We need to count how many times the digit $3$ appears when writing all integers from $1$ to $1000$.

First consider the numbers from $000$ to $999$ using three-digit representation.

• This includes numbers like $001, 002, \dots, 999$.
• In these $1000$ numbers, each digit position (hundreds, tens, units) takes each value $0$ to $9$ equally often.

Count occurrences of digit $3$ in each position:

• In the units place, digit $3$ appears once in every block of $10$ numbers. So it appears $100$ times.
• In the tens place, digit $3$ also appears $100$ times.
• In the hundreds place, digit $3$ also appears $100$ times.

Therefore total occurrences from $000$ to $999$ are $100+100+100=300.$

Now convert this to the range $1$ to $1000$:

• Writing $000$ contributes no digit $3$.
• Numbers $001$ to $999$ correspond exactly to $1$ to $999$ with the same count of digit $3$.
• The number $1000$ has no digit $3$.

Hence, the total number of times digit $3$ is written from $1$ to $1000$ is $300.$

Comparison with stored answer:

• Derived answer: $300$
• Stored correct answer: $300$
• They match.