How many different ways can an elf climb a ten-step staircase if he or she climbs only one of two steps a time?
89
This question
To find the number of different ways an elf can climb a ten-step staircase, where they can climb only one or two steps at a time, we can use a simple concept called "dynamic programming" to solve the problem.
One way to approach this is by breaking down the problem into smaller subproblems. Let's consider the last step the elf climbs. The elf can either take one step or two steps to reach this last step.
1. If the elf takes one step to reach the last step, then the remaining steps would be 9 (since the elf has taken one step).
2. If the elf takes two steps to reach the last step, then the remaining steps would be 8 (since the elf has taken two steps).
To find the total number of ways the elf can climb the ten-step staircase, we can sum up the number of ways for each of these two scenarios:
1. The number of ways to climb a nine-step staircase, which is a subproblem we can solve in a similar manner.
2. The number of ways to climb an eight-step staircase, also a subproblem we can solve similarly.
We continue this process until we reach a base case where the number of steps is zero or less.
Let's represent this process using a recursive function in Python:
```python
def count_ways(n):
if n <= 1:
return 1
else:
return count_ways(n-1) + count_ways(n-2)
num_ways = count_ways(10)
print(num_ways)
```
In this code, the `count_ways` function takes a parameter `n`, which represents the number of steps remaining. The function checks if `n` is less than or equal to 1 (the base case) and returns 1. Otherwise, it recursively calls itself passing `n-1` and `n-2` as arguments, representing the number of steps for the two scenarios mentioned above. Finally, the code calls the `count_ways` function with the number of steps set to 10 and prints the result.
When you run this code, it will output `89`, which means there are 89 different ways for the elf to climb the ten-step staircase.