Climbing Stairs Problem in DSA Go - 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 1 or 2 steps at a time.

beginner

What is the base case for the Climbing Stairs Problem?

For 1 step, there is 1 way to climb. For 2 steps, there are 2 ways: (1+1) or (2).

intermediate

How do you express the number of ways to climb n stairs using previous results?

ways(n) = ways(n-1) + ways(n-2). This means the ways to reach step n is the sum of ways to reach step n-1 and n-2.

intermediate

Why does the Climbing Stairs Problem relate to the Fibonacci sequence?

Because the number of ways to climb stairs follows the same pattern as Fibonacci numbers, where each number is the sum of the two before it.

advanced

What is the time complexity of a simple recursive solution without memoization for the Climbing Stairs Problem?

It is exponential, roughly O(2^n), because it recalculates the same subproblems many times.

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

A 3

B 4

C 2

D 5

The ways are: (1+1+1), (1+2), (2+1), so 3 ways.

What does ways(n) = ways(n-1) + ways(n-2) represent in the Climbing Stairs Problem?

A Number of ways to climb n-3 stairs

B Number of steps taken

C Number of stairs minus one

D Number of ways to climb n stairs

It shows the total ways to climb n stairs based on previous two steps.

Which of these is NOT a base case for the Climbing Stairs Problem?

A ways(1) = 1

B ways(2) = 2

C ways(0) = 0

D ways(3) = 3

ways(0) is usually considered 1 (one way to stand still), not 0.

What is the main drawback of a naive recursive solution for this problem?

A Recalculates subproblems many times

B Uses too much memory

C Does not work for large n

D Only works for even n

Naive recursion repeats calculations, causing exponential time.

Which technique can optimize the Climbing Stairs Problem to run in linear time?

A Binary Search

B Memoization

C Sorting

D Greedy Algorithm

Memoization stores results of subproblems to avoid repeated work.

Explain how the Climbing Stairs Problem can be solved using dynamic programming.

Think about how you can remember the number of ways to climb smaller steps to find the answer for bigger steps.

Describe why the Climbing Stairs Problem is similar to the Fibonacci sequence and how that helps in solving it.

Focus on how each step count depends on the two before it.

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 distinct ways to reach the top

B. The minimum number of steps to reach the top

C. The maximum number of steps you can take at once

D. The number of steps you skip

Solution

1. Step 1: Understand the problem statement

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

2. Step 2: Identify what is being counted

   We count all possible sequences of 1-step and 2-step moves that sum to n steps.

3. Final Answer:

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

4. Quick Check:

   Counting sequences of 1 or 2 steps = total ways [OK]

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

Common Mistakes:

• Confusing number of ways with minimum steps
• Thinking about maximum steps allowed
• Counting skipped steps instead of ways

2. Which Go function signature correctly represents a function to solve the Climbing Stairs problem for n steps?

easy

A. func climbStairs(n int) bool

B. func climbStairs(n string) string

C. func climbStairs() int

D. func climbStairs(n int) int

Solution

1. Step 1: Identify input and output types

   The input is the number of steps (integer), output is the count of ways (integer).

2. Step 2: Match function signature

   Only func climbStairs(n int) int matches input int and output int.

3. Final Answer:

   func climbStairs(n int) int -> Option D

4. Quick Check:

   Input int, output int = correct signature [OK]

Hint: Input and output both must be int for steps and ways [OK]

Common Mistakes:

• Using string instead of int for steps
• Missing input parameter
• Returning bool instead of int

3. What is the output of this Go code for n = 4?

func climbStairs(n int) int {
    if n <= 2 {
        return n
    }
    a, b := 1, 2
    for i := 3; i <= n; i++ {
        a, b = b, a+b
    }
    return b
}

fmt.Println(climbStairs(4))

medium

A. 7

B. 6

C. 5

D. 4

Solution

1. Step 1: Trace initial values and loop

   Start with a=1, b=2. Loop i=3 to 4:
   i=3: a=2, b=3
   i=4: a=3, b=5

2. Step 2: Return final value

   Return b=5 after loop ends.

3. Final Answer:

   5 -> Option C

4. Quick Check:

   Ways for 4 steps = 5 [OK]

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

Common Mistakes:

• Returning a instead of b
• Off-by-one error in loop range
• Incorrect initial values

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

func climbStairs(n int) int {
    if n == 1 {
        return 1
    }
    a, b := 1, 2
    for i := 2; i <= n; i++ {
        a, b = b, a+b
    }
    return b
}

medium

A. Return statement should return a instead of b

B. Loop starts at wrong index causing extra iteration

C. Initial values of a and b are incorrect

D. Missing base case for n == 2

Solution

1. Step 1: Analyze loop start and iterations

   Loop starts at i=2 but initial values already cover steps 1 and 2, causing one extra iteration.

2. Step 2: Correct loop start

   Loop should start at i=3 to avoid counting extra ways.

3. Final Answer:

   Loop starts at wrong index causing extra iteration -> Option B

4. Quick Check:

   Loop index off by one causes wrong count [OK]

Hint: Start loop at 3 when initial values cover steps 1 and 2 [OK]

Common Mistakes:

• Starting loop at 2 instead of 3
• Changing initial values incorrectly
• Returning wrong variable

5. You want to find the number of ways to climb 10 steps but can take 1, 2, or 3 steps at a time. How would you modify the climbing stairs approach?

hard

A. Use dynamic programming with dp[i] = dp[i-1] + dp[i-2] + dp[i-3]

B. Use the same formula dp[i] = dp[i-1] + dp[i-2]

C. Only count ways taking 3 steps at a time

D. Multiply the number of ways for 1 and 2 steps

Solution

1. Step 1: Understand the new step options

   Now you can take 1, 2, or 3 steps at a time, so ways to reach step i depend on previous 3 steps.

2. Step 2: Formulate recurrence relation

   Number of ways dp[i] = dp[i-1] + dp[i-2] + dp[i-3] to include all possibilities.

3. Final Answer:

   Use dynamic programming with dp[i] = dp[i-1] + dp[i-2] + dp[i-3] -> Option A

4. Quick Check:

   Include all step sizes in sum for dp [OK]

Hint: Sum ways from last 3 steps for 1,2,3 step moves [OK]

Common Mistakes:

• Using old formula for only 1 or 2 steps
• Ignoring 3-step moves
• Multiplying counts instead of adding