How many times does the digit 3 appear in the counts from 1 to 100?

The problem asks to count how many times the digit 3 appears when counting from 1 to 100.

The digit 3 can appear in two positions: the units place and the tens place.

Numbers with 3 in the units place:

3, 13, 23, 33, 43, 53, 63, 73, 83, 93

This gives 10 occurrences.

Numbers with 3 in the tens place:

30, 31, 32, 33, 34, 35, 36, 37, 38, 39

This gives 10 occurrences.

Total occurrences $= 10 + 10$

$= 20$

Note that 33 contains two 3s (one in the units place and one in the tens place), and both are counted separately.

Therefore, the digit 3 appears 20 times when counting from 1 to 100.