A person writes all the numbers from 0 to 99. The number of times digit 3 will…

Concept

In a complete set of fixed-position decimal strings, each digit occurs equally often in each position.

Across the 100 strings from 00 through 99, a chosen digit appears 10 times in the tens place and 10 times in the units place.

Application

1. View the numbers 0 through 9 as 00 through 09 only for positional counting; the added leading zero does not create or remove any occurrence of the digit 3.

2. In the units place, 3 occurs in 03, 13, 23, ..., 93, giving 10 occurrences.

3. In the tens place, 3 occurs in every number from 30 through 39, giving another 10 occurrences.

4. The number 33 contributes twice, once from each position, so the two positional counts may be added: 10 + 10 = 20.

Cross-check

There are 100 two-position strings, hence 200 digit positions in total. Symmetry distributes these positions equally among 10 digits, so each digit occurs 200 / 10 = 20 times.

Therefore, the digit 3 is written 20 times.