Lesson 14.2 — Construct Grids with Equal or Distinct Sums
第14讲 · 设计和相等或互不相同的和
Plan the possible totals first, then place 1s, 2s, and 3s so every required row, column, block, or edge behaves exactly as the rule demands.
Know whether the rule asks for equality or difference
A construction problem is much easier after you say exactly which totals must match and which totals must be different.
Equal-sum construction
Every required line has one common target.
- Find the target S.
- Fill one strong line first.
- Use the same total everywhere.
All-different construction
No two required totals may be the same.
- Find the possible range.
- Count how many distinct totals are needed.
- Keep a duplicate-warning ledger.
Translate the rule
Need a next step?
Hint
Say whether the totals must match or must differ.
Reasoning and worked explanation
Cells may repeat 1,2,3; it is the sixteen line totals that must be different. Diagonals are not part of this task.
Plan the complete range before filling the 8×8 grid
A row contains eight entries, and every entry is 1, 2, or 3. That gives a short, countable list of possible sums.
Minimum
All eight cells are 1.
Maximum
All eight cells are 3.
How many possibilities?
The possible integer sums are 8 through 24.
We need only 16 distinct totals—eight rows plus eight columns—so exactly one of the 17 possible values must be unused in a valid construction.
Plan the target set
Need a next step?
Hint
Use all 1s for the minimum and all 3s for the maximum.
Reasoning and worked explanation
The possible row sums are the seventeen integers 8–24. The sixteen line totals leave exactly one possible value unused.
Explore the worked example’s 8×8 staircase construction
The worked example gives one valid filling and explicitly says the answer is not unique. Select any row or column to inspect its entries and sum.
Complete original task: Fill every cell of an 8×8 grid with 1, 2, or 3 so that the sums of the eight rows and eight columns are all different.
On a narrow screen, swipe or scroll the grid horizontally to see the right-hand totals. The complete totals are also listed below.
Read the construction accurately
Need a next step?
Hint
Inspect one row and then one column.
Reasoning and worked explanation
The row totals are 8,10,12,14,17,19,21,23. The columns supply the other eight values between 8 and 23.
See how four carefully placed 2s control parity
Changing a 1 to a 3 raises a line sum by 2. That keeps each affected total’s parity (whether it is odd or even) unchanged. Each 2 supplies a one-step adjustment. The worked example uses four of them.
Focus on original Row 5
This row’s 2 creates the odd total 17. The four 2s are at (row 5,column 4), (row 6,column 3), (row 7,column 2), and (row 8,column 1). Starting with the same staircase using 1 instead of each 2, these changes add 1 to each of rows 5–8 and columns 1–4. That gives odd lower-row totals 17,19,21,23 and odd totals in the first four columns 9,11,13,15. The remaining totals are even. No totals repeat.
Understand the controlled changes
Need a next step?
Hint
A change from 1 to 2 adds one; from 1 to 3 adds two.
Reasoning and worked explanation
The four 2s sit at (5,4),(6,3),(7,2),(8,1). They make the lower four row totals odd and adjust the first four column totals by one.
Repair or create an 8×8 all-different construction
Tap a cell to cycle through blank, 1, 2, and 3. The checker accepts every valid grid—not only the worked example arrangement.
On a narrow screen, scroll the grid sideways to reach every cell and its totals. Keyboard: use arrow keys between cells and Space to change a value.
Live construction audit
A dash means that line still has a blank cell. Only complete lines contribute to the distinct-total and duplicate lists.
Need a next step?
Hint
Begin with a staircase of 1s and 3s, then audit both directions.
Reasoning and worked explanation
Use the repair challenge first. Each cell affects exactly its row and column. Fixing one duplicate must not create another. The extra independent check asks you to predict this two-line effect.
Your independent check
Build your arrangement and answer the prediction below, then press this mission’s Check button. After an attempt, you can review a worked example separately; your arrangement stays in place.
Explain this prediction
Each affected line loses 2: new totals 8 and 20, together 28.
Plan the guided 4×4 challenge using its exact sum range
Here a 2×2 block means two neighboring rows and two neighboring columns, with no skipped cells. Every such block contains four entries, each chosen from 1, 2, and 3. There are nine overlapping 2×2 blocks in a 4×4 grid.
