Class 8 · English Version · Mathematics
2026 Class 8 Mathematics · English Version · Chapter 1 · Chapter companion part
Explain pair-sum formulae with diagrams, distinguish terms from totals and construct checked magic squares.
Class 8 English Version: part B of the chapter 1 companion. Eight teaching sections, 30 written tasks and a notebook challenge. Selected companion coverage, not complete textbook coverage; use all three parts and review independent proofs and drawings with a teacher.
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For 1 + 2 + ... + 8, pair 1 with 8, 2 with 7, 3 with 6 and 4 with 5. Four pairs each total 9, giving 36. A second copy in reverse order produces eight columns of 9, which count the original sum twice, so divide by 2. This reverse-copy method also works for an odd number of terms without pretending there are a whole number of pairs in one copy.
For n terms with constant difference, S = n(first + last) ÷ 2. Both n and the endpoints matter. This formula sums an arithmetic sequence; it is not valid for every list with known endpoints. In the special list 1,2,...,n, first = 1 and last = n, giving n(n + 1) ÷ 2. A term is one entry; a sum adds all specified entries.
Begin with one unit square. Add an L-shaped border of 3 to make 2 × 2, then 5 to make 3 × 3, then 7 to make 4 × 4. At stage n the added border has n units along one side and n − 1 along the other, totalling 2n − 1 without double-counting the corner. The sum of the first n positive odd numbers is n². The nth odd number itself is 2n − 1, a different expression.
To write N = a² + b², choose positive integers a and b and calculate both squares. For 50, 1² + 7² = 50 and 5² + 5² = 50; swapping a and b is not a new unordered representation. State whether zero is allowed: here it is not. Not every positive integer can be written this way. The source’s count of 35 values from 1 to 100 uses positive summands and counts each resulting value once.
A normal order-m magic square uses each integer from 1 to m² exactly once, with every row, every column and both main diagonals having the same sum. Order means the number of rows or columns, not the total number of cells. Checking only the rows is insufficient. A grid with repeated numbers or a wrong diagonal is not a normal magic square just because one row adds correctly.
A normal 3 × 3 square uses 1 to 9 with centre 5 and line sum 15. One arrangement is 8,1,6 / 3,5,7 / 4,9,2. Corners are even in this arrangement and rotations or reflections give related valid arrangements. Check all three rows, all three columns and both diagonals. Adding the same constant c to every cell raises each line sum by 3c, but the new entries no longer form the normal 1-to-9 square.
Write 1 to 16 row by row. In our chosen method, keep the eight positions on the two main diagonals. Replace every other entry x by 17 − x. The completed square is 1,15,14,4 / 12,6,7,9 / 8,10,11,5 / 13,3,2,16. Each row, column and main diagonal totals 34. The source uses another valid diagonal-complement orientation; do not mix the two procedures halfway through.
The total of 1 to m² is m²(m² + 1) ÷ 2. Dividing it among m equal row sums gives the normal magic constant m(m² + 1) ÷ 2. For m = 5 it is 65. For an odd-order extension, place 1 in the top middle, move up-right with edge wrapping for the next value, and move down one from the previous cell when the next cell is occupied. This method is for odd orders; do not apply it to order four. Verify the supplied order-five table independently.
For the first 12 positive odd numbers, the last is 23. The arithmetic sum gives 12 × (1 + 23) ÷ 2 = 144. The square diagram gives 12² = 144. Two routes agree because they count the same specified total.
Natural number conventions differ. In the two-square search this companion requires a,b ≥ 1 and treats (a,b) and (b,a) as the same pair. The source count 35 is verified for that convention.
The source’s order-four diagonal replacement gives a rotated/complemented arrangement. This companion teaches a single keep-diagonals method and verifies it; either valid finished square still needs every check.
This counts all n terms in a constant-difference list, not a single nth term.
S = n(first + last) ÷ 2 1 + 2 + ... + n = n(n + 1) ÷ 2
A list with changing differences, such as square numbers, does not meet this arithmetic-sum condition.
The new L-shaped border has two arms sharing one corner. The full figure is a square.
nth positive odd = 2n − 1 1 + 3 + ... + (2n − 1) = n²
Do not substitute 2n − 1 when the task asks for the sum, or n² when it asks for one odd term.
Each pair is unordered and both entries must be at least 1.
| Target | One representation | Another representation |
|---|---|---|
| 50 | 1² + 7² = 1 + 49 | 5² + 5² = 25 + 25 |
| 65 | 1² + 8² = 1 + 64 | 4² + 7² = 16 + 49 |
| 130 | 3² + 11² = 9 + 121 | 7² + 9² = 49 + 81 |
| 170 | 1² + 13² = 1 + 169 | 7² + 11² = 49 + 121 |
| 185 | 4² + 13² = 16 + 169 | 8² + 11² = 64 + 121 |
| 325 | 1² + 18² = 1 + 324 | 6² + 17² = 36 + 289; also 10² + 15² |
Reversing the same pair is not a new representation. Not every integer has such a representation.
Sum the entries once, then divide by the number of equal row sums.
M = m(m² + 1) ÷ 2
Equal totals are necessary but not enough: check distinct entries and both main diagonals too.
Choose each stage or play the finite sequence. In this method the main-diagonal positions are kept and every other entry is complemented.
No autoplay. Step buttons and all three static stages remain available. The two-second teaching pace is illustrative; reduced motion disables automatic playback.
