## Task 4 - 300614

The pages of a book are numbered from 1 to 235.

a) How many times was the digit 4 used in total for the numbering?

b) How many times was the digit 0 used in total for the numbering?

c) How many digits in total need to be printed for this numbering?
a) The digit 4 is used exactly once in the units place for the numbers 1 to 10, 11 to 20, ..., 221 to 230, 231 to 235, i.e., a total of 24 times.

At the tens place, it is used exactly once for the numbers 40, 41, ..., 49, 140, 141, ..., 149, and not for any other numbers from 1 to 235, i.e., a total of 20 times.

At the hundreds place, it is not used for any numbers from 1 to 235. Therefore, the digit 4 is used a total of 44 times in the numbering.

b) The digit 0 is used exactly once in the units place for the numbers 1 to 10, 11 to 20, ..., 221 to 230, but not for the numbers 231 to 235, i.e., a total of 23 times.

At the tens place, it is used exactly once for the numbers 100, 101, ..., 109, 200, 201, ..., 209, and not for any other numbers from 1 to 235, i.e., a total of 20 times.

At the hundreds place, it is not used for any numbers from 1 to 235. Therefore, the digit 0 is used a total of 43 times in the numbering.

c) Among the numbers from 1 to 235, there are exactly 9 single-digit numbers \(1, 2, \ldots, 9\), exactly 90 two-digit numbers \(10, 11, \ldots, 99\), and exactly 136 three-digit numbers \(100, 101, \ldots, 235\).

Therefore, a total of \(9 + 90 \cdot 2 + 136 \cdot 3 = 9 + 180 + 408 = 597\) digits need to be printed for the numbering.

Solutions of the I. Round 1960 taken from \([5]\)

### 3.31.2 II. Round 1990, Class 6