Climbing Stairs with a Broken Step: A Mathematical Exploration

Have you ever encountered a staircase with a broken step? In this article, we'll explore a mathematical problem where a staircase has 9 steps, with the 5th step being broken and cannot be stepped on. We'll dive into the detailed steps to figure out how many ways one can climb the staircase if you can only move 1 or 2 steps at a time.

Problem Statement and Interpretation

The staircase has a total of 9 steps, numbered from 1 to 9. The 5th step (5th) is broken and cannot be stepped on. You can move either 1 or 2 steps at a time. The climb should be from step 1 to step 9 without any reversals.

Breakdown of the Problem

To solve this problem effectively, we can break it down into smaller segments:

Climb from step 1 to step 4. Climb from step 4 to step 6. Climb from step 6 to step 9.

Since the 5th step is broken, you must reach the 6th step from the 4th step and then proceed to the 9th step without stepping on the 5th one.

Recurrence Relation and Boundary Conditions

Let's define ( T(n) ) as the number of ways to climb ( n ) steps. We can use the following recurrence relation:

[ T(n) T(n-1) T(n-2) ]

Base Cases:

( T(1) 1 ): There is only one way to climb one step, which is by taking one step. ( T(2) 2 ): There are two ways to climb two steps: 1-2 or 2-2.

Solving for Specific Cases

To solve for ( T(4) ) and ( T(6) ):

( T(3) T(2) T(1) 2 1 3 ) ( T(4) T(3) T(2) 3 2 5 ) ( T(5) ): Since the 5th step is broken, we cannot step on it, so we go from the 4th step directly to the 6th step. ( T(6) T(5) T(4) 1 5 6 ) ( T(7) T(6) T(5) 6 1 7 ) ( T(8) T(7) T(6) 7 6 13 ) ( T(9) T(8) T(7) 13 7 20 )

Total Ways to Climb the Steps

Given that the total number of ways to climb from step 1 to step 9 can be calculated by:

[ T(9) T(4) times T(2) times T(3) ]

Substituting the values we computed:

[ T(9) 5 times 1 times 5 25 ]

Hence, the total number of ways to climb the staircase is 25.

Conclusion: We have explored a detailed mathematical approach to solving the problem of climbing a staircase with a broken step. By breaking down the problem into smaller, manageable segments and using a recurrence relation, we were able to determine that there are 25 unique ways to climb from step 1 to step 9, given the constraints.