The number of times the digit 3 will be written when listing the integers from 1 to 1000 is ________.

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

We consider all numbers from 000 to 999 (padding with leading zeros to make them 3-digit). This gives us 1000 numbers. The number 1000 has no digit 3, so it does not contribute.

Each number has 3 digit positions: units, tens, and hundreds. Across all 1000 numbers, there are $$1000 \times 3 = 3000$$ total digit positions.

By symmetry, each digit from 0 to 9 appears equally often in each position. In the units place, among the numbers 000 to 999, the digit 3 appears in every 10th number, so it appears $$1000/10 = 100$$ times in the units place.

Similarly, the digit 3 appears 100 times in the tens place and 100 times in the hundreds place.

Therefore, the total number of times the digit 3 is written is $$100 + 100 + 100 = 300$$.