The problem asks how many numbers are in the sequence $0, 1, 2, 10, 11, 12, 20, \dots, 10000$ when written in base 3.

Key Insights:

1. Base 3 System:

   • The numbers are expressed in base 3, where the digits can only be $0, 1, 2$.

2. General Form:

   • In base 3, a number with $n$ digits can have up to $3^n$ different values (from $0$ to $3^n - 1$).

3. Sequence Stops at $10000$ (in base 3):

   • The largest number in the sequence is $10000$ in base 3, which is $3^4 = 81$ in decimal.

4. Counting All Numbers:

   • The sequence starts at $0$ and goes up to $10000$ (inclusive). This corresponds to $81 + 1 = 82$ numbers (since $0$ is included).

Conclusion:

There are 82 numbers in this sequence.

Would you like more details or further explanations on any part of this solution?

Follow-up Questions:

1. What is the significance of the base 3 system compared to base 10?
2. How is the number $10000$ in base 3 converted to decimal?
3. Can this method be applied to any other base system?
4. What are the first 10 numbers in base 3, and how do they compare to base 10?
5. How can you quickly identify the number of digits needed to represent a given number in base 3?

Tip:

When working with numbers in different bases, practice converting them between base 10 and the desired base to build familiarity.