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)?

3

4,5
12,15
13,14,16,17
39,42,48,51
40,41,43,44,49,50,52,53
120,123,129,132,147,150,156,159
121,122,124,125,130,131,133,134,148,149,151,152,157,158,160,161

363,366,372,375,390,393,399,402,444,447,453,456,471,474,480,483

32 numbers, all above 363, so the next step will exceed 1000

Looks like 1+2+2+4+4+8+8+16+16 = 61 numbers

I guess you could show that there are never any duplicates in the list (since adding 1 or 2 to a multiple of 3 spreads out the results enough)

Then show the number of reachable numbers at each step,
then show when the minimum number at a step will exceed 1000.

Not quite sure what principle was to be studied here.

oops. forgot to add in the 32 for the last line

To find the number of reachable numbers between 1 and 1000, we can simulate the game and keep track of all possible numbers that could be reached.

We will start with the number 1 and initialize a set to store all reachable numbers. Then, we will iterate over the numbers from 1 to 1000, simulating the game rules for each number.

For each number n, we will multiply it by 3 (in Alan's turn) and check all possible outcomes of Bob's turn. Bob can either add 1 or 2 to the current number. We will add these new numbers to the set of reachable numbers.

Here is the Python code to do the simulation:

```python
reachable_numbers = set()
reachable_numbers.add(1)

for n in range(1, 1001):
# Alan's turn: multiply by 3
reachable_numbers.add(n * 3)

# Bob's turn: add 1 or 2
reachable_numbers.add(n + 1)
reachable_numbers.add(n + 2)

# Count the number of reachable numbers between 1 and 1000
count = len([n for n in reachable_numbers if 1 <= n <= 1000])

print(count)
```

Running this code will give the output as the number of reachable numbers between 1 and 1000.