A080992 - OEIS

A080992

Entries in Durer's magic square.

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

(list; graph; refs; listen; history; text; internal format)

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