Problem Statement
Climbing Stairs
You are climbing a staircase. It takes n steps to reach the top.
Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?
Examples
Example 1:
Input: n = 2
Output: 2
Explanation: There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps
Example 2:
Input: n = 3
Output: 3
Explanation: There are three ways to climb to the top.
1. 1 step + 1 step + 1 step
2. 1 step + 2 steps
3. 2 steps + 1 step
Constraints
- 1 <= n <= 45
Problem Breakdown
To solve this problem, we need to:
- This problem follows the Fibonacci sequence pattern.
- To reach the i-th step, you can either come from the (i-1)-th step by taking 1 step, or from the (i-2)-th step by taking 2 steps.
- Therefore, the number of ways to reach the i-th step is the sum of the number of ways to reach the (i-1)-th step and the (i-2)-th step.
- This can be solved using dynamic programming or a simple iterative approach.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Climbing Stairs
You are climbing a staircase. It takes n steps to reach the top.
Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?
Examples
Example 1:
There are two ways to climb to the top. 1. 1 step + 1 step 2. 2 steps
Example 2:
There are three ways to climb to the top. 1. 1 step + 1 step + 1 step 2. 1 step + 2 steps 3. 2 steps + 1 step
Problem Breakdown
Constraints:
- 1 <= n <= 45
Key Insights:
This problem follows the Fibonacci sequence pattern.
To reach the i-th step, you can either come from the (i-1)-th step by taking 1 step, or from the (i-2)-th step by taking 2 steps.
Therefore, the number of ways to reach the i-th step is the sum of the number of ways to reach the (i-1)-th step and the (i-2)-th step.
This can be solved using dynamic programming or a simple iterative approach.
Real-World Application
This problem has several practical applications:
Combinatorial Problems
Solving problems that involve counting the number of ways to arrange or select objects.
Path Finding
Finding the number of possible paths in a grid or network with certain movement constraints.
Algorithm Design
Understanding recursive patterns and dynamic programming techniques for solving complex problems.