TRUE if a hypercube is semimagic, magic, perfectis.semimagichypercube(a, give.answers=FALSE, FUN=sum, boolean=FALSE)
is.diagonally.correct(a, give.answers = FALSE, FUN=sum, boolean=FALSE)
is.magichypercube(a, give.answers = FALSE, FUN=sum, boolean=FALSE)
is.perfect(a, give.answers = FALSE, FUN=sum, boolean=FALSE)
is.latinhypercube(a, give.answers=FALSE)TRUE meaning to also return
the sumsTRUE meaning that the hypercube is
deemed magic, semimagic, etc, if all applications of FUN
evaluate to TRUE. If boolean is FALSE, the
hypercube is magic etc if al A semimagic hypercube has all is.semimagichypercube(), if give.answers
is TRUE, the sums returned are in the form of an array of
dimension c(rep(n,d-1),d). The first d-1 dimensions are
the coordinates of the projection of the summed elements onto the surface
hypercube. The last dimension indicates the dimension along which the
sum was taken over.
Optional argument FUN, defaulting to sum(), indicates
the function to be taken over each of the d dimensions.
Currently requires FUN to return a scalar.
A Latin hypercube is one in which each line of elements whose
coordinates differ in only one dimension comprises the numbers
$1$ to $n$ (or $0$ to $n-1$), not
necessarily in that order. Each integer thus appears
$n^{d-1}$ times.
A magic hypercube is a semimagic hypercube with the additional
requirement that all $2^{d-1}$ long (ie extreme
point-to-extreme point) diagonals sum correctly. Correct diagonal
summation is tested by is.diagonally.correct(); by specifying a
function other than sum(), criteria other than the diagonals
returning the correct sum may be tested.
In is.magichypercube(), if argument give.answers=TRUE
then a list is returned. The first element of this list is Boolean
with TRUE if the array is a magic hypercube. The second
element and third elements are answers
fromis.semimagichypercube() and is.diagonally.correct()
respectively.
In is.diagonally.correct(), if argument
give.answers=TRUE, the function also returns an array of
dimension c(q,rep(2,d)) (that is, $q\times 2^d$
elements), where $q$ is the length of FUN() applied to a
long diagonal of a (if $q=1$, the first dimension is
dropped). If $q=1$, then in dimension d having
index 1 means FUN() is applied to elements of a with the
$d^{\rm th}$ dimension running over 1:n; index 2
means to run over n:1. If $q>1$, the index of the
first dimension gives the index of FUN(), and subsequent
dimensions have indices of 1 or 2 as above and are interpreted in the
same way.
An example of a function for which these two are not identical is given below.
If FUN=f where f is a function returning a vector of
length i, is.diagonally.correct() returns an array
out of dimension c(i,rep(2,d)), with
out[,i_1,i_2,...,i_d] being f(x) where
x is the appropriate long diagonal. Thus the $2^d$
equalities out[,i_1,i_2,...,i_d]==out[,3-i_1,3-i_2,...,3-i_d]
hold if and only if identical(f(x),f(rev(x))) is TRUE
for each long diagonal (a condition met, for example, by sum()
but not by the identity function or function(x){x[1]}).
A perfect magic hypercube (use is.perfect()) is a magic
hypercube with all nonbroken diagonals summing correctly. This is a
seriously restrictive requirement for high dimensional hypercubes. As
yet, this function does not take a give.answers argument.
The terminology in this area is pretty confusing.
"Generalized magic cubes", Dana Richards. Mathematics Magazine, volume 53, number 2, March 1980.