Use the up-right wrapping rule for odd order; if the next cell is occupied, move down one from the previous cell.
| Row | Column 1 | Column 2 | Column 3 | Column 4 | Column 5 |
|---|---|---|---|---|---|
| 1 | 17 | 24 | 1 | 8 | 15 |
| 2 | 23 | 5 | 7 | 14 | 16 |
| 3 | 4 | 6 | 13 | 20 | 22 |
| 4 | 10 | 12 | 19 | 21 | 3 |
| 5 | 11 | 18 | 25 | 2 | 9 |
Each row and column and both main diagonals total 65, with 1 to 25 used once. This extension method is not for even order.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original values and layout. The method generalises to an arithmetic list; it is not an endpoint formula for arbitrary non-linear lists.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original unit-square count models, not a measured physical area. The new border is the bottom row plus right column with their corner counted once.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original square-area extension to the source numerical examples. Both side counts are positive integers and all cell units are equal within this plate.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original orientation and layout of a standard magic square. Rotations and reflections can also be valid; order three means three rows, not three cells.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original arrangement using the standard complement property. Entries 1 to 16 appear once; the uncompleted seed grid is not magic.
Show your working on paper. Check your answer, then choose Next.
36
8 × 9 ÷ 2 = 36.
120
15 × 16 ÷ 2 = 120.
155
10 × (2 + 29) ÷ 2 = 155.
144
The sum is 12².
23
2 × 12 − 1 = 23; it is not the sum 144.
9
25 − 16 = 9, the fifth odd border.
7
50 − 1 = 49 = 7².
7
65 − 16 = 49 = 7².
15
Each row, column and main diagonal has this sum.
5
The centre is 5 when the entries are 1 to 9.
34
The total 136 is divided among four equal rows.
5
17 − 12 = 5; diagonal positions are kept in this method.
16
4 × 4 = 16, not four.
65
5 × (25 + 1) ÷ 2 = 65.
27
Each line contains three entries, so it increases by 3 × 4 = 12.
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Opposite endpoints make four pairs of 9, giving 36. Alternatively, two opposite copies give eight columns of 9, so 2S = 72 and S = 36.
Explain why division by 2 is needed.
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Write one copy forwards and one backwards. Each of n columns sums to n + 1, so 2S = n(n + 1) and S = n(n + 1)/2.
Two copies make the derivation work for odd as well as even n.
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These are squares, with non-constant differences, so the arithmetic-sum formula does not apply. Direct addition gives 1 + 4 + 9 + 16 = 30.
Check the formula’s conditions before calculating.
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Successive L-shaped borders of 1,3,5,... build a square with n unit cells along each side. Its area count is n × n. The new border adds n + (n − 1) cells, avoiding double-counting the corner.
Connect the border count to the total square, not just memorise a formula.
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The nth term is 2n − 1, so the eighth odd number is 15. The sum is n², so the first eight total 64.
Use both expressions and say what each measures.
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130 = 3² + 11² = 7² + 9². Each pair sums to 130, and swapping the order of one pair would not create a third representation.
Check all four squares and distinguish a new pair from a swap.
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325 = 1² + 18² = 6² + 17² = 10² + 15². The totals are 1 + 324, 36 + 289 and 100 + 225.
All summands must be positive and pairs distinct.
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List a² + b² for positive a ≤ b while the sum is at most 100, then remove duplicate totals. Allowing zero would introduce some perfect-square values that the positive-only definition excludes.
Use an explicit convention and count values rather than ordered pairs.
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For 8,1,6 / 3,5,7 / 4,9,2, every row and column sums to 15; diagonals 8 + 5 + 2 and 6 + 5 + 4 also sum to 15. Every number 1 to 9 occurs once.
Check all eight lines and distinct entries.
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Its columns or main diagonals may differ, or entries may repeat or lie outside 1 to 9. Check every defining condition.
Give a concrete possible failure, not an unsupported certificate.
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Fill 1 to 16 rowwise, identify both diagonals and keep those eight positions. Complement each other value using 17 − x, then check all rows, columns and both main diagonals total 34.
Do not complement diagonal entries in a method that says keep them.
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Each three-cell line gains 12, so its sum changes from 15 to 27. Equal line sums are preserved but the entries are now 5 to 13, so it is not normal using 1 to 9.
Separate the magic property from the normal-entry convention.
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The total of 1 to m² is m²(m² + 1)/2. Divide by m equal row sums to obtain m(m² + 1)/2. At m = 4, 4 × 17 / 2 = 34.
Name order, total entries and row count.
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Use up-right wrapping for an empty next cell and one step down from the previous cell if blocked. A supplied valid result is 17,24,1,8,15 / 23,5,7,14,16 / 4,6,13,20,22 / 10,12,19,21,3 / 11,18,25,2,9. Check all rows, columns and both main diagonals total 65.
Also check the integers 1 to 25 each occur exactly once; this odd-order method is not for order four.
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Choose a swap, recompute affected rows, columns and diagonals, and identify a failed condition. Some special swaps can preserve some lines, so check rather than assume.
Different valid counterexamples are accepted after human review.
Make a pairing proof, a coloured odd-border square and a checked magic square on paper.
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This original Kidemy practice follows the topic order of 2026 Class 8 Mathematics · English Version, chapter 1, printed pages 1–11. It does not replace the complete chapter or its practical teaching.
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