Description Usage Arguments Details Value Note Author(s) References Examples

Implementation of various properties presented in a paper by Arthur T. Benjamin and K. Yasuda

1 2 3 | ```
is.square.palindromic(m, base=10, give.answers=FALSE)
is.centrosymmetric(m)
is.persymmetric(m)
``` |

`m` |
The square to be tested |

`base` |
Base of number expansion, defaulting to 10; not relevant for the “sufficient” part of the test |

`give.answers` |
Boolean, with |

The following tests apply to a general square matrix `m`

of size
*n*n*.

A centrosymmetric square is one in which

`a[i,j]=a[n+1-i,n+1-j]`

; use`is.centrosymmetric()`

to test for this (compare an*associative*square). Note that this definition extends naturally to hypercubes: a hypercube`a`

is centrosymmetric if`all(a==arev(a))`

.A persymmetric square is one in which

`a[i,j]=a[n+1-j,n+1-i]`

; use`is.persymmetric()`

to test for this.A matrix is square palindromic if it satisfies the rather complicated conditions set out by Benjamin and Yasuda (see refs).

These functions return a list of Boolean variables whose value depends
on whether or not `m`

has the property in question.

If argument `give.answers`

takes the default value of
`FALSE`

, a Boolean value is returned that shows whether the
sufficient conditions are met.

If argument `give.answers`

is `TRUE`

, a detailed list is
given that shows the status of each individual test, both for the
necessary and sufficient conditions. The value of the second element
(named `necessary`

) is the status of their Theorem 1 on page 154.

Note that the necessary conditions do not depend on the base `b`

(technically, neither do the sufficient conditions, for being a square
palindrome requires the sums to match for *every* base `b`

.
In this implementation, “sufficient” is defined only with
respect to a particular base).

Every associative square is square palindromic, according to Benjamin and Yasuda.

Function `is.square.palindromic()`

does not yet take a
`give.answers`

argument as does, say, `is.magic()`

.

Robin K. S. Hankin

Arthur T. Benjamin and K. Yasuda. *Magic
“Squares” Indeed!*, American Mathematical Monthly, vol
106(2), pp152-156, Feb 1999

1 2 3 4 5 6 | ```
is.square.palindromic(magic(3))
is.persymmetric(matrix(c(1,0,0,1),2,2))
#now try a circulant:
a <- matrix(0,5,5)
is.square.palindromic(circulant(10)) #should be TRUE
``` |

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