VectorPaint order-three magic squares: check nine unique values and eight sums of fifteen

English translation of the Chinese original, translated on 2026-11-11. Original publication dates, attribution, code and version notes are retained.

A 3×3 number grid is not a magic square merely because its total is 45. Arrange a classical order-three pattern manually, checking values 1–9 once each and eight line sums. Unlike a Latin square, this uses nine distinct numbers and diagonal-sum conditions. VectorPaint provides grid/text objects, not automatic solving, grading or uniqueness enumeration.

Give all nine cells fixed identities

Draw three equal rows/columns, numbered top-to-bottom and left-to-right in the checking view. Retain the declared orientation after screen rotation rather than reading from the bottom under old coordinates. A cell side of 160 gives width 480 in drawing units, excluding stroke width. Group text separately so centring glyphs does not move the grid. Verify nine blank cells, excluding outside titles. Manual lines suffice without a table control or invented magic-square tool; count and orientation remain your checks.

Enter a classical arrangement without discovery claims

Enter rows 8,1,6; 3,5,7; 4,9,2, with centre 5. Cambridge NRICH’s introduction supplies equal row/column/diagonal terminology. This is a classical arrangement; the recording and controls are original exercises, not a new discovery. Use independent text objects rather than misplacing a whole line into one cell. Compare with an external nine-cell list; neat alignment does not validate values. Enlarge potentially confused 8/3 and 6/9 glyphs.

Check membership before sums

Sort the nine values externally into 1–9, each once. Nine objects alone could contain duplicate 5 and a missing value. Total 45 is necessary, not sufficient. Replacing 2 with 3 gives 46, duplicate 3 and no 2: repair membership instead of compensating elsewhere. Do not insert 0 or 10 to force sums. Distinguish this ordinary order-three case from generalized squares using other values; not every object called a magic square must use 1–9.

Write all row and column calculations

Rows 8+1+6, 3+5+7, 4+9+2 each give 15. Columns 8+3+4, 1+5+9, 6+7+2 also give 15. Record cell positions and result for each, not one checked row for the whole grid. Add values rather than concatenating 816. For a wrong column inspect its original three objects instead of stretching the grid to reassign columns visually. Six checks still leave two diagonals. Correct rows/columns do not imply every arbitrary path totals 15.

Complete both main diagonals

The main diagonals 8+5+2 and 6+5+4 both give 15. Centre 5 belongs to two different checks; that does not duplicate its membership. Eight lines mean three rows, three columns and two main diagonals, not bent routes, perimeter paths or arbitrary triples. Temporary patterned guides can aid tracing but must not obscure 5; hide them when unwanted. Two-cell sloping neighbours are not the defined three-cell diagonals. Record exact positions instead of relying on remembered colours in small output.

Swap two cells within one row

In a copy swap first-row 1 and 6, giving 8,6,1 with other rows unchanged. Membership, total and first-row sum remain correct, but column two 6+5+9=20 and column three 1+7+2=10. Diagonal 8+5+2 remains 15, while 1+5+4=10. Record changed and preserved conditions rather than saying every sum fails. Restore by positions and recheck all eight, not by relabelling columns. This edits number objects without automatic detection and says nothing against the original classical pattern.

Inspect glyphs, positions and records after export

Save and read all nine numbers in an actual PNG at use size, then compare the eight calculations. Resolve rotated 6/9 ambiguity through font or orientation, not reversed glyphs. Hide unwanted row/column numbers while retaining necessary explanations. Thin lines obscuring values or enlarge presentation instead of deleting a number. Inspect other formats independently; reposition substituted text without changing values. Monochrome remains checkable by position, and image export itself is no software mathematical validation.

Troubleshoot membership, rows, columns and diagonals

For nine entries with duplicates inspect membership; for one correct row but a failed grid inspect columns/diagonals; for total 45 alone inspect positional constraints; for errors after restoration inspect overlapping objects. Recheck affected lines after each cell repair and finally all eight. Retain original grid, membership list, six row/column equations, two diagonals and swap control. This is a reproducible example, not automatic solving, proof of uniqueness among every arrangement or classification of any nine-cell table. Other orders/number sets need new conditions and target sums.