Count How Often a Chosen Digit Appears
Lesson 13.3 — 统计指定数字在页码中出现的次数
Learn to count a digit by place value, distinguish digit occurrences from page labels containing the digit, and work backward from a total occurrence count.
Answers saved before this lesson update
Earlier answers — original questions
These answers belong to the earlier question wording. They are kept for reference; try the updated questions above and recheck your missions.
Count digit occurrences—not merely page labels
One page number can contain the chosen digit more than once.
One printed page number.
Page 44 is one label containing 4.
Both written positions are counted.
Keep the units separate
Hint 1 — choose a starting point
Count each matching position.
Hint 2 — take the next step
One label may contribute more than one occurrence.
Count by place value
Track the chosen digit separately in the ones, tens, hundreds, and higher places.
Digit 1 from page 1 through 99
Digit 0 from page 1 through 99
Read the place-value ledgers
Hint 1 — choose a starting point
Make a separate list for each place.
Hint 2 — take the next step
A leading zero is never printed.
Worked example: count zeros from page 1 through 320
The worked example solves this by counting zeros in each written place.
Complete problem: A storybook has 320 pages. How many times does the digit 0 appear in all page numbers from 1 through 320?
Page 100
Page 205
Page 320
Rebuild the worked example count
Hint 1 — choose a starting point
Ones-place zeros occur at multiples of ten.
Hint 2 — take the next step
Tens-place zeros occur in the 100–109-style blocks.
Guided practice: count digit 1 from page 1 through 131
A page such as 111 contributes three occurrences—one in every place.
Complete practice problem
A Grade 5 mathematics book has 131 pages. How many times does the digit 1 appear in all page numbers from 1 through 131?
Complete the place-value total
Hint 1 — choose a starting point
Include both 1s in 101 and all three in 111.
Hint 2 — take the next step
Count ones, tens, and hundreds separately, then add.
Use a live chosen-digit occurrence laboratory
Choose a whole-number final page from 1 through 99,999 and a digit. The laboratory counts visible occurrences by place value.
| Place | Occurrences | Examples (highlight only this place) |
|---|
Sample page labels containing the digit (first 18 matches at most)
Use the laboratory to investigate original-style counts
Hint 1 — choose a starting point
Choose the requested endpoint and digit.
Hint 2 — take the next step
Compare place totals rather than just sample labels.
Distinguish occurrences from page labels containing the digit
These two counts differ whenever one page label contains the digit more than once.
occurrences of digit 4
page labels containing digit 4
Count page labels that avoid digit 5
Count the allowed labels directly, grouped by width.
There are nine allowed digits (0–9 except 5), but zero cannot lead a label. A three-digit label here starts with 1, 2, 3, or 4. Page 500 contains a 5 and is excluded:
Do not mix the questions
Occurrences: How many written positions equal the chosen digit?
Containing labels: How many page numbers contain the chosen digit at least once?
Avoiding labels: How many page numbers contain none of that digit?
Choose the correct count
Hint 1 — choose a starting point
First decide: occurrences, containing labels, or avoiding labels?
Hint 2 — take the next step
For containing labels, count each label only once.
Work backward: exactly 39 zero digits
The worked example gives an occurrence total and asks for the final page number.
Complete practice problem: The page numbers of a book contain exactly 39 zero digits. How many pages does the book have?
Page 209 would add another zero, so 208 is the endpoint with exactly 39 zeros.
Reconstruct the cumulative count
Hint 1 — choose a starting point
Start with the cumulative total through 199.
Hint 2 — take the next step
Page 200 contributes two zeros; inspect each following page.
Work backward: exactly 71 zeros—and test feasibility
Occurrence totals can jump by more than one on a single page, so some targets are impossible.
original guided practice
A book’s page numbers contain exactly 71 zero digits. How many pages does it have?
Inverse occurrence laboratory
An occurrence total may give one endpoint, several, or none. For example, exactly nine zeros allows every endpoint from 90 through 99. Page 100 adds two at once. Try target 9 and digit 0 below.
Audit the guided practice
Hint 1 — choose a starting point
Inspect the jump at page 400.
Hint 2 — take the next step
A count may have one endpoint, several endpoints, or no endpoint.
Independent workshop
Try all eight questions before opening the explanations. Correct all eight answers to complete this mission.
Hint
Count ones, tens, and hundreds separately.
Worked explanation — open after trying
Ones: 13. Tens: 10 in 20–29 and 6 in 120–125. Total: 13 + 16 = 29.
Hint
A label with two 2s still counts once.
Worked explanation — open after trying
There are 29 occurrences, but 22 and 122 each contribute an extra occurrence. Thus 29 − 2 = 27 labels.
Hint
Count labels, not occurrences; consider six blocks of 100.
Worked explanation — open after trying
Within each block 00–99, 19 endings contain 6: ten in 60–69 plus nine other endings in 6. Six blocks through 599 give 114; 600 adds one. Total 115. Temporary 00 is bookkeeping and never contains 6.
Hint
Count ones-place and tens-place zeros; inspect 600 separately.
Worked explanation — open after trying
Ones-place zeros: 60. Tens-place zeros: 50 in 100–109 through 500–509, plus one in 600. Total 60 + 51 = 111.
Hint
Count the zeros through the last complete hundred before the target, then inspect how each following page changes the total.
Worked explanation — open after trying
300 adds two, reaching 51. Pages 301–309 add nine, reaching 60. Page 310 adds another, so the unique endpoint is 309.
Hint
Page 111 contributes three occurrences.
Worked explanation — open after trying
Ones: 12; tens: 12; hundreds: 12. Total 36.
Hint
Count every visible zero.
Worked explanation — open after trying
1000 has three written zeros.
Hint
After page 90, when does the next zero appear?
Worked explanation — open after trying
Page 90 supplies the ninth zero. Pages 91–99 add none; page 100 adds two. Every endpoint 90–99 works.
Fresh exit ticket
Use the method on new problems. All five answers must be correct. Complete all ten missions to earn the certificate.
Hint
Count positions.
Worked explanation — open after trying
Each of its three positions contains 2.
Hint
Count multiples of ten and blocks with a tens-place zero.
Worked explanation — open after trying
25 ones-place zeros + 20 tens-place zeros = 45.
Hint
Include the entire hundreds-place run.
Worked explanation — open after trying
12 ones-place + 20 tens-place + 21 hundreds-place = 53.
Hint
Count page 11 only once.
Worked explanation — open after trying
1, the ten labels 10–19, and 21 give 12 labels.
Hint
Find the fifth multiple of ten.
Worked explanation — open after trying
10, 20, 30, 40, 50 supply the first five zeros. The smallest endpoint is 50; 51–59 also retain five.
Lesson checkpoints completed
Page-Digit Occurrence Detective
This certifies that Learner can count a chosen digit by place value, distinguish occurrences from page labels containing the digit, and work backward from an occurrence total.