Block 1 of 9
The range is exactly large enough
Therefore, if all nine sums are different, they must be exactly:
Build the target list
Need a next step?
Hint
Count possible starting positions of the top-left corner.
Reasoning and worked explanation
There are 3×3=9 blocks. Their sums lie from 4 to 12; nine different sums must use every integer in that range.
Solve the 4×4 guided-practice construction
Tap cells to cycle through blank, 1, 2, and 3. The live ledger shows the nine overlapping block sums and warns about duplicates.
Nine 2×2 block sums
Need a next step?
Hint
The 4-sum block must be four 1s; the 12-sum block must be four 3s.
Reasoning and worked explanation
Those extreme blocks cannot share a cell. Place them apart, then use the live ledger to fill the middle sums. A changed cell can affect several overlapping blocks.
Your independent check
Build your arrangement and answer the prediction below, then press this mission’s Check button. After an attempt, you can review a worked example separately; your arrangement stays in place.
Explain this prediction
That interior cell belongs to four blocks. Each increases by 2, so the total increase is 8.
Transfer the idea to twelve distinct edge sums
This practice problem places 1 through 9 at the nine nodes of a 3×3 lattice. Only segments joining immediately neighboring nodes horizontally or vertically count. There are twelve; diagonals and longer segments do not count. Each endpoint sum is the sum of the two numbers at that segment’s ends, and all twelve must differ.
Select one node, then another, to swap their numbers.
Live endpoint-sum ledger
Need a next step?
Hint
List horizontal and vertical edges separately.
Reasoning and worked explanation
There are six horizontal and six vertical edges. Swapping two nodes changes only edges touching those nodes. Audit all twelve totals after the swap.
Your independent check
Build your arrangement and answer the prediction below, then press this mission’s Check button. After an attempt, you can review a worked example separately; your arrangement stays in place.
Explain this prediction
Four edges touch the center, so 4×2=8. This prediction is separate from the 1–9 construction rule.
Explore further: equal sums, two values, and proving a minimum
Two possible values. If a five-cell line uses only x and 2x, and t cells use 2x, its total is (5+t)x. For x=3 and target 21, solve 15+3t=21: exactly two cells must be 6. Count how many larger entries each required line needs before filling the grid.
A minimum needs two parts. Draw two three-position lines sharing one center: a—c—b and d—c—e. Use 1–5 once across the five positions. Both line sums must be S. The pool totals 15; adding the two lines counts the center c twice, so 2S=15+c. Since c≥1, S≥8. This is the lower bound.
Now achieve it: use center 1 and outside pairs(2,5)and(3,4). Both lines total 8. The lower bound and this actual construction together prove that 8 is the minimum. A bound alone would not prove a valid arrangement exists.
Fresh strategy workshop
Try these new situations before opening the explanations. Correct all eight answers to complete this workshop.
Worked explanation — try first
Six 1s total 6.
Worked explanation — try first
Six 3s total 18.
Worked explanation — try first
18−6+1=13.
Worked explanation — try first
Six rows plus six columns give twelve totals.
Worked explanation — try first
Each replacement adds 2; 13+2+2=17.
Worked explanation — try first
There are four starting rows and four starting columns: 4×4=16.
Worked explanation — try first
1+2+3+3=9.
Worked explanation — try first
Two rows each have 3 horizontal segments, plus 4 vertical ones: 6+4=10.
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
Five 3s total 15.
Worked explanation — try first
15−5+1=11.
Worked explanation — try first
(3−1)×(5−1)=2×4=8.
Worked explanation — try first
Subtract 2 and 1: 18−3=15.
Worked explanation — try first
The smallest is 4, the largest 8. Only 4,5,6,7,8 are possible, so nine different block sums are impossible.
0 / 5
Certificate of Completion
Controlled-Sum Grid Architect
This certifies that
Grade 5 Mathematician
planned sum ranges, built non-unique constructions, and verified equal or all-different totals.
Teaching notes
The worked example explicitly says that the 8×8 answer is not unique. This page therefore validates mathematical conditions rather than one exact grid.