Read Page Numbers as Digits and Solve Digit Clues
Lesson 13.1 · 把页数看成数位，并解数字线索题
A page count is a number built from digits. Learn to separate pages, page numbers, and digit symbols; translate clues into place-value relationships; test every possible hundreds digit; and prove when an answer is—or is not—unique.
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.
Separate page count, page number, and digit
These three ideas sound similar, but they count different things. Naming the right object is the first step in every page-number problem.
How many pages the book contains.
The printed label on one page, or the final label showing the book's page count.
One symbol from 0 through 9. The page number 195 uses three digit symbols.
Classify each item
Hint 1 — choose a starting point
Name the object being counted: pages, a whole label, or one symbol.
Hint 2 — take the next step
A digit is one of 0–9. A page label may have several digits.
Build a three-digit page count from place value
A three-digit number has a hundreds digit, a tens digit, and a ones digit. Each position contributes a different value.
Read the fixed challenge number 472
Hint 1 — choose a starting point
Read positions from left to right: hundreds, tens, ones.
Hint 2 — take the next step
Multiply the tens digit by ten to find its contribution.
Translate the worked example's clues into equations
The chapter's first worked problem describes a three-digit page count. Let H, T, and O stand for its hundreds, tens, and ones digits.
A three-digit book page count
The ones digit is 4 greater than the hundreds digit. The tens digit is 4 greater than the ones digit. How many pages are in the book?
Match each verbal clue to its equation
Hint 1 — choose a starting point
Translate “A is 4 greater than B” as A = B + 4.
Hint 2 — take the next step
Combine the two +4 steps; remember H cannot be zero.
Find the only possible hundreds digit
Digits must stay between 0 and 9, and the hundreds digit must be at least 1. Test every possible H rather than guessing.
Test one hundreds digit
Why only H = 1?
The tens digit is:
Because T cannot exceed 9:
Because a three-digit number cannot begin with zero:
Both conditions force:
Complete candidate audit
Record the forced digits
Hint 1 — choose a starting point
Start with T = H + 8.
Hint 2 — take the next step
Use T ≤ 9 and H ≥ 1 together.
Assemble 195 and verify every clue
A candidate is not finished until it passes the place-value rules and every stated relationship.
Four-part verification
Finish the worked example
Hint 1 — choose a starting point
Write digits in H, T, O order.
Hint 2 — take the next step
Check each difference and explain why no second candidate survives.
Solve a clearly stated mean clue
The arithmetic mean of two numbers is their sum divided by two.
Adapted guided practice
A book has from 800 through 899 pages, inclusive. Its hundreds digit H is 6 greater than its ones digit O. Its tens digit T is the arithmetic mean of H and O. Find the page count.
Use both the page range and the digit relationships; together they must determine the page count.
Worked explanation — open after trying
The range fixes H = 8. Then O = 8 − 6 = 2 and T = (8 + 2) ÷ 2 = 5. Assemble H, T, O to get 852. Each digit fits and the range leaves only this candidate.
Optional: how extra clues narrow the candidates
A book has a three-digit page count. The hundreds digit is 6 greater than the ones digit. The tens digit is the arithmetic mean of the ones and hundreds digits. How many pages does the book have?
Without a page-range clue
Four page counts satisfy this reading.
Add a clue: the tens digit is odd
Two page counts still survive, so the result is not unique.
Finding a candidate proves possibility. A unique answer requires exactly one survivor. The actual question above instead supplies the page range 800–899, which leaves exactly one candidate.
Hint 1 — choose a starting point
The range 800–899 fixes H.
Hint 2 — take the next step
Use O = H − 6, then T = (H + O) ÷ 2.
Use a candidate laboratory
Change the relationships and let the page enumerate every valid three-digit number. Here enumeration means testing every allowed hundreds digit from 1 through 9, calculating the other two digits, and keeping only those that satisfy every clue. This complete search proves that none was skipped.
Load “Equal steps of 3,” then answer
Use the preset button. The rules become O = H + 3 and T = O + 3.
Hint 1 — choose a starting point
With equal steps of 3, T = H + 6.
Hint 2 — take the next step
Test H = 1 through 9 and keep only whole digits at most 9.
Catch common digit-clue errors
A candidate can fail because of place value, digit range, an unchecked clue, or missing uniqueness.
Four verification questions
- Is every symbol a digit from 0 through 9?
- Is the hundreds digit nonzero?
- Does the candidate satisfy every relationship?
- Does exactly one candidate survive?
“Possible” is not “proved”
One working candidate proves only that the puzzle is possible. A unique answer requires showing that all other candidates fail.
Diagnose each claim
Hint 1 — choose a starting point
Check whether each proposed symbol is a single allowed digit.
Hint 2 — take the next step
A unique solution requires testing every possible candidate.
Independent workshop
Try all eight questions before opening the explanations. Correct all eight answers to complete this mission.
Hint
Read hundreds, tens, ones from left to right.
Worked explanation — open after trying
638 = 600 + 30 + 8, so its tens digit is 3.
Hint
A zero can hold the tens place.
Worked explanation — open after trying
700 + 0 + 4 = 704.
Hint
Find O, then T. Write the digits in H, T, O order.
Worked explanation — open after trying
O = 4 and T = 7, so 200 + 70 + 4 = 274.
Hint
Use T = H + 7 and T ≤ 9.
Worked explanation — open after trying
H can be 1 or 2 only. The candidates are 183 and 294.
Hint
Start with the smallest allowed H.
Worked explanation — open after trying
H = 1 gives O = 3 and T = 8. In H, T, O order, the number is 183.
Hint
Use the largest H that keeps T ≤ 9.
Worked explanation — open after trying
H = 2 forces O = 4 and T = 9, giving 294.
Hint
A leading zero is not printed.
Worked explanation — open after trying
086 is normally written 86, which has two digits.
Hint
The range fixes H. The arithmetic mean means add and divide by two.
Worked explanation — open after trying
H = 7, O = 3, and T = (7 + 3) ÷ 2 = 5. The page count is 753.
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 visible positions, including zero.
Worked explanation — open after trying
6, 0, and 8 are three written symbols.
Hint
Digit and place value are different.
Worked explanation — open after trying
The tens digit 6 contributes 6 × 10 = 60.
Hint
Find both forced digits, then use H, T, O order.
Worked explanation — open after trying
O = 6 and T = 7, giving 276.
Hint
A three-digit number places tens before ones.
Worked explanation — open after trying
O = 4, T = 8, so the page count is 184.
Hint
Unique means exactly one.
Worked explanation — open after trying
Testing every allowed candidate and finding exactly one survivor proves uniqueness.
Certificate of Completion
Page-Number Digit Detective
Learner translated digit clues, tested all possible hundreds digits, verified the page count 195, and recognized when wording does not determine a unique answer.
Lesson 13.1 · Grade 5 Math Education
Teaching notes
The worked example's relationships and answer 195 are preserved. The guided practice uses a phrase translated literally here as “odd average,” but that phrase is not defined on the worked example page and does not determine one unique number under the reasonable interpretations tested above. The lesson therefore uses it as a clue-quality investigation and labels the added 800–899 clue as new instructional scaffolding, not as a worked solution.