Magic Squares

Excerpts

• Magic Squares

  For simplicity I shall apply this method to construct a magic square of only the sixth order, though an exactly similar method will apply to any even square of an order higher than the second.

• Magic Squares

  Following the analogy of the notation used above, two numbers which are equidistant from the ends of the series $1,2,3,\dotsc,n$ are said to be *complementary*.

• Magic Squares

  we begin by constructing two subsidiary squares, one of the unit-digits, $1,2,3,\dotsc,n$, and the other of the radix-digits $0,n,2n,\dotsc,(n-1)n$.

• Magic Squares

  I do not know to whom the modification is due.

• Magic Squares

  I confine my account to such magic squares, that is, to squares formed with consecutive integers, from $1$ upwards.

• Magic Squares

  If the integers are the consecutive numbers from $1$ to $n^2$ the square is said to be of the $n$th order, and it is easily seen that in this case the sum of the numbers in any row, column, or diagonal is equal to $\frac{1}{2}n(n^2 +1)$: this number may be denoted by $N$.

• Magic Squares

  The formation of these squares is an old amusement, and in times when mystical philosophical ideas were associated with particular numbers it was natural that such arrangements should be deemed to possess magical properties.

• Magic Squares

  Magic squares of an odd order were constructed in India before the Christian era according to a law of formation which is explained hereafter.

• Magic Squares

  He taught that a square of one cell, in which unity was inserted, represented the unity and eternity of God; while the fact that a square of the second order could not be constructed illustrated the imperfection of the four elements, air, earth, fire, and water; and later writers added that it was symbolic of original sin.

• Magic Squares

  The reason why such a square is magic can be explained best by expressing the numbers in the scale of notation whose radix is $5$ (or $n$, if the magic square is of the order $n$), except that $5$ is allowed to appear as a unit-digit and $0$ is not allowed to appear as a unit-digit.

• Magic Squares

  The cells filled by the same number form a *broken diagonal*.

• Magic Squares

  It is unfortunate that no more obvious rule---such, for instance, as one for bordering a doubly-even square---can be suggested for writing down instantly and without thought singly-even magic squares.

• Magic Squares

  The majority of the medieval astrologers and physicians were much impressed by such arrangements.

• Magic Squares

  The square so formed is necessarily magic in rows, columns, and diagonals.

• Magic Squares

  two numbers which are equidistant from the ends of the series $1,2,3,\dotsc,n$ are said to be *complementary*.

• Magic Squares

  In the case of a singly even square, that is, one in which $n$ is divisible by $2$, but not by $4$, we cannot satisfy the proviso if any horizontal row in the first square has all its vertically related squares, other than the two squares in the diagonals, filled with complementary numbers.

• Magic Squares

  In this manner from the magic square of the $3$rd order we can build up successively squares of the orders $5$, $7$, $9$, &c., that is, any odd magic square.

• Magic Squares

  By reciprocating the figures composed of the points on which the numbers are placed we obtain a collection of lines forming pencils, and, if these lines be numbered to correspond with the points, the pencils will be magic

• Magic Squares

  I believe that with a little patience a magic square of any order can be thus built up, and of course it will have the property that, if each border is successively stripped off, the square will still remain magic.

Equations

• Magic Squares

  N = \frac{1}{2}n (n^2 + 1)

  The magic sum N, the common total of every row, column and diagonal of a magic square of order n filled with the integers 1 to n^2, equals one half of n times (n^2 + 1).

• Magic Squares

  \frac{1}{2}n(n-1)

  Each of the n radix-digits in a diagonal of the square built by De la Hire's method has the value one half of n(n-1), so the diagonal sums to that multiple of n.

• Magic Squares

  \frac{1}{2}(n+1)

  Each of the n unit-digits in a diagonal of an odd magic square built by De la Loubère's method equals one half of (n + 1).

• Magic Squares

  n(n-2x + 1)

  In the x-th row from the top, the number in a cell is less than the number in the vertically related cell in the complementary row by n(n - 2x + 1).

• Magic Squares

  n-2y+1

  In the y-th column from the left, the number in a cell is less than the number in the horizontally related cell in the complementary column by n - 2y + 1.

• Magic Squares

  N-\frac{1}{2}n^2(n-2x+1)

  The sum of the numbers in the x-th row from the top of a square of order n, before interchanges, is N minus one half of n^2 times (n - 2x + 1).

• Magic Squares

  N + \frac{1}{2}n^2(n-2x + 1)

  The sum of the numbers in the complementary row, the x-th row from the bottom, before interchanges, is N plus one half of n^2 times (n - 2x + 1).

• Magic Squares

  N-\frac{1}{2}n (n-2y + 1)

  The sum of the numbers originally in the y-th column from the left, of a square of order n in natural order, is N minus one half of n times (n - 2y + 1).

• Magic Squares

  N + \frac{1}{2}n(n-2y +1)

  The sum of the numbers originally in the complementary column, the y-th column from the right, is N plus one half of n times (n - 2y + 1).

• Magic Squares

  \frac{1}{2}(n^2 + 1)

  The average number in a magic square of the nth order is one half of n squared plus one.

• Magic Squares

  \frac{1}{2}(n-2)\{(n-2)^2+1\}

  The sum of the numbers in each line of a magic square of order n-2 (the inner square of a bordered square) is one half of (n-2) times ((n-2) squared plus one).

• Magic Squares

  \frac{1}{2}\{(n-2)^2+1\}

  The average number in a magic square of order n-2 is one half of ((n-2) squared plus one).

• Magic Squares

  2(n-1)

  The difference between the average number of an nth-order square and that of an (n-2)th-order square is two times (n-1), so every number of the inner square is raised by this amount.

• Magic Squares

  n^2 + 1-p

  The number that must be placed opposite the number p in a bordered magic square of n squared cells is n squared plus one minus p; the book denotes it by p with a bar.

• Magic Squares

  \frac{1}{2}(n+1)(n+2)

  A double-domino set running from double zero to double n contains one half of (n+1)(n+2) dominoes.

Problems

No exercises in this chapter.