Problem Description
 Dürer's Magic Square is contained in a famous copper engraving, "Melancholia",
created in 1514 by German artist Albrecht Dürer.
There are 86 different combinations of four numbers from the square that sum to
it's magic number, 34!
How many can you find?
Background & Techniques
Albrecht Dürer is generally considered to be Germany's most
famous Renaissance artist. He was about 20 years younger than
Leonardo da Vinci and around 1500 became greatly interested in the
relationship between mathematics and art. Leonardo and his
contemporary, mathematician Pacioli, almost certainly influenced Dürer in
these studies. In 1514 he created the copperplate engraving
named "Melancholia I" which contained this magic square - the
first magic square published in Europe. (Notice that the creation
date of the picture, 1514, is contained in the bottom row of the
square.)
The program will sequentially display all 86 ways to pick four numbers
that sum to 34. The button to start this display is hidden until the
user has identified 20 solutions. The first 20 or so are not too
difficult to find. After that, finding solutions by inspection
becomes increasingly difficult. I have yet to find more than 40
solutions - and I have watched the program solution several
times!
The programming for this problem is fairly
straightforward. A StringGrid is used to display the square
and an OnDrawCell exit to highlight the current set of selected
cells, whether selected by the user or the program. To accomplish
this, an array of strings, S, contains the selected cell
values. As each cell is drawn, its value is compared to the
values in S and if a match is found, the cell is
highlighted. User solutions are placed in a Listbox
Stringlist. In order to block duplicate solutions, we sort the four
numbers before adding them to the list so that Items.IndexOf
procedure can be used to check for duplicate
solutions.
Addendum September 8,. 2003:
Viewer Don Rowland recently noticed a minor problem while finding
solutions - the program allowed the same square to be selected more
than once. Together we came with the fix which was posted today.
.
Running/Exploring the Program
Suggestions for Further Explorations
I wonder - how many of the 86 solutions are
symmetric? I haven't really made a study of symmetry but it
seems that a solution would be symmetrical if you could fold the square
along a straight line so that the solution squares overlapped
exactly. Is flipping and rotating the square the same
logical operation as folding?
| Original Date: January 19,
2003 |
Modified: September 08, 2006
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