How many ways are there of spelling out “ABRACADABRA” by traversing the following diamond, always going from one letter to an adjacent one? One of the ways is shown.
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To find the number of ways to spell out "ABRACADABRA" by traversing the diamond, we can use a technique called backtracking or recursion.
First, let's understand the rules of traversing the diamond:
1. You always start at the topmost letter.
2. From a given letter, you can move to one of the two letters in the row below, directly below the current letter.
3. Once you reach the bottom row, you can move to the adjacent letter in the row above.
To count the number of ways, we can follow these steps:
1. Define a recursive function that takes the current position and the current string formed as inputs.
2. Start with an empty string and begin the recursion from the topmost letter, 'A'.
3. In each recursive call, check if the current string matches the desired string, "ABRACADABRA".
4. If the current string matches the desired string, increment a counter variable by one.
5. If the current position is in the bottom row, move to the adjacent letter in the row above and make a recursive call.
6. If the current position is not in the bottom row, make two recursive calls by moving to the two possible adjacent letters in the row below.
7. Return the counter variable after all recursive calls.
Here's a Python code implementation that solves the problem:
```python
def count_ways(position, current_string):
if current_string == "ABRACADABRA": # Condition to check if string matches desired string
return 1
count = 0
if position >= 10: # Check if current position is in the bottom row
count += count_ways(position - 10, current_string + "A") # Move to adjacent letter in the row above
else:
count += count_ways(position + 2, current_string + "R") # Move to the two possible letters in the row below
return count
total_ways = count_ways(0, "")
print("Total ways to spell 'ABRACADABRA':", total_ways)
```
The code will output the total number of ways to spell out "ABRACADABRA" by traversing the given diamond.