Climbing Stairs

class Solution {
public:
    int climbStairs(int steps) {
        double sqrtOf5 = sqrt(5);
        double golden = (1 + sqrtOf5) / 2;
        double inverseGolden = (1 - sqrtOf5) / 2;
        return (int)((pow(golden, steps + 1) - pow(inverseGolden, steps + 1)) / sqrtOf5);
    }
};

The problem "Climbing Stairs" involves calculating the number of distinct ways to climb a given number of stairs where you can advance one or two steps at a time. The given C++ solution utilizes a mathematical approach involving the Fibonacci sequence due to the nature of the problem, where the number of ways to reach the nth step is the sum of the ways to reach the (n-1)th and (n-2)th steps.

Here's a breakdown of what the code does:

• Calculate values needed from the Fibonacci formula:
  • sqrtOf5: the square root of 5.
  • golden: Phi (the golden ratio), calculated as (1 + sqrtOf5) / 2.
  • inverseGolden: The conjugate of the golden ratio, (1 - sqrtOf5) / 2.
• For the given number of steps, the function climbStairs determines the number of ways to climb using the formula derived from Binet's Fibonacci number formula.
• It returns the result by calculating ((pow(golden, steps + 1) - pow(inverseGolden, steps + 1)) / sqrtOf5) which effectively calculates the nth Fibonacci number where n is steps + 1 to account for starting on the ground (0th step).

The approach ensures efficient calculation leveraging the closed-form expression of Fibonacci, often called Binet's Formula, which allows for direct computation without iterative or recursive methods that could be less efficient for large inputs.

public class Solution {
    public int climbStairs(int steps) {
        double squareRootOf5 = Math.sqrt(5);
        double goldenRatio = (1 + squareRootOf5) / 2;
        double inverseGoldenRatio = (1 - squareRootOf5) / 2;
        return (int) ((Math.pow(goldenRatio, steps + 1) - Math.pow(inverseGoldenRatio, steps + 1)) / squareRootOf5);
    }
}

The "Climbing Stairs" solution in Java leverages the Fibonacci series to determine the number of ways to reach the top of a stairwell given a number of steps. The method climbStairs uses a formula derived from Binet's Fibonacci number formula:

• Calculate the square root of 5.
• Define the golden ratio and its inverse.
• Use these values to compute the Fibonacci sequence formula that applies to climbing stairs: (Math.pow(goldenRatio, steps + 1) - Math.pow(inverseGoldenRatio, steps + 1)) / squareRootOf5.

This formula directly calculates the correct Fibonacci value for the number of ways to climb the stairs using a combination of exponentiation and natural constants to achieve an efficient and clear solution. This method handles the input steps effectively, offering a mathematically sound approach to solving the problem without iterative or recursive methods that are often less efficient.

int calculateSteps(int steps) {
    double root5 = sqrt(5);
    double golden = (1 + root5) / 2;
    double silver = (1 - root5) / 2;
    return (int)((pow(golden, steps + 1) - pow(silver, steps + 1)) / root5);
}

The provided C function calculateSteps computes the number of ways to climb a given number of stairs using a mathematical approach based on the Fibonacci sequence. Each step can be calculated with the formula involving the golden ratio (approximately 1.618) and its conjugate, the silver ratio.

• The function accepts an integer steps as its parameter, representing the total number of steps.

• It starts by computing the square root of 5 and storing it in root5.

• It then calculates the golden ratio (golden) and the silver ratio (silver).

• The result is achieved using a direct application of Binet's Formula where:

  • It calculates the power of golden and silver raised to the parameter steps+1.
  • The difference of these powers divided by root5 gives the result.

• The function finally casts this result to an integer and returns it.

This efficient approach ensures that you can compute the number of ways to climb stairs without iterative or recursive calculations, making it computationally efficient even for large numbers of steps.

var countWays = function (steps) {
    var root5 = Math.sqrt(5);
    var goldenRatio = (1 + root5) / 2;
    var silverRatio = (1 - root5) / 2;
    return Math.floor((Math.pow(goldenRatio, steps + 1) - Math.pow(silverRatio, steps + 1)) / root5);
};

The provided JavaScript function countWays solves the problem of determining the number of ways to climb a staircase with a given number of steps, utilizing the Fibonacci sequence.

• The function receives an integer steps as its parameter, representing the total number of steps in the staircase.
• It initially calculates the square root of 5 and stores this value in root5.
• The function defines goldenRatio which symbolizes the (1 + root5)/2, commonly related to the positive part of the Fibonacci closed-form expression.
• Similarly, silverRatio is calculated as (1 - root5)/2, representing the negative part of the Fibonacci formula.
• Lastly, the function computes the number of ways by using the formula of the Fibonacci sequence applied directly to the staircase problem: (goldenRatio^(steps + 1) - silverRatio^(steps + 1)) / root5.
• Math.floor is used to ensure that the result is returned as an integer, which is the expected format for counting distinct ways.

By using the Fibonacci sequence's properties, this function efficiently calculates the number of distinct ways to climb the stairs with complexity essentially linear in terms of power calculation time, making it very efficient for large numbers of steps. The Fibonacci-based approach is both elegant and effective for the climbing stairs problem, exploiting the natural pattern in which climbing configurations grow.

class Solution:
    def countWays(self, steps: int) -> int:
        root5 = 5 ** 0.5
        golden_ratio = (1 + root5) / 2
        inv_golden_ratio = (1 - root5) / 2
        return int((golden_ratio ** (steps + 1) - inv_golden_ratio ** (steps + 1)) / root5)

The provided Python solution addresses the problem of determining the number of ways to climb stairs, using the Fibonacci sequence formula. The method countWays in the Solution class calculates the number of distinct ways to reach the top of a staircase with a given number of steps, referred to as steps.

The solution utilizes the mathematical concept of the golden ratio and its inverse to compute the Fibonacci number that corresponds to the provided number of steps plus one. This is achieved through the following process:

• Calculate the square root of 5.
• Derive the golden ratio as (1 + root5) / 2.
• Calculate the inverse of the golden ratio as (1 - root5) / 2.
• Use these constants to apply the formula for finding the nth Fibonacci number: ((golden_ratio ** (steps + 1) - inv_golden_ratio ** (steps + 1)) / root5).
• Convert the result into an integer which represents the total ways to climb the staircase.

This approach effectively harnesses a mathematical shortcut (Binet's Fibonacci Number Formula) to achieve a result without the need for iterative looping or recursion, thus optimizing performance by directly computing the result through mathematical expressions.