Alan and Bob are playing with numbers. Starting with the number n=1 they take turns performing arithmetic manipulations with n. Alan goes first and always multiplies n by 3. Bob adds 1 or 2 to the number n, depending on a coin toss. A number is called reachable, if it can possibly appear in this game. How many reachable numbers are there between 1 and 1000 (inclusive)?
To determine the number of reachable numbers between 1 and 1000, we can use a dynamic programming approach.
First, create an array of boolean values of size 1001 (to include the range 1 to 1000). Initialize all values in the array as false.
Next, set the first value, index 1, in the array as true because it is given in the problem statement that the initial number, n=1, is reachable.
Then, iterate from 1 to 1000. For each index i, check if it is reachable by multiplying i by 3 and updating the corresponding index in the array to true.
After that, iterate from 1 to 1000 again and check if each index i is reachable through the operations performed by Bob. If i-1 or i-2 is reachable, update the corresponding index in the array to true.
Finally, count the number of true values in the array from index 1 to 1000, which represents the total number of reachable numbers.
Here is a Python code snippet that implements the above approach:
```python
def count_reachable_numbers():
reachable = [False] * 1001
reachable[1] = True
for i in range(1, 1001):
if reachable[i]:
reachable[i * 3] = True
for i in range(1, 1001):
if i - 1 <= 1000 and reachable[i - 1]:
reachable[i] = True
if i - 2 <= 1000 and reachable[i - 2]:
reachable[i] = True
return sum(reachable)
print(count_reachable_numbers())
```
Running this code will output the total number of reachable numbers between 1 and 1000.