Fill Connected Regions Using Frequency and Adjacency
Lesson 14.3 · 用数量与相邻关系填连通区域
Build regions whose labels tell their exact sizes, keep every copy edge-connected, and use givens, path lengths, and careful backtracking to solve a complete practice problem.
Turn each label into an exact inventory
In this kind of puzzle, the number written in a region tells both its name and its required number of cells.
The frequency rule
There is one 1, two 2s, three 3s, and so on through nine 9s. Every copy of one label must belong to the same connected region.
The complete 1–9 inventory
Check the inventory
Need a next step?
Hint
A label is also its exact cell budget.
Reasoning and worked explanation
The budgets add to 1+2+…+9=45. Count all cells of a label, not only the visible largest piece.
Know what “connected” means
Two cells are adjacent only when they share a full edge. Touching at one corner does not connect a region.
Example A
The marked cells share an edge.
Example B
The marked cells touch only at a corner.
Example C
Every marked cell can be reached through shared edges.
Classify the three examples
Need a next step?
Hint
Trace a route through full shared edges.
Reasoning and worked explanation
Diagonal contact is not an edge. A bent route is connected if each step crosses a shared edge.
Read the complete original board
All information needed is included here. The board has 45 usable cells—the exact total required by the 1–9 inventory.
original puzzle. Fill the board with one 1, two 2s, three 3s, …, eight 8s, and nine 9s. All copies of the same number must form one edge-connected region. Eight cells are already given. The seven marked cells A–G must all contain different numbers. Find the five-digit number formed by reading the values in C, D, E, F, G in that order; this is not their sum or product.
What is fixed?
The printed givens are:
The letters A–G mark the second row. They are positions, not extra values.
The top and bottom rows each have five usable cells; the five middle rows each have seven.
Read before solving
Need a next step?
Hint
Read fixed clues before painting any cells.
Reasoning and worked explanation
The shape has 5+7+7+7+7+7+5=45 cells. The seven marked positions are all in the second row and must differ.
Eliminate labels that cannot appear in A–G
Use exact frequency and the shortest possible connecting path. A region containing two cells must include a full edge-connected path between them.
Why 1 cannot appear
The single required 1 is already printed in the board.
No other cell—including A–G—may contain 1.
Why 5 cannot reach A–G
Other fixed labels act as barriers. The table shows the smallest possible number of cells in a path from each marked position to the given 5.
Even the shortest choices, B and F, need 6 cells. A 5-region has only 5 cells, so none of A–G can be 5.
Record the eliminations
Need a next step?
Hint
Count the starting and finishing cells of a connecting path.
Reasoning and worked explanation
The single 1 is already used. A path to the given 5 needs at least six cells, exceeding the five-cell budget.
Force the smallest unfinished regions
After excluding 1 and 5, the seven distinct A–G cells must use exactly the remaining seven labels: 2, 3, 4, 6, 7, 8, and 9.
Find the A–G copy of 2
Only D can connect to the given 2 using exactly two cells. Therefore:
Find the A–G copy of 3
Only C can reach the given 3 in a connected region of exactly three cells. The cell between them must also be 3.
Check the forced cells
Need a next step?
Hint
The seven marked cells must use the seven labels left after excluding 1 and 5.
Reasoning and worked explanation
D is the only marked position next to the given 2. C reaches the given 3 in exactly three cells, forcing the cell between them to 3.
Use a branch ledger instead of random guessing
The worked example next says to continue by trial. Here, every trial is recorded and checked against all remaining frequency and connectivity conditions.
Branch 1
Temporarily place 4 at E.
Branch 2
Temporarily place 4 at F.
Reconstruct the worked A–G row
After studying the full worked completion below, record its marked row. The next mission checks every rule on your own board.
Need a next step?
Hint
A failed trial needs a specific contradiction, not just a red message.
Reasoning and worked explanation
Use the trial diagrams below. One short route for F=4 traps too few cells above it. Undo that route; a worked valid completion is then revealed region by region. One failed route alone is not a proof against every route.
