Chapter 14 · Number-Filling Arrays and Diagrams

Count Shared Positions and Discover Magic-Square Invariants

Lesson 14.1 · 统计重复位置，发现三阶幻方不变量

Learn to count a value once, twice, or many times depending on how a diagram uses its position. Then use the same idea to uncover powerful facts that hold in every 3 × 3 magic square.

Count participation before solving individual cells.

Self-contained lessonInteractive diagramsMultiple valid constructions respectedObjective checks only

Mission 1

Count how often each position participates

Learn to notice repeated numbers when adding totals. Here we count only the three straight lines drawn: horizontal, vertical, and diagonal. Each complete line passes through C; it is not two separate lines at C.

Not complete

Trace a complete line: choose a button, or use Tab and Enter.

The three lines are A–C–B, D–C–E, and F–C–G. A is the leftmost outer point.

Participation ledger

Center C

Each outer

Weighted counting: if a position lies on three lines, its value appears three times when all three line sums are added.

“Unknown” does not mean “count once.” The diagram—not the value—determines the weight.

Check the participation count

How many of the three drawn straight lines pass through C? Choose the leftmost outer point A. How many of the three drawn straight lines pass through A? When you add all three line totals, how many times do you include C’s number? What determines a position's weight?

Need a next step?

Hint

Start at A and trace the horizontal line through C to B. Now inspect the vertical and diagonal lines: does either pass through A? Count only lines drawn in this diagram.

Reasoning and worked explanation

Only A–C–B passes through A, so A’s number is counted once. All three lines pass through C, so C’s number is counted three times.

For example, suppose each outer point contains 1 and C contains 5. Each line total is 1 + 5 + 1 = 7. Adding the three totals gives 21: six outer 1s counted once each, plus the center 5 counted three times. That is 6 × 1 + 3 × 5 = 21. The letters label positions; they are not numbers to add.

Mission 2

Explore the worked example's six-triangle diagram

Count only the six triangular sectors formed by the center and two neighboring outer points. Larger triangles made by joining sectors are not included. Select a triangle to see its three vertices, then compare the participation of the center and outer positions.

Not complete

Select any triangular sector.

Participation counts

Each outer

Center C

6 triangles × 3 vertices = 18 vertex appearances

Six outer points each contribute two appearances:

6 × 2 = 12

The center contributes the remaining six appearances.

Read the diagram

Number of triangles Total vertex appearances Times each outer point is used Times the center is used

Need a next step?

Hint

A sector has three vertices. Trace all sectors around one outside point.

Reasoning and worked explanation

Six sectors give 18 vertex appearances. The six outside positions each occur twice; the center occurs six times.

Mission 3

Use weighted counting to find the worked example center

The numbers 1 through 7 are used once. Let v be the number placed at the center C. The sum of the vertex numbers over all six triangles is 64.

Not complete

Worked example 1: Put 1, 2, …, 7 into the seven circles. If the six triangle vertex sums total 64, what number is in the center?

28sum of 1 through 7

56if every number were counted twice

4vfour extra copies of the center

64given weighted total

Test a center candidate

Center value v: 1

Each outer value is counted twice. The center is counted six times, so it has four extra copies beyond the baseline.

Predicted total

56 + 4 × 1 = 60

The target is 64.

2(28 − v) + 6v = 56 + 4v = 64

Complete the worked example reasoning

Sum 1 + 2 + ··· + 7 Baseline if every value is counted twice Extra copies of the center Center value v

Need a next step?

Hint

Begin with every number counted twice. What extra copies remain?

Reasoning and worked explanation

The pool totals 28. Twice that is 56. The center needs four more copies, so 56+4v=64.

Mission 4

Meet the eight lines of a 3 × 3 magic square

In this lesson, a magic square is a 3 × 3 array with one common sum S on all three rows, all three columns, and both corner-to-corner diagonals. Different cells need not contain different numbers unless a question explicitly requires it.

Not complete

Choose a row, column, or diagonal.

