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Various tests for the magicness of a square

is.magic.Rd

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, func=sum, boolean=FALSE) 
is.panmagic(m, give.answers = FALSE, func=sum, boolean=FALSE) 
is.pandiagonal(m, give.answers = FALSE, func=sum, boolean=FALSE) 
is.semimagic(m, give.answers = FALSE, func=sum, boolean=FALSE) 
is.semimagic.default(m)
is.associative(m)
is.normal(m)
is.sparse(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)
is.antimagic(m, give.answers = FALSE, func=sum) 
is.totally.antimagic(m, give.answers = FALSE, func=sum)
is.heterosquare(m, func=sum) 
is.totally.heterosquare(m, func=sum)
is.sam(m)
is.stam(m)

Arguments

m

The square to be tested

give.answers

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

func

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 func evaluate to TRUE. If boolean is FALSE, square m is magic etc if all applications of func are identical

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).

  • A sparse matrix is one whose nonzero entries are consecutive integers, starting at 1.

  • An associative square (also regular 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.

  • An ultramagic square is pandiagonal and associative.

  • 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.

  • An antimagic square is one whose row sums and column sums are consecutive integers; a totally antimagic square is one whose row sums, column sums, and two unbroken diagonals are consecutive integers. Observe that an antimagic square is not necessarily totally antimagic, and vice-versa.

  • A heterosquare has all rowsums and column sums distinct; a totally heterosquare [NB nonstandard terminology] has all rowsums, columnsums, and two long diagonals distinct.

  • A square is sam (or SAM) if it is sparse and antimagic; it is stam (or STAM) if it is sparse and totally antimagic. See documentation at SAM.

Value

Returns TRUE if the square is semimagic, etc. and FALSE if not.

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.

Function is.semimagic.default() returns TRUE if the argument is semimagic [with respect to sum()] using a faster method than is.semimagic().

Note

Functions that take a func argument apply that function to each row, column, and diagonal as necessary. If func takes its default value of sum(), the sum will be returned; if prod(), the product will be returned, and so on. There are many choices for this argument that produce interesting tests; consider func=max, for example. With this, a “magic” square is one whose row, sum and (unbroken) diagonals have identical maxima. Thus diag(5) is magic with respect to max(), but diag(6) isn't.

Argument boolean is designed for use with non-default values for the func argument; for example, a latin square is semimagic with respect to func=function(x){all(diff(sort(x))==1)}.

Function is.magic() is vectorized; if a list is supplied, the defaults are assumed.

References

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

Author

Robin K. S. Hankin

See also

minmax,is.perfect,is.semimagichypercube,sam

Examples

is.magic(magic(4))
#> [1] TRUE

is.magic(diag(7),func=max)  # TRUE
#> [1] TRUE
is.magic(diag(8),func=max)  # FALSE
#> [1] FALSE

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

is.panmagic(panmagic.4())
#> [1] TRUE
is.panmagic(panmagic.8())
#> [1] TRUE

data(Ollerenshaw)
is.mostperfect(Ollerenshaw)
#> [1] TRUE

proper.magic <- function(m){is.magic(m) & is.normal(m)}
proper.magic(magic(20))
#> [1] TRUE