If one asks how many times the digit 9 appears in the sequence of numbers from 1 to 100, the initial answer might be 10 or 11. However, for the correct solution, it is necessary to consider all numbers containing this digit, not just those ending with it, which requires careful consideration.
Let's first examine the count from 1 to 9. In this range, the digit 9 appears only once, in the number 9.
Next, in the range from 10 to 19, the digit 9 is present in the number 19, meaning once.
When analyzing numbers from 20 to 29, the digit 9 appears in the number 29. Similarly, in the ranges from 30 to 39, 40 to 49, 50 to 59, 60 to 69, 70 to 79, and 80 to 89, the digit 9 appears once in each last number (39, 49, 59, 69, 79, 89). Thus, between 20 and 89, the digit 9 appears only 7 times.
The most complex section is the range from 90 to 99. Special attention is required here because the digit 9 is present in the tens place in all ten numbers in this range (90, 91, 92, 93, 94, 95, 96, 97, 98, and 99). Although it appears once in each of these numbers, in the number 99, the digit 9 is written twice. Consequently, in this segment, the digit 9 appears a total of 11 times.
Finally, the number 100 does not contain the digit 9, so it does not add any additional occurrences of this digit to the count.
In summary, in the count from 1 to 9, the digit 9 appears once; from 10 to 19—once; from 20 to 89—seven times; and from 90 to 99—eleven times. Summing all these values (1 + 1 + 7 + 11), the total number of times the digit 9 appears in the count from 1 to 100 is 20. It is important to remember that carelessness about the fact that the digit 9 appears twice in the number 99 can lead to an incorrect answer.


