A080992 Entries in Dürer's magic square.

16, 3, 2, 13, 5, 10, 11, 8, 9, 6, 7, 12, 4, 15, 14, 1

Offset: 1

Author

David W. Wilson, Feb 26 2003

Keywords

• fini
• full
• nonn

Comments

4 X 4 magic square included in Albrecht Dürer's 1514 engraving "Melancolia". 15 and 14 appear in the bottom row, giving the date.

A006003(4) = 34 is the magic constant, occurring 23 times as sum of exactly 4 distinct numbers 1..16 with regular patterns in the 4 X 4 square(see also link): - Reinhard Zumkeller, Jun 20 2013

sum(T(k,i): i = 1..4) = sum(T(i,k): i = 1..4) = 34, for k = 1..4;

sum(T(k,k): k = 1..4) = sum(T(k,5-k): k = 1..4) = 34;

T(1,1) + T(1,2) + T(2,1) + T(2,2) = 16 + 3 + 5 + 10 = 34;

T(1,3) + T(1,4) + T(2,3) + T(2,4) = 2 + 13 + 11 + 8 = 34;

T(3,1) + T(3,2) + T(4,1) + T(4,2) = 9 + 6 + 4 + 15 = 34;

T(3,3) + T(3,4) + T(4,3) + T(4,4) = 7 + 12 + 14 + 1 = 34;

T(1,1) + T(1,4) + T(4,1) + T(4,4) = 16 + 13 + 4 + 1 = 34;

T(2,2) + T(2,3) + T(3,2) + T(3,3) = 10 + 11 + 6 + 7 = 34;

T(1,2) + T(2,4) + T(4,3) + T(3,1) = 3 + 8 + 14 + 9 = 34;

T(1,3) + T(3,4) + T(4,2) + T(2,1) = 2 + 12 + 15 + 5 = 34;

T(1,2) + T(2,3) + T(4,2) + T(2,1) = 3 + 11 + 15 + 5 = 34;

T(1,3) + T(2,4) + T(4,3) + T(2,2) = 2 + 8 + 14 + 10 = 34;

T(1,2) + T(3,3) + T(4,2) + T(3,1) = 3 + 7 + 15 + 9 = 34;

T(1,3) + T(3,4) + T(4,3) + T(3,2) = 2 + 12 + 14 + 6 = 34;

T(1,2) + T(1,3) + T(4,2) + T(4,3) = 3 + 2 + 15 + 14 = 34;

T(4,2)*100 + T(4,3) = 1514, the year of the engraving and the pair (T(4,4),T(4,1)) = (1,4) corresponds to Albrecht Dürer's coded initials.

The square has its magic constant (34) equal to one of its eigenvalues (34, 8, -8, 0) like any other normal magic square of order n > 2. - Michal Paulovic, Mar 14 2021

Examples

			.          1    2    3    4
.       +----+----+----+----+
.    1  | 16 |  3 |  2 | 13 |
.       +----+----+----+----+
.    2  |  5 | 10 | 11 |  8 |
.       +----+----+----+----+
.    3  |  9 |  6 |  7 | 12 |
.       +----+----+----+----+
.    4  |  4 | 15 | 14 |  1 |
.       +----+----+----+----+
.          D   ^^   ^^    A
		

References

• Hossin Behforooz, "Permutation-free magic squares", J. Recreational Mathematics, vol. 33, (2004-2005), pp. 103-106.

Links

• History 291, Princeton University, Durer's Melancolia
• Laurence Eaves and Brady Haran, Magic square - Sixty Symbols
• A. Skalli, Magic cube with Dürer's square
• Torsten "Kermit", Die Rolle Dürers in Thomas Manns Doktor Faustus (in German).
• Eric Weisstein's World of Mathematics, Dürer's Magic Square
• Eric Weisstein's World of Mathematics, Gnomon Magic Square
• Wikipedia, Albrecht Dürer's magic square
• Reinhard Zumkeller, The 23 sums in Albrecht Dürer's magic square
• Index entries for sequences related to magic squares

Extensions

Extended by Reinhard Zumkeller, Jun 20 2013