The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order n – that is, a magic square which contains the numbers 1, 2, ..., n2 – the magic constant is .
For normal magic squares of orders 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). For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order n is n3 + n/2. The largest magic constant of normal magic square which is also a:
triangular number is 15 (solve the Diophantine equation x2 = y3 + 16y + 16, where y is divisible by 4);
square number is 1 (solve the Diophantine equation x2 = y3 + 4y, where y is even);
generalized pentagonal number is 171535 (solve the Diophantine equation x2 = y3 + 144y + 144, where y is divisible by 12);
Note that 0 and 1 are the only normal magic constants of rational order which are also rational squares.
However, there are infinitely many rational triangular numbers, rational generalized pentagonal numbers and rational tetrahedral numbers which are also magic constants of rational order.
The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes. Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give polyiamond magic constant.[1]
The magic constant of an n-pointed normal magic star is .
Magic seriesedit
In 2013 Dirk Kinnaes found the magic series polytope. The number of unique sequences that form the magic constant is now known up to .[2]
Moment of inertiaedit
In the mass model, the value in each cell specifies the mass for that cell.[3] This model has two notable properties. First it demonstrates the balanced nature of all magic squares. If such a model is suspended from the central cell the structure balances. (consider the magic sums of the rows/columns .. equal mass at an equal distance balance). The second property that can be calculated is the moment of inertia. Summing the individual moments of inertia (distance squared from the center × the cell value) gives the moment of inertia for the magic square, which depends solely on the order of the square.[4]
magic, constant, unnamed, numerical, constants, magic, number, programming, unnamed, numerical, constants, magic, constant, magic, magic, square, numbers, column, diagonal, magic, square, example, magic, square, shown, below, magic, constant, normal, magic, sq. For unnamed numerical constants see Magic number programming Unnamed numerical constants The magic constant or magic sum of a magic square is the sum of numbers in any row column or diagonal of the magic square For example the magic square shown below has a magic constant of 15 For a normal magic square of order n that is a magic square which contains the numbers 1 2 n2 the magic constant is M n n2 12 displaystyle M n cdot frac n 2 1 2 For normal magic squares of orders 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 For example a normal 8 8 square will always equate to 260 for each row column or diagonal The normal magic constant of order n is n3 n 2 The largest magic constant of normal magic square which is also a triangular number is 15 solve the Diophantine equation x2 y3 16y 16 where y is divisible by 4 square number is 1 solve the Diophantine equation x2 y3 4y where y is even generalized pentagonal number is 171535 solve the Diophantine equation x2 y3 144y 144 where y is divisible by 12 tetrahedral number is 2925 Note that 0 and 1 are the only normal magic constants of rational order which are also rational squares However there are infinitely many rational triangular numbers rational generalized pentagonal numbers and rational tetrahedral numbers which are also magic constants of rational order The term magic constant or magic sum is similarly applied to other magic figures such as magic stars and magic cubes Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give polyiamond magic constant 1 Contents 1 Magic stars 2 Magic series 3 Moment of inertia 4 See also 5 References 6 External linksMagic stars editThe magic constant of an n pointed normal magic star is M 4n 2 displaystyle M 4n 2 nbsp Magic series editIn 2013 Dirk Kinnaes found the magic series polytope The number of unique sequences that form the magic constant is now known up to n 1000 displaystyle n 1000 nbsp 2 Moment of inertia editIn the mass model the value in each cell specifies the mass for that cell 3 This model has two notable properties First it demonstrates the balanced nature of all magic squares If such a model is suspended from the central cell the structure balances consider the magic sums of the rows columns equal mass at an equal distance balance The second property that can be calculated is the moment of inertia Summing the individual moments of inertia distance squared from the center the cell value gives the moment of inertia for the magic square which depends solely on the order of the square 4 See also editMagic number physics References edit A303295 Oeis Walter Trump http www trump de magic squares Heinz http www magic squares net ms models htm A 3 dimensional magic square Peterson http www sciencenews org view generic id 7485 description Magic Square Physics External links edit260 as a magic constant for the 8 queens problem and 8x8 magic square Hypercube Math formulae Retrieved from https en wikipedia org w index php title Magic constant amp oldid 1135527104, wikipedia, wiki, book, books, library,
