressed some frequently asked questions related to Hailstone sequences and provided additional insights. We hope that this Q&A article has helped to clarify any doubts and provided a better understanding of the Hailstone sequence and its related methods.