(sold for $136.0)
1500s, Austria/Germany. Copper "Magic Square of Saturn" Token (Pfennig) Coin. R! Condition: F-VF Mint Place: Hall (Tirol) Mint Period: 1500s-1580s Denomination: Magic Square of Saturn (also known as the Lo Shu Square in ancient Chinese mathematics) - Counter Token (Raitpfennig) References: Joseph Appel (Münzen und Medaillen der Republiken, Städte Ortschaften, Gymnasien etc. Wien, 1829), Himmelszeichen, Amulete, etc., p. 1099, No. 4031. var. R! Material: Copper Diameter: 25mm Weight: 3.7gm Obverse: Mathematical magic square (4-9-2/3-5-7/8-1-6) of Saturn , which add up to 15 in all directions. Legend: ZEITLICH . WOLL . GERAIT . (castle?) . Reverse: Abacus with five counters to left and eight to right. Legend: + MACHT . GVETE . RICHTIGKAIT The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection. The Square of Lo Shu is also referred to as the Magic Square of Saturn. em>. In recreational mathematics, a magic square is an arrangement of distinct numbers (i.e. each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the main and secondary diagonals, all add up to the same number. A magic square has the same number of rows as it has columns, and in conventional math notation, "n" stands for the number of rows (and columns) it has. Thus, a magic square always contains n2 numbers, and its size (the number of rows [and columns] it has) is described as being "of order n". A magic square that contains the integers from 1 to n2 is called a normal magic square. (The term "magic square" is also sometimes used to refer to any of various types of word squares.) Normal magic squares of all sizes except 2 × 2 (that is, where n = 2) can be constructed. The 1 × 1 magic square, with only one cell containing the number 1, is trivial. The smallest (and unique up to rotation and reflection) nontrivial case, 3 × 3. Any magic square can be rotated and reflected to produce 8 trivially distinct squares. In magic square theory all of these are generally deemed equivalent and the eight such squares are said to comprise a single equivalence class. The constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, M. Every normal magic square has a constant dependent on n, calculated by the formula M = [n(n2 + 1)] / 2. For normal magic squares of order n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). Magic squares have a long history, dating back to 650 BC in China. At various times they have acquired magical or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.
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