magic (version 1.3-20)

is.magic: Various tests for the magicness of a square

Description

Returns TRUE if the square is magic, semimagic, panmagic, associative, normal. If argument give.answers is TRUE, also returns additional information about the sums.

Usage

is.magic(m, give.answers = FALSE, FUN=sum, boolean=FALSE) 
is.panmagic(m, give.answers = FALSE, FUN=sum, boolean=FALSE) 
is.semimagic(m, give.answers = FALSE, FUN=sum, boolean=FALSE) 
is.associative(m)
is.normal(m)
is.mostperfect(m,give.answers=FALSE)
is.2x2.correct(m,give.answers=FALSE)
is.bree.correct(m,give.answers=FALSE)
is.latin(m,give.answers=FALSE)

Arguments

m

The square to be tested

give.answers

Boolean, with TRUE meaning return additional information about the sums (see details).

FUN

A function that is evaluated for each row, column, and unbroken diagonal.

boolean

Boolean, with TRUE meaning that the square is deemed magic, semimagic, etc, if all applications of FUN evaluate to TRUE. If boolean is FALSE, square m is magic etc

Value

• Returns TRUE if the square is semimagic, etc. If give.answers is taken as an argument and is TRUE, return a list of at least five elements. The first element of the list is the answer: it is TRUE if the square is (semimagic, magic, panmagic) and FALSE otherwise. Elements 2-5 give the result of a call to allsums(), viz: rowwise and columnwise sums; and broken major (ie NW-SE) and minor (ie NE-SW) diagonal sums.

  Function is.bree.correct() also returns the sums of elements distant $n/2$ along a major diagonal (diag.sums); and function is.2x2.correct() returns the sum of each $2\times 2$ submatrix (tbt.sums); for other size windows use subsums() directly. Function is.mostperfect() returns both of these.

Details

A semimagic square is one all of whose row sums equal all its columnwise sums (ie the magic constant).

A magic square is a semimagic square with the sum of both unbroken diagonals equal to the magic constant.

A panmagic square is a magic square all of whose broken diagonals sum to the magic constant. Ollerenshaw calls this a ``pandiagonal'' square.

A most perfect square has all 2-by-2 arrays anywhere within the square summing to $2S$ where $S=n^2+1$; and all pairs of integers $n/2$ distant along the same major (NW-SE) diagonal sum to $S$ (note that the $S$ used here differs from Ollerenshaw's because her squares are numbered starting at zero). The first condition is tested by is.2x2.correct and the second by is.bree.correct. All most perfect squares are panmagic.

A normal square is one that contains $n^2$ consecutive integers (typically starting at 0 or 1).

An associative square is a magic square in which $a_{i,j}+a_{n+1-i,n+1-j}=n^2+1$. Note that an associative semimagic square is magic; see also is.square.palindromic(). The definition extends to magic hypercubes: a hypercube a is associative if a+arev(a) is constant.

A latin square of size $n\times n$ is one in which each column and each row comprises the integers 1 to n (not necessarily in that order). Function is.latin() is a wrapper for is.latinhypercube() because there is no natural way to present the extra information given when give.answers is TRUE in a manner consistent with the other functions documented here.

References

http://mathworld.wolfram.com/MagicSquare.html

See Also

minmax,is.perfect,is.semimagichypercube

Examples

is.magic(magic(4))

is.magic(diag(9),FUN=max)  #should be TRUE

stopifnot(is.magic(magic(3:8)))

is.panmagic(panmagic.4())
is.panmagic(panmagic.8())

data(Ollerenshaw)
is.mostperfect(Ollerenshaw)

proper.magic <- function(m){is.magic(m) & is.normal(m)}
proper.magic(magic(20))