The digit 1 appears 136 times when all the page numbers of a book are written…

Concept: To count how many times a particular digit appears across all numbers from 1 to N, break the range into blocks of 10 and 100. In every complete block of 10 consecutive numbers, each digit from 0-9 appears exactly once in the units place, and in every complete run of 100 numbers headed by a fixed leading digit, that leading digit appears in the hundreds place for all 100 numbers in that block.

Applying this to the digit 1:

1. From page 1 to 99, the digit 1 appears as a units digit 10 times (1, 11, 21, ..., 91) and as a tens digit 10 times (10-19), giving 20 occurrences.

2. From page 100 to 199, the hundreds digit is 1 for every one of these 100 pages, contributing 100 occurrences, plus the same tens/units pattern as 0-99 repeating within each hundred, contributing another 20 occurrences — 120 occurrences in total.

3. So pages 1 to 199 together contain 20 + 120 = 140 occurrences of the digit 1.

4. The book is stated to contain only 136 occurrences, which is 140 - 136 = 4 fewer than the count through page 199.

5. Pages 196, 197, 198 and 199 each contain exactly one occurrence of the digit 1 (its hundreds digit), so these four pages together account for exactly the extra 4 occurrences.

6. Removing these four pages means the book must stop at page 195, leaving exactly 140 - 4 = 136 occurrences of the digit 1.

Cross-check: page 194 gives 140 - 5 = 135 occurrences (one short), and page 200 already gives 140 occurrences (four too many), so 195 is the unique page count that produces exactly 136 occurrences.