Step by Step solution about How to resolve the algorithm Lychrel numbers step by step in the Java programming language

The provided Java code investigates the Lychrel numbers, integers that do not become palindromes after a finite number of iterations of the sum of the number and its reverse. The code uses dynamic programming with memoization to efficiently determine whether a given number is a Lychrel number.

Key Concepts:

1. Lychrel Numbers: Numbers that never become palindromes after a finite number of iterations of adding the number and its reverse.

2. Memoization: A technique used to improve the performance of a function by storing its results for later reuse.

Implementation Details:

1. Class Definition:

• The Lychrel class encapsulates the logic for determining if a given number is a Lychrel number.

2. Tuple Class:

• The Tuple class represents a pair of values: a flag and a BigInteger.
• The flag indicates whether the number is a Lychrel number, and the BigInteger represents the number itself.

3. rev() Method:

• The rev() method calculates the reverse of a BigInteger by converting it to a String, reversing the string, and converting it back to a BigInteger.

4. lychrel() Method:

• The lychrel() method is the core of the algorithm and performs the following steps:
  • Checks if the number is already in the cache map, avoiding redundant calculations.
  • Calculates the reverse of the number.
  • Initializes a tuple and a list to track seen numbers.
  • Executes a loop that iteratively adds the number to its reverse and checks if either the sum becomes a palindrome or the number was previously seen.
  • Updates the cache based on the outcome of the loop.

5. Main Method:

• Sets up lists to collect seeds (numbers that become Lychrel numbers), related (numbers that become palindromes after joining other Lychrel numbers), and palindromes (numbers that become palindromes after adding to themselves).
• Loops through numbers from 1 to 10,000 and determines their Lychrel status.
• Prints statistics on the number of Lychrel seeds, related numbers, and palindromes found.

By leveraging memoization, the code efficiently determines Lychrel numbers, providing insights into this fascinating mathematical concept.

import java.math.BigInteger;
import java.util.*;

public class Lychrel {

    static Map<BigInteger, Tuple> cache = new HashMap<>();

    static class Tuple {
        final Boolean flag;
        final BigInteger bi;

        Tuple(boolean f, BigInteger b) {
            flag = f;
            bi = b;
        }
    }

    static BigInteger rev(BigInteger bi) {
        String s = new StringBuilder(bi.toString()).reverse().toString();
        return new BigInteger(s);
    }

    static Tuple lychrel(BigInteger n) {
        Tuple res;
        if ((res = cache.get(n)) != null)
            return res;

        BigInteger r = rev(n);
        res = new Tuple(true, n);
        List<BigInteger> seen = new ArrayList<>();

        for (int i = 0; i < 500; i++) {
            n = n.add(r);
            r = rev(n);

            if (n.equals(r)) {
                res = new Tuple(false, BigInteger.ZERO);
                break;
            }

            if (cache.containsKey(n)) {
                res = cache.get(n);
                break;
            }

            seen.add(n);
        }

        for (BigInteger bi : seen)
            cache.put(bi, res);

        return res;
    }

    public static void main(String[] args) {

        List<BigInteger> seeds = new ArrayList<>();
        List<BigInteger> related = new ArrayList<>();
        List<BigInteger> palin = new ArrayList<>();

        for (int i = 1; i <= 10_000; i++) {
            BigInteger n = BigInteger.valueOf(i);

            Tuple t = lychrel(n);

            if (!t.flag)
                continue;

            if (n.equals(t.bi))
                seeds.add(t.bi);
            else
                related.add(t.bi);

            if (n.equals(t.bi))
                palin.add(t.bi);
        }

        System.out.printf("%d Lychrel seeds: %s%n", seeds.size(), seeds);
        System.out.printf("%d Lychrel related%n", related.size());
        System.out.printf("%d Lychrel palindromes: %s%n", palin.size(), palin);
    }
}