A080992 Entries in Durer's magic square.

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

Offset

1,1

Comments

4 X 4 magic square included in Albrecht Durer'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

References

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

Links

Table of n, a(n) for n=1..16.

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

Example

.          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

Crossrefs

Sequence in context: A031323 A095838 A040248 * A040249 A099143 A070709

Adjacent sequences: A080989 A080990 A080991 * A080993 A080994 A080995

Keyword

fini,full,nonn

Author

David W. Wilson, Feb 26 2003

Extensions

Extended by Reinhard Zumkeller, Jun 20 2013

Status

approved