Step by Step solution about How to resolve the algorithm Juggler sequence step by step in the C++ programming language

The provided C++ code explores the Juggler sequence, which is defined as follows:

Given a positive integer n, the Juggler sequence starts with n. In each subsequent step, if the current number is even, it is replaced by its square root, and if it is odd, it is replaced by the square root of its cube. The sequence continues until the number becomes 1.

The code defines a function called juggler that takes a positive integer n as input and returns a tuple containing the following information about the Juggler sequence starting with n:

• count: The number of steps required to reach 1.
• max_count: The maximum number of steps in the sequence.
• max: The maximum value reached in the sequence.
• digits: The number of digits in the maximum value reached in the sequence.

The main function of the code demonstrates the use of the juggler function by printing the results for a range of positive integers (n = 20 to n = 39) and a set of specific large positive integers.

Here's a detailed explanation of the code:

1. Function juggler:

   • It takes a positive integer n as input.
   • It initializes three variables:
     • count to 0 (to count the steps in the sequence).
     • max_count to 0 (to track the maximum number of steps in the sequence).
     • a to n (to represent the current number in the sequence).
     • max to n (to track the maximum value reached in the sequence).
   • It enters a loop that continues until a becomes 1.
   • Inside the loop:
     • If a is even, it updates a to its square root (a = sqrt(a)).
     • If a is odd, it updates a to the square root of its cube (a = sqrt(big_int(a * a * a)) using GMP's big_int class for large integer arithmetic).
     • It increments count by 1 to track the steps.
     • It checks if the current a is greater than the previously recorded maximum max. If so, it updates max and max_count to reflect the new maximum value and the number of steps it took to reach that maximum.
   • Finally, it returns a tuple containing count, max_count, max, and the number of digits in max.

2. Main Function (main):

   • It sets the output locale to enable locale-specific formatting.
   • It prints the header line: "n l[n] i[n] h[n]" to indicate the columns in the output: n (number), l[n] (steps to reach 1), i[n] (maximum steps in the sequence), and h[n] (maximum value reached).
   • It enters a loop to print the results for integers n from 20 to 39.
   • For each n, it calls the juggler function and unpacks the returned tuple into variables (count, max_count, max, and digits).
   • It prints the formatted results.
   • After the first loop, it prints a new header line: "n l[n] i[n] d[n]" to indicate the columns in the output: n (number), l[n] (steps to reach 1), i[n] (maximum steps in the sequence), and d[n] (number of digits in the maximum value reached).
   • It enters a second loop to print the results for a set of predefined large integers (n).
   • For each n, it calls the juggler function and unpacks the returned tuple into variables (count, max_count, max, and digits).
   • It prints the formatted results.

#include <cassert>
#include <iomanip>
#include <iostream>
#include <string>

#include <gmpxx.h>

using big_int = mpz_class;

auto juggler(int n) {
    assert(n >= 1);
    int count = 0, max_count = 0;
    big_int a = n, max = n;
    while (a != 1) {
        if (a % 2 == 0)
            a = sqrt(a);
        else
            a = sqrt(big_int(a * a * a));
        ++count;
        if (a > max) {
            max = a;
            max_count = count;
        }
    }
    return std::make_tuple(count, max_count, max, max.get_str().size());
}

int main() {
    std::cout.imbue(std::locale(""));
    std::cout << "n    l[n]  i[n]   h[n]\n";
    std::cout << "--------------------------------\n";
    for (int n = 20; n < 40; ++n) {
        auto [count, max_count, max, digits] = juggler(n);
        std::cout << std::setw(2) << n << "    " << std::setw(2) << count
                  << "    " << std::setw(2) << max_count << "    " << max
                  << '\n';
    }
    std::cout << '\n';
    std::cout << "       n       l[n]   i[n]   d[n]\n";
    std::cout << "----------------------------------------\n";
    for (int n : {113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443,
                  275485, 1267909, 2264915, 5812827, 7110201, 56261531,
                  92502777, 172376627, 604398963}) {
        auto [count, max_count, max, digits] = juggler(n);
        std::cout << std::setw(11) << n << "    " << std::setw(3) << count
                  << "    " << std::setw(3) << max_count << "    " << digits
                  << '\n';
    }
}