Various tests for the magicness of a square
Returns TRUE if the square is magic, semimagic, panmagic, associative,
normal. If argument give.answers is TRUE, also returns
additional information about the sums.
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)
m |
The square to be tested |
give.answers |
Boolean, with |
func |
A function that is evaluated for each row, column, and unbroken diagonal |
boolean |
Boolean, with |
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-by-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 consecutiveintegers. 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.
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 2x2 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().
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.
Robin K. S. Hankin
is.magic(magic(4))
is.magic(diag(7),func=max) # TRUE
is.magic(diag(8),func=max) # FALSE
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))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.