Your independent check
Answer the prediction as well as the main task, then press this mission’s Check button. Worked examples are for studying the method; explain the prediction using its own numbers.
Explain this prediction
It needs 7 cells but can reach at most 6: one cell short.
Complete and audit the full practice problem
Choose a number in the palette, then tap editable cells. The checker validates the mathematical conditions, not a picture match.
Live inventory and connectivity
Need a next step?
Hint
Check budgets before connectivity, then check the marked row.
Reasoning and worked explanation
Use one 1, two 2s, and so on. Fixed clues cannot move. Every same-label cell must connect by edges, and A–G must all differ. Study the worked regions, then try building the board yourself.
Your independent check
Answer the prediction as well as the main task, then press this mission’s Check button. Worked examples are for studying the method; explain the prediction using its own numbers.
Explain this prediction
Keep the exact seven-cell budget. Adding an eighth violates the frequency rule.
Transfer the method to a smaller region puzzle
Solve this new 3×5 practice board using the same rules. Use one 1, two 2s, three 3s, four 4s, and five 5s. Each label must form one edge-connected region.
Transfer inventory
Need a next step?
Hint
Start with already completed labels and the smallest unfinished region.
Reasoning and worked explanation
The 1 and two 2s are fixed. The 3-region needs one more cell. Complete its edge connection, then use four 4s and five 5s to fill the remaining space.
Your independent check
Answer the prediction as well as the main task, then press this mission’s Check button. Worked examples are for studying the method; explain the prediction using its own numbers.
Explain this prediction
Four moves visit 5 cells. The path uses the entire budget.
Explore further: groups with forbidden differences
A different kind of grouping puzzle says that the positive difference (larger minus smaller) between any two different numbers in a group may not also belong to that group. Test group{1,4,7}:7−4=3,7−1=6,4−1=3; neither 3 nor 6 is in the group, so it passes. Group{1,3,4} fails because 4−3=1. When adding a candidate, check differences between it and every old member, and check whether the candidate equals an old difference. For example adding 3 to{1,4,7} fails both ways. Record the exact conflicting pair when rejecting a trial.
Fresh strategy workshop
Try these new situations before opening the explanations. Correct all eight answers to complete this workshop.
Worked explanation — try first
1+2+3+4+5+6=21.
Worked explanation — try first
7−4=3.
Worked explanation — try first
Count the starting cell as well: four moves visit five cells.
Worked explanation — try first
5−4=1, so that proposed connection is impossible.
Worked explanation — try first
Every copy of one label must lie in one edge-connected region; the right count alone is insufficient.
Worked explanation — try first
The one-cell inventory is already complete.
Worked explanation — try first
The region must use exactly six cells, not every cell it could potentially reach.
Worked explanation — try first
The available labels are 2,3,5,6.
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
Pair 1+7,2+6,3+5 and add 4: 28.
Worked explanation — try first
8−5=3.
Worked explanation — try first
Include the starting cell: 6+1=7.
Worked explanation — try first
The new bridging cell gives a shared-edge path through all three cells.
Worked explanation — try first
The count is correct; connectivity fails. The two pieces must be joined without exceeding four cells.
0 / 5
Certificate of Completion
Connected-Region Constraint Architect
This certifies that
Grade 5 Mathematician
used exact frequencies, edge adjacency, path lengths, and systematic backtracking to solve connected-region puzzles.
Teaching notes
The worked example explicitly states two deductions: A–G cannot contain 1 or 5, and the cell directly above the printed 2 must be 2. It then presents a completed filling obtained by trial. This lesson preserves those original statements and adds shortest-path tables, a branch ledger, a full-condition validator, and the smaller 3×5 transfer puzzle as new instructional scaffolding.
The worked example does not explicitly discuss uniqueness. A complete case audit used to design this page leaves one completion under the stated conditions. The student checker nevertheless validates the rules themselves rather than comparing only with a picture.