Place the numbers 1 through 9 on a $3 \times 3$ grid so that the sum of each row, each column, and each diagonal is exactly 15. This is the legendary magic square of Lo Shu, discovered according to legend in the shell of a divine turtle. What is the solution? How many unique solutions are there (not counting rotations and reflections)?
Chinese mathematics, ~2200 BC
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The magic square of Lo Shu (China)
Hints
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- With those restrictions, only rotation/reflection equivalent configurations remain.
- There is 1 essential solution. The rest are its 8 symmetries of the square (4 rotations and their reflections).
- A fundamental solution (unique except for symmetries) is: Rows: 2+7+6=15, 9+5+1=15, 4+3+8=15.
Solution
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Answer: A fundamental solution (unique except for symmetries) is:
1) Direct verification
- Rows: $2+7+6=15$, $9+5+1=15$, $4+3+8=15$.
- Columns: $2+9+4=15$, $7+5+3=15$, $6+1+8=15$.
- Diagonals: $2+5+8=15$, $6+5+4=15$.
2) Why the structure is forced
For numbers 1 to 9, the total sum is 45, so each of the 3 rows must add up to 15.
The center participates in 4 lines (row, column and two diagonals), and is forced to be 5 (average value).
Furthermore, pairs opposite the center must add up to 10: $(1,9)$, $(2,8)$, $(3,7)$, $(4,6)$.
With those restrictions, only rotation/reflection equivalent configurations remain.
3) Uniqueness
There is 1 essential solution. The rest are its 8 symmetries of the square (4 rotations and their reflections).
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