A217568 Rows of the 8 magic squares of order 3 and magic sum 15, lexicographically sorted.

2, 7, 6, 9, 5, 1, 4, 3, 8, 2, 9, 4, 7, 5, 3, 6, 1, 8, 4, 3, 8, 9, 5, 1, 2, 7, 6, 4, 9, 2, 3, 5, 7, 8, 1, 6, 6, 1, 8, 7, 5, 3, 2, 9, 4, 6, 7, 2, 1, 5, 9, 8, 3, 4, 8, 1, 6, 3, 5, 7, 4, 9, 2, 8, 3, 4, 1, 5, 9, 6, 7, 2

Offset: 1

Author

Jean-François Alcover, Oct 08 2012

Keywords

• easy
• fini
• nonn
• full
• tabf

Comments

See A320871, A320872 and A320873 for the list of all 3 X 3 magic squares of distinct integers, primes, resp. consecutive primes. In all these, only the lexicographically smallest of the eight "equivalent" squares are listed. Note that the terms are not always in the order that corresponds to the terms of this sequence. For example, in row 3 of A320871 and row 11 of A320873, the second term is smaller than the third term. However, when this is not the case, then row n of the present sequence is the list of indices which gives the n-th variant of the square from the (ordered) set of 9 elements: e.g., (2, 7, 6, ...) means that the 2nd, 7th and 6th of the set of 9 numbers yield the first row of the square. For example, A320873(n) = A073519(a(n)), 1 <= n <= 9. - M. F. Hasler, Nov 04 2018

Examples

			The first such magic square is
2, 7, 6
9, 5, 1
4, 3, 8
From _M. F. Hasler_, Sep 23 2018: (Start)
The complete table reads:
[2, 7, 6, 9, 5, 1, 4, 3, 8]
[2, 9, 4, 7, 5, 3, 6, 1, 8]
[4, 3, 8, 9, 5, 1, 2, 7, 6]
[4, 9, 2, 3, 5, 7, 8, 1, 6]
[6, 1, 8, 7, 5, 3, 2, 9, 4]
[6, 7, 2, 1, 5, 9, 8, 3, 4]
[8, 1, 6, 3, 5, 7, 4, 9, 2]
[8, 3, 4, 1, 5, 9, 6, 7, 2] (End)
		

Links

• Eric Weisstein, MathWorld: Magic Square
• Wikipedia, Magic Square
• Index entries for sequences related to magic squares

Crossrefs

Cf. A320871, A320872, A320873: inequivalent 3 X 3 magic squares of distinct integers, primes, consecutive primes.

Programs

Mathematica

squares = {}; a=5; Do[m = {{a + b, a - b - c, a + c}, {a - b + c, a, a + b - c}, {a - c, a + b + c, a - b}}; If[ Unequal @@ Flatten[m] && And @@ (1 <= #1 <= 9 & ) /@ Flatten[m], AppendTo[ squares, m]], {b, -(a - 1), a - 1}, {c, -(a - 1), a - 1}]; Sort[ squares, FromDigits[ Flatten[#1] ] < FromDigits[ Flatten[#2] ] & ] // Flatten

PARI

A217568=select(S->Set(S)==[1..9],concat(vector(9,a,vector(9,b,[a,b,15-a-b,20-2*a-b,5,2*a+b-10,a+b-5,10-b,10-a])))) \\ Could use that a = 2k, k = 1..4, and b is odd, within max(1,7-a)..min(9,13-a). - M. F. Hasler, Sep 23 2018