Lesson 14.5 — Arrange Number Rings under Neighbor Rules
第14讲 · 填数阵图 · 数字环
Treat a ring as a closed chain, test every neighboring pair—including the pair across the starting point—and use compatibility, parity, and systematic enumeration to build valid arrangements.
Translate the rule → plan difficult neighbors → close the loop → audit every edge.
A ring is a closed chain
In a row, the endpoints each have one neighbor. In a ring, every position has two neighbors because the final position connects back to the first.
Count edges, not gaps on a line
For six positions, the pairs are:
The closing pair 6–1 is where many almost-correct rings fail.
Check the closed-loop idea
Need a next step?
Hint
Draw the connection from the last position back to the first.
Reasoning and worked explanation
A ring with n positions has n edges; every position has two distinct neighboring positions.
Translate “difference at least 3” into a neighbor test
Every edge joins two numbers whose absolute difference is at least 3. Absolute difference means larger minus smaller; it is a difference between number values, not a distance measured on the drawing.
The exact rule
A valid-looking pattern is not enough. Each of the ten edge differences must be checked.
Compatibility explorer
Allowed neighbors
Forbidden neighbors
Read the compatibility rule
Need a next step?
Hint
Subtract the smaller value from the larger.
Reasoning and worked explanation
The absolute difference must be at least 3 on every edge. A difference of exactly 3 is allowed.
Place close-number pairs opposite as a planning strategy
The worked example begins by keeping especially close pairs far apart: 1 with 2, 3 with 4, and so on. On a ten-position ring, opposite positions are five one-edge moves apart in either direction (for example, Position 1 to Position 6). There are four positions strictly between them along either half of the ring.
Five original pairs
Separate planning from verification
Need a next step?
Hint
Separate close pairs, but do not stop there.
Reasoning and worked explanation
Placing close pairs opposite is only a plan. Other forbidden pairs, including those with difference 2, may still touch.
Audit the four original arrangements
This worked example displays four valid rings and explicitly says the answer is not unique. View each model and inspect all ten differences, including the closing difference.
Fill 1 through 10 once each
Arrange the ten numbers around the ring so that every pair of adjacent values differs by at least 3.
Viewed 0 of 4 original models.
Complete edge audit
Read the worked example models
Need a next step?
Hint
Read all ten difference labels, including the closing one.
Reasoning and worked explanation
All four models use 1–10 once and every edge is at least 3. A valid rotation or reflection obeys the same conditions.
Build and verify your own difference ring
Choose one number for each position. The checker accepts any arrangement that uses 1 through 10 exactly once and passes all ten edge tests.
Live mathematical audit
You can also click two numbered circles in the diagram to swap them.
Need a next step?
Hint
Try the partly filled starter and fill the most restricted gap first.
Reasoning and worked explanation
A gap between 3 and 1 needs a value at least 3 away from both. Compare unused candidates, continue gap by gap, and finish with a full edge audit.
Use parity before building a prime-sum ring
The guided practice changes the rule: select six different whole numbers from 1 through 9 so every adjacent pair has a prime sum. A prime is a whole number greater than 1 with exactly two positive divisors, 1 and itself. Parity means whether a whole number is odd or even.
Possible prime sums
Every listed prime is odd. The only even prime is 2, but two different positive numbers from 1–9 cannot sum to 2.
The ring must alternate parity
A six-position ring must therefore use exactly three odd and three even numbers.
Use the parity filter
Need a next step?
Hint
Can two different positive numbers total the even prime 2?
Reasoning and worked explanation
No. Every prime neighboring sum is odd, so one label is odd and the other even. A six-ring alternates three of each.
Build a six-number prime-sum ring
Choose six distinct numbers from 1 through 9 and arrange them so all six neighboring sums are prime. The closing sum is part of the test.
Live prime-sum audit
Click two filled circles to swap them, or use the position selectors.
Need a next step?
Hint
Each selected number needs two different allowed neighbors.
Reasoning and worked explanation
For set 1,2,3,4,6,9, number 9 can only neighbor 2; that set fails. Set 1,2,3,4,8,9 works in order 1,4,3,8,9,2: sums 5,7,11,17,11,3.
Enumerate the guided-practice sets completely
The worked example counts two arrangements as the same whenever they use the same six selected numbers, even if the circular order differs. We therefore enumerate six-number sets, then test whether each set has at least one valid ring order.
| # | Selected set | One working circular order | Status |
|---|
Read the complete enumeration
Need a next step?
Hint
Count selected sets, then test whether each can close into a ring.
Reasoning and worked explanation
Choose three of five odd labels and three of four even labels: 10×4=40 candidates. Parity is necessary, not sufficient. The full cycle check leaves 17 sets.
Fresh strategy workshop
Try these new situations before opening the explanations. Correct all eight answers to complete this workshop.
Worked explanation — try first
Eight successive edges include the closing edge.
Worked explanation — try first
Their difference is 3, below 4.
Worked explanation — try first
Allowed: 1,2,3,9,10.
Worked explanation — try first
The final 2 meets the first 1, giving difference 1. The other nine edges pass, but the ring fails.
Worked explanation — try first
Each prime sum must be odd because distinct positive values cannot total 2. Parity alternates: four odd and four even.
Worked explanation — try first
The six sums are 5,7,11,17,11,3.
Worked explanation — try first
Count selected sets, not their arrangements.
Worked explanation — try first
Only 2 gives a prime sum with 9. A cycle needs two different neighbors, so this set cannot work.
0 / 8
Independent exit ticket
Try these new situations before opening the explanations. All five correct completes this mission. The certificate requires all ten missions.
Worked explanation — try first
Every position contributes the edge to the next position, including the closing one.
Worked explanation — try first
|2−7|=5, so this edge passes.
Worked explanation — try first
Opposite parity at every edge forces five odd and five even values.
Worked explanation — try first
The final 1 meets the first 2: 1+2=3.
Worked explanation — try first
All eight orders use one identical selected set.
0 / 5
Certificate of Completion
Number-Ring Neighbor-Rule Architect
This certifies that
Grade 5 Mathematician
closed every loop, audited every edge, used compatibility and parity, and distinguished circular arrangements from selected-number sets.
Explain one useful strategy
This writing is saved but is not automatically graded.
Teaching notes
The original asks students to place 1 through 10 in a ten-position ring so every adjacent difference is at least 3, recommends putting the close pairs (1,2), (3,4), (5,6), (7,8), and (9,10) opposite, and displays four valid arrangements.It explicitly states that the answer is not unique.
The original asks students to choose six different values from 1 through 9 so every adjacent sum is prime, and specifies that arrangements using the same six values count as the same filling regardless of circular order.The parity explanation, live validators, and complete enumeration are added scaffolds.The enumeration groups by six-number set and finds 17 feasible sets.
All student builders validate the mathematical conditions rather than comparing with one hard-coded original picture.