What stays the same?

required lines

common line sum S

center value e

The famous 1–9 square is one example, but the invariants in this lesson hold for every additive 3 × 3 magic square, even when the entries are not 1 through 9.

Important: knowing the center does not always determine every cell. It determines powerful totals and pair relationships.

Read the magic square

Number of required lines Common line sum Center value Sum of all nine cells

Need a next step?

Hint

Include both diagonals as well as rows and columns.

Reasoning and worked explanation

A 3×3 magic square has eight required lines. Checking only rows does not prove the square is magic.

Mission 5

Count the square two ways to prove S = 3e

Let S be the common line sum and e the center value. The center participates in more of the specially chosen lines than every other cell.

Not complete

Ledger A: all three rows

Three row sums3S

What is counted?All 9 cells once

So the total of all nine cells is 3S.

Ledger B: four lines through the center

Middle row + middle column + 2 diagonals4S

Every non-center cellonce

Center efour times

Compared with all cells once, this ledger has three extra copies of e.

4Sfour center-line sums

3S + 3eall cells once plus three extra centers

S = 3esubtract 3S

e = S ÷ 3the center is one third of the line sum

4S = 3S + 3e ⟹ S = 3e

Use the invariant

If e = 7, then S = The four center lines count e how many times? That is how many extra copies beyond counting all cells once? If S = 42, then e =

Need a next step?

Hint

Compare the four center lines with all three rows.

Reasoning and worked explanation

The four center lines count each outside cell once and the center four times. Subtract all cells once: the excess is 3e, so S=3e.

Mission 6

Discover opposite-pair and whole-grid invariants

Opposite cells lie directly across the center: the two ends of the middle row, middle column, or either corner-to-corner diagonal. Each of these four lines contains one opposite pair plus the center value e.

Not complete

S = 3e

common line sum

opposites = 2e

because pair + e = 3e

all 9 = 9e

because total = 3S

Magic-square invariant machine

Center eTop-left aTop-middle b

The remaining six cells are generated so every required line has sum 3e. Enter whole numbers from −1,000,000 through 1,000,000; entries need not be positive or distinct. The questions below use their stated center values, independently of the machine settings.

line sum

each opposite pair

all nine cells

Use the three invariants

If e = 12, line sum S If e = 12, each opposite pair If e = 12, total of all nine cells If opposite cells sum to 30, center e

Need a next step?

Hint

Each opposite pair shares a line with the center.

Reasoning and worked explanation

Pair+e=3e, giving pair=2e. Three rows total 3S=9e.

Mission 7

original exercise: recover b and h

The worked example labels the bottom-right cell ℓ. Every row, column, and diagonal has the same sum.

Not complete

original Exercise 1: In the magic square, a = 4, d = 19, and ℓ = 22. Find b and h.

a = 4

b = ?

d = 19

e = ?

h = ?

ℓ = 22

Staged deduction

a + e + ℓ = S = 3e

4 + e + 22 = 3e ⟹ e = 13

S = 3 × 13 = 39

g = 39 − 4 − 19 = 16

c + g = 2e = 26 ⟹ c = 10

b = 39 − 4 − 10 = 25

h = 26 − 25 = 1

Reconstruct the worked solution

Center e Common sum S Top-middle b Bottom-middle h

Need a next step?

Hint

The known diagonal puts 4 and 22 opposite each other.

Reasoning and worked explanation

Their total 26 is 2e, so e=13. Use S=39 and the remaining row and opposite-pair totals.

Mission 8

Solve without knowing the center—and transfer to overlaps

Some original clues determine a requested cell even when the center remains unknown. The same “count shared positions twice” idea also works in overlapping circles.

Not complete

original Exercise 2

Given:

b = ?

f = 19

g = 96

ℓ

Top row and left column have equal sums:

a+b+c = a+d+g ⟹ b = d+g−c

Opposite pairs have equal sum 2e:

c+g = d+f ⟹ d−c = g−f

b = g + (g−f) = 2g−f

b = 2×96−19 = 173

Overlap transfer: count all eight positions

Use each of 1–8 once in the eight small circles. Each large circle contains five small circles and has total 22. A and B belong to both large circles; the other six positions belong to only one. Which proposed pair cannot be A,B?

