Find Missing or Repeated Pages from Sums
从页码和中找出漏页、重页与缺失纸张
Build the expected sum of all page numbers, compare it with the reported sum, and use the difference as evidence. Then use the odd–even structure of a physical leaf to decide whether a missing-sheet report is possible.
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.
Build the expected sum of all page numbers
In this lesson, page labels are being added as numbers. We are not counting digit symbols.
Pair the first and last page
For pages 1 through 10:
General rule
The first and last pages add to:
For any n, even or odd, write two copies in opposite directions:
1 + 2 + … + n
n + (n−1) + … + 1
There are n columns, each totaling n + 1. This counts the original sum twice, so divide by two:
Page-sum laboratory
Check the expected-total rule
Hint 1 — choose a starting point
Pair two copies of the sum in opposite order.
Hint 2 — take the next step
Each column totals n + 1; divide the double total by two.
Worked example: one page number was counted twice
A repeated page label makes the reported total larger than the correct total by exactly that page number.
Complete practice problem: A book has 62 pages. When all page numbers were added, one page number was accidentally included twice. The reported total was 2,000. Which page number was repeated?
Rebuild the worked example reasoning
Hint 1 — choose a starting point
Find the correct total before comparing.
Hint 2 — take the next step
Repeated label = reported total − correct total.
Guided practice: one page number was omitted
An omitted page label makes the reported total smaller than the correct total by exactly that page number.
Complete practice problem: A book has pages 1 through 80. When all page numbers were added, one page number was omitted. The reported total was 3,182. Which page number was left out?
Complete the guided practice
Hint 1 — choose a starting point
An omission makes the report smaller.
Hint 2 — take the next step
Missing label = correct total − reported total.
Use a one-page error laboratory
This laboratory generates a reported sum from the chosen error; the fixed questions below ask you to recover the label from the supplied reports. Assume exactly one label is added twice or left out, with no other counting or arithmetic errors.
Test 13, Question 3
Pages 1–60 were added. One page was included twice, giving 1,845.
Test 13, Question 4
Pages 1–55 were added. One page was omitted, giving 1,500.
Use the ledger on two new original problems
Hint 1 — choose a starting point
Choose the book length and whether a label is repeated or omitted.
Hint 2 — take the next step
Check that the proposed label is inside the book.
Understand a physical leaf: two related page labels
A missing page label and a missing physical leaf are not the same event. One leaf has a front and a back. Number the leaves k = 1, 2, 3, … . Leaf k has front 2k − 1 and back 2k. Their sum is 4k − 1, one less than a multiple of four: 3, 7, 11, 15, … .
Odd page followed by the next even page
Every same-leaf sum is 3 more than a multiple of 4
The pair sum leaves remainder 3 when divided by 4
Turn a missing-leaf sum into its two pages
Read the leaf structure
Hint 1 — choose a starting point
A leaf carries an odd label and the next even one.
Hint 2 — take the next step
Its sum is one less than a multiple of four.
original exercise: use a digit-count loss to find a missing leaf
This problem combines the digit-block ideas from Lessons 13.2–13.4 with the same-leaf rule.
Complete practice problem: A 58-page book is missing one physical leaf from somewhere inside it. The remaining page numbers contain 104 digit symbols. What are the two page numbers on the missing leaf?
Digit symbols in the complete book
Digit symbols removed
A missing leaf must contribute two page labels. The only same-leaf pair using one one-digit label and one two-digit label is:
Reconstruct the practice problem
Hint 1 — choose a starting point
Compare digit counts, not sums of label values.
Hint 2 — take the next step
A loss of three digits needs one one-digit and one two-digit label.
Recover a missing leaf from the remaining page-number sum
First find the total value removed. Then check whether that value can be the sum of one odd–even leaf pair.
Missing-leaf extension: A 90-page book is missing one physical leaf. The sum of all remaining page numbers is 4,012. Which leaf was torn out?
Solve and audit the missing leaf
Hint 1 — choose a starting point
Find the removed sum first.
Hint 2 — take the next step
Split it into consecutive labels and verify odd front/even back.
Detect impossible missing-sheet reports
A difference can look like a pair of consecutive numbers and still fail the physical-leaf rule.
Exercise 11
A 48-page book is missing one leaf. A student says the remaining page-number sum is 1,131.
Although 45 = 22 + 23, pages 22 and 23 are on different leaves. Also:
Test 13, Question 11
A 300-page book loses 15 physical leaves. Could the sum of all 30 missing page numbers be 2,014?
A simple odd/even proof: each leaf has an odd + even label, giving an odd sum. Fifteen odd sums have an odd total; 2014 is even. It is impossible.
Optional remainder check: the symbol ≡ below means “has the same remainder after division by 4.”
Use structure, not just subtraction
Hint 1 — choose a starting point
Test the physical structure as well as the arithmetic.
Hint 2 — take the next step
An odd number of leaves has an odd total label sum.
Independent workshop
Try all eight questions before opening the explanations. Correct all eight answers to complete this mission.
Hint
Use n(n+1)÷2.
Worked explanation — open after trying
36 × 37 ÷ 2 = 666.
Hint
Subtract the correct sum from the report.
Worked explanation — open after trying
Correct sum 44 × 45 ÷ 2 = 990. Extra = 1007−990=17.
Hint
Subtract the report from the correct sum.
Worked explanation — open after trying
70 × 71 ÷ 2 = 2485; 2485−2459=26.
Hint
Find consecutive labels, then check odd first.
Worked explanation — open after trying
(123−1)÷2=61; the pair is 61 and 62.
Hint
Find the removed sum, then check one leaf.
Worked explanation — open after trying
5050−4955=95; 95=47+48. Back label 48.
Hint
Consecutive labels need the correct odd/even order.
Worked explanation — open after trying
820−775=45=22+23. Those labels face each other; they are not one leaf.
Hint
Which leaf crosses the one-digit/two-digit boundary?
Worked explanation — open after trying
Only the leaf 9/10 has one one-digit and one two-digit label. Its front is 9.
Hint
Subtract one, halve, and check odd.
Worked explanation — open after trying
(159−1)÷2=79; back 80, sum 159.
Fresh exit ticket
Use the method on new problems. All five answers must be correct. Complete all ten missions to earn the certificate.
Hint
The formula works for odd page counts too.
Worked explanation — open after trying
Two reversed copies make 61 pairs of 62. Divide by two: 61×62÷2=1891.
Hint
Find the extra amount.
Worked explanation — open after trying
52×53÷2=1378; 1401−1378=23.
Hint
Find the missing amount.
Worked explanation — open after trying
66×67÷2=2211; 2211−2162=49.
Hint
Find the odd member of the consecutive pair.
Worked explanation — open after trying
(143−1)÷2=71; back 72.
Hint
Each leaf has an odd label sum.
Worked explanation — open after trying
Nine odd numbers add to an odd total, while 1000 is even. Impossible.
Page-Sum & Missing-Leaf Detective
This certifies that Learner can build expected page-number sums, identify repeated or omitted page labels, reconstruct a missing physical leaf, and reject structurally impossible reports.
Optional reflection — not automatically graded
This response is saved for printing or discussion. The lesson does not keyword-match or score it.