Use Facing Pages, Sheets, and Odd–Even Structure
用相对页、纸张与奇偶结构解决页码问题
Read the physical structure of a book: which pages face each other, which two labels lie on one leaf, and how an article’s length changes the parity of the next starting page. Then use those rules to prove a maximum.
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 three book structures
“Consecutive,” “facing,” and “on the same physical leaf” do not mean the same thing.
Two pages visible at once.
The front and back of one piece of paper.
Two numbers differ by one, but structure still matters.
Classify the page relationships
Hint 1 — choose a starting point
Decide whether the object is a spread or one leaf.
Hint 2 — take the next step
A spread is even/odd; one leaf is odd/even.
Explore facing pages
Under this lesson’s numbering convention, an open spread is determined by its even left page. The pair checker accepts the two page numbers in either order.
Move through the book
Pair checker
Read the facing-page rule
Hint 1 — choose a starting point
The facing labels differ by one.
Hint 2 — take the next step
The smaller label must be even.
original guided practice: facing pages with product 1,806
Translate “two facing pages” into two consecutive numbers with the smaller one even.
Complete practice problem: A mathematics book is opened to two facing pages. The product of the page numbers is 1,806. Find the two page numbers.
Build the model
Look for two neighboring factors. The smaller factor must be even.
Verify every condition
Consecutive-product explorer
Complete the worked example reasoning
Hint 1 — choose a starting point
Try neighboring factors near the square root.
Hint 2 — take the next step
Check product, adjacency, and even-left parity.
Track one physical leaf—and the middle leaf
The two sides of Leaf k carry page labels 2k − 1 and 2k. A “central leaf” means one leaf with equally many leaves before and after it; both sides of every leaf are numbered.
Leaf-index laboratory
Test 13 transfer: missing leaf sum 95
One physical leaf is missing, and its two page labels add to 95.
The front is page 47 and the back is page 48.
Test 13 transfer: central leaf product 2,450
Pages 49 and 50 are Leaf 25. If this is the central leaf, there are 49 leaves altogether, so the book has 98 numbered pages.
Use the leaf structure
Hint 1 — choose a starting point
Leaf k has labels 2k−1 and 2k.
Hint 2 — take the next step
A central leaf has the same number of leaves before and after it.
See how article length changes the next starting page
Parity means whether a number is odd or even. An article starting on page s and using L pages occupies s through s + L − 1, including both endpoints. The next article begins on s + L.
Parity machine
Apply the parity machine
Hint 1 — choose a starting point
The article occupies start through start + length − 1.
Hint 2 — take the next step
The next start is start + length.
Worked example: prove the maximum number of odd starting pages
Fifteen articles have lengths 1, 2, 3, …, 15 pages. They are bound back-to-back with continuous page numbering.
Complete practice problem: Use each article length 1 through 15 exactly once, in any order. The first article starts on page 1; each next article starts immediately after the preceding article, with no blank pages. What is the greatest possible number of articles whose first page has an odd page number?
Why four odd lengths must start even
Each odd-length article flips the parity. Ignoring the even lengths between them, the eight odd-length articles start:
Why 11 can actually be reached
The seven even-length articles do not change parity, so all seven can be placed during an odd-start phase. The worked example gives this order:
Only the articles of lengths 3, 7, 11, and 15 begin on even pages. The other 11 begin on odd pages.
Rebuild the proof
Hint 1 — choose a starting point
Ignore even lengths temporarily and follow odd lengths.
Hint 2 — take the next step
Prove an upper bound, then construct an order that reaches it.
Build and audit an article arrangement
Each chip shows an article’s length in pages, not its starting page. Select a chip and move it earlier or later in the reading order. The table recalculates every starting page.
| Position | Article length | Starting page | Start parity | Next start |
|---|
Verify an optimal construction
Build your own order and check it together with the answers below. After an attempted check, you can open a separate worked example without changing your order. The Move buttons also work with a keyboard; the selected chip stays focused.
Mission completion also requires the current arrangement studio to show exactly 11 odd starts.
Hint 1 — choose a starting point
Put even-length articles where the start is odd.
Hint 2 — take the next step
Check the table after moving a chip; the current order must reach 11.
Transfer the structure to a February calendar
Find a 2 × 2 block whose four dates have written digits with the greatest possible sum.
Test 13 transfer: Use the displayed 28-day February, whose first day is Saturday. A valid rectangle uses two neighboring weekday columns and two neighboring week rows, with a date in all four cells. It cannot include blank cells or wrap from Sunday to Monday. The example block 4, 5, 11, 12 has digit sum 14. What is the greatest possible digit sum?
Current block
Find and verify the maximum
Hint 1 — choose a starting point
Add digits, not date values.
Hint 2 — take the next step
Visit every valid 2×2 block and compare its digit sum.
Independent workshop
Try all eight questions before opening the explanations. Correct all eight answers to complete this mission.
Hint
A spread is even then next odd.
Worked explanation — open after trying
76 + 1 = 77.
Hint
One leaf has odd front, next even back.
Worked explanation — open after trying
85 + 1 = 86.
Hint
Try consecutive factors near the square root of 3192.
Worked explanation — open after trying
56 × 57 = 3192; 56 is even, so the facing pair is valid.
Hint
Next start = start + length.
Worked explanation — open after trying
25 + 14 = 39; parity is preserved.
Hint
The next start is one beyond the last occupied page.
Worked explanation — open after trying
25 + 13 = 38; parity flips.
Hint
Count odd lengths, then place all even lengths during an odd-start phase.
Worked explanation — open after trying
Five odd lengths alternate odd/even starts, giving three odd starts. All four even lengths can start odd: maximum 7. Order 2,4,6,8,1,3,5,7,9 reaches it.
Hint
Find the leaf number and count equal numbers of leaves on both sides.
Worked explanation — open after trying
This is Leaf 12. Eleven leaves on each side give 23 leaves, hence 46 pages.
Hint
Add individual digits.
Worked explanation — open after trying
1+7+1+8+2+4+2+5 = 30.
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 smaller page must be even.
Worked explanation — open after trying
38 is even and 39 is the next odd page.
Hint
The odd front is one less.
Worked explanation — open after trying
64 − 1 = 63.
Hint
Add an even number to either an odd or even start.
Worked explanation — open after trying
An even length preserves parity.
Hint
Give both a bound and a construction.
Worked explanation — open after trying
Four odd lengths force two even starts. The three even lengths can all start odd. Maximum 5; order 2,4,6,1,3,5,7 reaches it.
Hint
Find the leaf’s position, then put equally many leaves before and after it.
Worked explanation — open after trying
15+1+15=31 leaves; 31×2=62 pages.
Book-Structure & Parity Navigator
This certifies that Learner can distinguish facing pages from one physical leaf, use even–odd page structure, solve consecutive-page products, and prove a maximum with parity and a construction.