22 + 22 = (1+2+···+8) + (A+B)

44 = 36 + (A+B) ⟹ A+B = 8

Among the worked example choices, the pair 4 and 8 cannot fill A and B because its sum is 12, not 8.

Apply both shared-position arguments

For f = 19 and g = 96, b = The shortcut expression for b is In the overlapping circles, A + B = Which original pair cannot be A and B?

Need a next step?

Hint

In the overlap, ask which labels were counted twice.

Reasoning and worked explanation

All eight labels total 36. Adding the large-circle sums counts A and B one additional time. Subtract 36 to find the shared-pair total.

Explore further: a magic square with equal products

Start with the additive magic square 8,1,6 / 3,5,7 / 4,9,2. Replace each entry n by 2ⁿ (n factors of 2). Its rows become 256,2,64 / 8,32,128 / 16,512,4. All nine values differ. Every row, column, and diagonal product is 2¹⁵=32768, because multiplying powers of 2 adds their exponents. For example 256×2×64=32768. The same additive structure now controls products.

Mission 9

Fresh strategy workshop

Try these new situations before opening the explanations. Correct all eight answers to complete this workshop.

Not complete

1. Six triangular sectors use the numbers 1–7 once. Their vertex sums total 76. What is the center?

Worked explanation — try first

The number pool totals 28. Count all labels twice, then four extra centers: (76−56)÷4=5.

2. In a 3×3 magic square, the center is 8. What is its line sum?

Worked explanation — try first

A line sum is three times the center: 3×8=24.

3. A magic square has line sum 51. Find its center.

Worked explanation — try first

51÷3=17.

4. A magic square has center 11. An opposite pair contains 7. Find its partner.

Worked explanation — try first

Opposites total 2×11=22, so the partner is 22−7=15.

5. All nine cells of a magic square total 126. Find its center.

Worked explanation — try first

The whole grid totals 9e. Therefore e=126÷9=14.

6. Eight circles use 1–8 once. Two large circles each contain five of them; A and B belong to both. Each large-circle sum is 24. Find A+B.

Worked explanation — try first

The two totals count all eight labels once and A,B one extra time: 48−36=12.

7. In a magic square labeled a,b,c / d,e,f / g,h,ℓ, f=12 and g=30. Find b.

Worked explanation — try first

Equal line sums and opposite pairs give b=2g−f=60−12=48.

8. Three lines share one center; all six outer points lie on one line each. Labels 1–7 are used once and the three line totals add to 36. Find the center.

Worked explanation — try first

The total 28 counts the center once; the three lines count it three times. The extra 8 equals two centers: 8÷2=4.

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Mission 10

Independent exit ticket

Try these new situations before opening the explanations. All five correct completes this mission. The certificate requires all ten missions.

Not complete

1. The six-sector 1–7 diagram has vertex-sum total 80. Find its center.

Worked explanation — try first

80−56=24 accounts for four extra centers. 24÷4=6.

2. A magic square has center 16. Find its line sum.

Worked explanation — try first

3×16=48.

3. An opposite pair in that square contains 9. Find the other number.

Worked explanation — try first

2×16−9=23.

4. A magic square has line sum 33. Find the total of all nine cells.

Worked explanation — try first

Add the three rows: 3×33=99.

5. Use numbers 1–8 once in the same eight-position overlap diagram. The large-circle totals are 23 and 25. Find the shared-pair total.

Worked explanation — try first

23+25−36=12. The large-circle totals need not be equal for this counting argument.

0 / 5

Optional reflection — saved, not automatically graded

Describe a strategy you can now use on a new puzzle. Explain one check that prevents a mistake.

Certificate of mastery

Shared-Position & Magic-Square Architect

This certifies that

Grade 5 Mathematician

counted shared positions correctly and used the invariants S = 3e, opposite pairs = 2e, and total = 9e.

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