A320871 List of all inequivalent 3 X 3 magic squares made of distinct positive integers, sorted by increasing sum. For each equivalence class modulo symmetries of the square, the lexicographically smallest representative is shown.

2, 7, 6, 9, 5, 1, 4, 3, 8, 2, 9, 7, 11, 6, 1, 5, 3, 10, 3, 7, 8, 11, 6, 1, 4, 5, 9, 3, 8, 7, 10, 6, 2, 5, 4, 9, 2, 11, 8, 13, 7, 1, 6, 3, 12, 3, 10, 8, 12, 7, 2, 6, 4, 11, 4, 8, 9, 12, 7, 2, 5, 6, 10, 4, 9, 8, 11, 7, 3, 6, 5, 10, 2, 13, 9, 15, 8, 1, 7, 3, 14, 3, 11, 10, 15, 8, 1, 6, 5, 13

Offset: 1

Author

M. F. Hasler, Oct 28 2018

Keywords

• nonn
• tabf

Comments

"Symmetries of the square" means the symmetry group D4 consisting of reflections on any of the 4 symmetry axes of the square (horizontal H, vertical V, 2 diagonals D & A), which also generates the rotations around the center by multiples of 90°, R1, R2, R3 (and R0 = id): e.g., H o D = R1, where D means to transpose the 3 X 3 matrix, H means reversal of the rows, etc.

The 8 ("equivalent") variants of the first square are listed in A217568.

Examples

			The first five inequivalent magic squares (with magic sums 15, 18, 18, 18, 21) are
   [2 7 6]    [ 2 9  7]    [ 3 7 8]    [ 3 8 7]    [ 2 11  8]
   [9 5 1]    [11 6  1]    [11 6 1]    [10 6 2]    [13  7  1]
   [4 3 8]    [ 5 3 10]    [ 4 5 9]    [ 5 4 9]    [ 6  3 12]
They are listed as rows of the 9 elements of each square, so the first row is:
  [2, 7, 6; 9, 5, 1; 4, 3, 8],
the second row is:
  [2, 9, 7; 11, 6, 1; 5, 3, 10], and so on.
		

Crossrefs

Cf. A217568: the 8 equivalent variants of the first row.

Cf. A320872: subsequence of rows that consist only of primes; A268790 lists their magic sums with duplicates removed.

Cf. A320873: the first row that consists of a set of consecutive primes; it has magic sum = 4440084513 = A270305(1) = A073520(3).

Programs

PARI

A320871_row(N=10,show_all=1,c=3)={for(c=c,oo, forstep(d=c-1,2,-1, for(b=max(2*d+1-c,1), d-1, d!=2*b&& S=[c-d,c+b,c+d-b;c+2*d-b,c,c-2*d+b;c-d+b,c-b,c+d]; !(show_all&&print(S))&& !N--&& return(S))))} \\ The third (optional) argument allows starting the list with the first square(s) having the central element >= c, i.e., magic sum >= 3c.