Climbing Stairs Problem in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Climbing Stairs Problem?

It is a problem where you count how many distinct ways you can climb to the top of a staircase with n steps, taking either 1 or 2 steps at a time.

beginner

What is the base case for the Climbing Stairs Problem when using recursion?

When n = 1, there is only 1 way to climb (one step). When n = 2, there are 2 ways (1+1 steps or a single 2-step).

intermediate

How can the Climbing Stairs Problem be related to the Fibonacci sequence?

The number of ways to climb n stairs equals the sum of ways to climb (n-1) stairs and (n-2) stairs, just like Fibonacci numbers add the two previous numbers.

intermediate

Why is dynamic programming useful for the Climbing Stairs Problem?

Dynamic programming stores results of smaller problems to avoid repeated calculations, making the solution efficient and fast.

intermediate

What is the time complexity of the dynamic programming solution for the Climbing Stairs Problem?

The time complexity is O(n) because it calculates the number of ways for each step from 1 to n once.

How many ways can you climb 3 stairs if you can take 1 or 2 steps at a time?

A 3

B 2

C 4

D 1

The ways are: 1+1+1, 1+2, and 2+1, totaling 3 ways.

What does the Climbing Stairs Problem count?

A Number of steps in the staircase

B Number of ways to climb stairs

C Time taken to climb stairs

D Height of the staircase

It counts the distinct ways to reach the top by taking 1 or 2 steps.

Which approach avoids repeated calculations in the Climbing Stairs Problem?

A Greedy

B Brute Force

C Dynamic Programming

D Divide and Conquer

Dynamic Programming stores intermediate results to avoid repeats.

What is the relation between Climbing Stairs and Fibonacci numbers?

A Ways to climb n stairs = n * 2

B No relation

C Ways to climb n stairs = n squared

D Ways to climb n stairs = ways to climb (n-1) + ways to climb (n-2)

The problem follows the Fibonacci pattern of summing previous two results.

What is the minimum number of steps you can take at a time in the Climbing Stairs Problem?

A 1

B 2

C 3

D 0

You can take either 1 or 2 steps at a time, so minimum is 1.

Explain how to solve the Climbing Stairs Problem using dynamic programming.

Think about how you can use answers for smaller steps to find the answer for n steps.

Describe the connection between the Climbing Stairs Problem and the Fibonacci sequence.

Focus on how the number of ways to reach step n depends on the two previous steps.

Practice

1. What does the Climbing Stairs problem ask you to find?
Given n steps, how many distinct ways can you climb to the top if you can take 1 or 2 steps at a time?

easy

A. The total number of steps climbed

B. The maximum number of steps you can climb at once

C. The minimum number of steps to reach the top

D. The number of distinct ways to reach the top

Solution

1. Step 1: Understand the problem statement

   The problem asks for the count of different ways to climb to the top using 1 or 2 steps at a time.

2. Step 2: Identify what is being asked

   We want the total number of distinct sequences of 1-step and 2-step moves that sum to n.

3. Final Answer:

   The number of distinct ways to reach the top -> Option D

4. Quick Check:

   Count ways to climb = number of sequences [OK]

Hint: Count sequences of 1 or 2 steps to reach n [OK]

Common Mistakes:

• Confusing number of ways with minimum steps
• Thinking about maximum steps instead of ways
• Counting total steps instead of sequences

2. Which of the following Python function signatures correctly represents a function to solve the Climbing Stairs problem?

easy

A. def climb_stairs(n: int) -> int:

B. def climb_stairs(n): return str

C. def climb_stairs(n: str) -> int:

D. def climb_stairs() -> int:

Solution

1. Step 1: Check parameter and return types

   The input n is an integer representing steps, and output is an integer count of ways.

2. Step 2: Match function signature

   def climb_stairs(n: int) -> int: correctly uses int input and int output with type hints.

3. Final Answer:

   def climb_stairs(n: int) -> int: -> Option A

4. Quick Check:

   Input int, output int function signature [OK]

Hint: Input is int steps, output is int ways [OK]

Common Mistakes:

• Using string type for input
• Returning string instead of int
• Missing input parameter

3. What is the output of this code?

def climb_stairs(n: int) -> int:
    if n <= 2:
        return n
    a, b = 1, 2
    for _ in range(3, n + 1):
        a, b = b, a + b
    return b

print(climb_stairs(5))

medium

A. 5

B. 8

C. 13

D. 10

Solution

1. Step 1: Trace initial values and base cases

   For n=5, start with a=1, b=2. Loop runs for i=3 to 5.

2. Step 2: Calculate loop iterations

   Iteration 3: a=2, b=3; Iteration 4: a=3, b=5; Iteration 5: a=5, b=8.

3. Final Answer:

   8 -> Option B

4. Quick Check:

   Ways to climb 5 steps = 8 [OK]

Hint: Use Fibonacci-like addition for steps count [OK]

Common Mistakes:

• Returning n for all inputs
• Off-by-one errors in loop
• Mixing a and b values

4. Find the error in this code snippet for climbing stairs:

def climb_stairs(n: int) -> int:
    if n == 1:
        return 1
    a, b = 1, 2
    for i in range(2, n):
        a = b
        b = a + b
    return b

print(climb_stairs(4))

medium

A. Variables a and b are updated incorrectly

B. Loop range should start from 3

C. Base case for n == 2 is missing

D. Return statement should be 'return a'

Solution

1. Step 1: Analyze variable updates inside loop

   Inside loop, a is set to b, then b = a + b uses updated a, causing wrong sum.

2. Step 2: Correct update order

   We must update both simultaneously like a, b = b, a + b to avoid using updated a prematurely.

3. Final Answer:

   Variables a and b are updated incorrectly -> Option A

4. Quick Check:

   Simultaneous update needed for correct Fibonacci logic [OK]

Hint: Update a,b together to avoid wrong sums [OK]

Common Mistakes:

• Updating variables sequentially causing wrong values
• Wrong loop range causing missed steps
• Missing base cases

5. You want to find the number of ways to climb 10 steps but can take 1, 2, or 3 steps at a time. Which modification to the standard climbing stairs solution is correct?

hard

A. Use dp[i] = dp[i-1] * dp[i-2] * dp[i-3] with base dp[0]=1, dp[1]=1, dp[2]=2

B. Use dp[i] = dp[i-1] + dp[i-2] with base dp[0]=1, dp[1]=1

C. Use dp[i] = dp[i-1] + dp[i-2] + dp[i-3] with base dp[0]=1, dp[1]=1, dp[2]=2

D. Use dp[i] = dp[i-1] + dp[i-3] with base dp[0]=1, dp[1]=1, dp[2]=2

Solution

1. Step 1: Understand step options and recurrence

   With steps 1, 2, or 3 allowed, ways to reach i = sum of ways to reach i-1, i-2, and i-3.

2. Step 2: Set correct base cases

   dp[0]=1 (one way to stand still), dp[1]=1, dp[2]=2 (1+1 or 2 steps).

3. Final Answer:

   Use dp[i] = dp[i-1] + dp[i-2] + dp[i-3] with base dp[0]=1, dp[1]=1, dp[2]=2 -> Option C

4. Quick Check:

   Sum previous three dp values for 3-step climb [OK]

Hint: Sum ways from last 3 steps for 1-2-3 step climb [OK]

Common Mistakes:

• Multiplying dp values instead of adding
• Using only two previous dp values
• Incorrect base cases