Alright, let's break down how to compute the total number of possible states in the Vacuum Cleaner World—this becomes straightforward once you map out all the variables that define a single state.

Key Components of a State #

Every state in this environment is defined by two distinct pieces of information:

• Vacuum Cleaner Position: The current location of the vacuum. If there are N total positions in the world, there are N possible choices for where the vacuum can be.
• Cleanliness of Each Position: Every position is either dirty or clean. For each position, that's 2 possible states. With N positions, the total number of cleanliness combinations is 2^N (since each position's state is independent of the others).

Total State Count Formula #

To get the total number of unique states, multiply the number of possible vacuum positions by the number of possible cleanliness combinations:

Total States = N × 2^N

Example: 2-Position Vacuum World #

Let's use the common example with N=2 positions (say, Position A and Position B):

1. Vacuum positions: 2 options (A or B)
2. Cleanliness combinations: 2^2 = 4 (both dirty, A dirty/B clean, A clean/B dirty, both clean)
3. Total states: 2 × 4 = 8

You can list out all 8 states to confirm:

• Vacuum at A, A dirty, B dirty
• Vacuum at A, A dirty, B clean
• Vacuum at A, A clean, B dirty
• Vacuum at A, A clean, B clean
• Vacuum at B, A dirty, B dirty
• Vacuum at B, A dirty, B clean
• Vacuum at B, A clean, B dirty
• Vacuum at B, A clean, B clean

内容的提问来源于stack exchange，提问作者Samiha