Pandiagonal magic squares due to Hudson

Returns a regular pandiagonal magic square of order \(6m\pm 1\) using a method developed by Hudson.

Usage

hudson(n = NULL, a = NULL, b = NULL)

Arguments

n

Order of the square, \(n=6m\pm 1\). If NULL, use the length of a

a

The first line of Hudson's \(A\) matrix. If NULL, use Hudson's value of c(n-1,0:(n-2))

b

The first line of Hudson's \(B\) matrix. If NULL, use Hudson's value of c(2:(n-1),n,1). Using default values for a and b gives an associative square

Details

Returns one member of a set of regular magic squares of order \(n=6m\pm 1\). The set is of size \((n!)^2\).

Note that n is not checked for being in the form \(6n\pm 1\). If it is not the correct form, the square is magic but not necessarily normal.

References

C. B. Hudson, On pandiagonal squares of order 6t +/- 1, Mathematics Magazine, March 1972, pp94-96

Author

Robin K. S. Hankin

See also

recurse

Examples

hudson(n=11)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#>  [1,]  112    3   15   27   39   51   63   75   87    99   100
#>  [2,]   92  104  116    7   19   31   43   55   56    68    80
#>  [3,]   72   84   96  108  120   11   12   24   36    48    60
#>  [4,]   52   64   76   88   89  101  113    4   16    28    40
#>  [5,]   32   44   45   57   69   81   93  105  117     8    20
#>  [6,]    1   13   25   37   49   61   73   85   97   109   121
#>  [7,]  102  114    5   17   29   41   53   65   77    78    90
#>  [8,]   82   94  106  118    9   21   33   34   46    58    70
#>  [9,]   62   74   86   98  110  111    2   14   26    38    50
#> [10,]   42   54   66   67   79   91  103  115    6    18    30
#> [11,]   22   23   35   47   59   71   83   95  107   119    10
magicplot(hudson(n=11))

is.associative(hudson(n=13))
#> [1] TRUE
hudson(a=(2*1:13)%%13 ,  b=(8*1:13)%%13)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#>  [1,]   34   55   89  110  131  165   17   51   72    93   127   148     0
#>  [2,]  154    6   27   61   82  116  137  158   23    44    65    99   120
#>  [3,]   92  126  147   12   33   54   88  109  130   164    16    50    71
#>  [4,]   43   77   98  119  153    5   26   60   81   115   136   157    22
#>  [5,]  163   15   49   70   91  125  146   11   32    53    87   108   142
#>  [6,]  114  135  156   21   42   76   97  118  152     4    38    59    80
#>  [7,]   52   86  107  141  162   14   48   69  103   124   145    10    31
#>  [8,]    3   37   58   79  113  134  168   20   41    75    96   117   151
#>  [9,]  123  144    9   30   64   85  106  140  161    13    47    68   102
#> [10,]   74   95  129  150    2   36   57   78  112   133   167    19    40
#> [11,]   25   46   67  101  122  143    8   29   63    84   105   139   160
#> [12,]  132  166   18   39   73   94  128  149    1    35    56    90   111
#> [13,]   83  104  138  159   24   45   66  100  121   155     7    28    62
all(replicate(10,is.magic(hudson(a=sample(13),b=sample(13)))))
#> [1] TRUE