R26. Magic Function in R

Magic matrices have sum of all rows and sum of all columns equal to some number. In addition, the sum of the primary and secondary cross diagonal equal that number, called the magic number. The magic number is only dependent on size of matrix.

The program will solve magic matrices for odd size, such as 11 by 11 (N = 11).

For a magic matrix, such as that of size 11 by 11, we have elements from 1 to 11 square. The magic number is the mean of all elements from 1 to N square, multiplied by N.

We use the matrix function to create a matrix and then loop, over the 2-dimensions, to get the individual elements, {1,2,3,...N^2}, into their correct positions. Magic matrices are not unique, there are many others, dependent on the algorithm, with the same sum property, however the magic number for a particular N is unique.

Lastly, we check all the relevant sums.

# ex26.R

N <- 11

magic_num <- sum(1:N^2)/N

mat <- function(i,j){
  i <- (i-1) %% N
  j <- (j-1) %% N
  c(x = i+1, y = j+1)
}
    
A <- matrix(numeric(N^2), nrow = N)
st <- c(N-(N-1)/2,(N+3)/2)

for (k2 in 1:N) {
  st <- c(st[1]+1,st[2]-1)
  for (k in 1:N) {
    cxy <- mat(st[1]+k-1,st[2]+k-1)
    A[cxy["x"],cxy["y"]] <- k+(k2-1)*N
  }
}

AR <- matrix(numeric(N^2),nrow = N)
for (k2 in 1:N) {
  AR[k2,] <- rev(A[k2,])
}

cat('\nmagic(', N, ') = \n', sep = '')
print(A)
cat('\nmagic number =', magic_num)
cat('\nsum primary diag =', sum(diag(A)))
cat('\nsum secondary diag =', sum(diag(AR)))
for (k2 in 1:N) {
  cat('\nsum row', k2, '=', sum(A[k2,]))
  cat('\nsum col', k2, '=', sum(A[,k2]))
}
# magic(11) = 
#   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
# [1,]   56  117   46  107   36   97   26   87   16    77     6
# [2,]    7   57  118   47  108   37   98   27   88    17    67
# [3,]   68    8   58  119   48  109   38   99   28    78    18
# [4,]   19   69    9   59  120   49  110   39   89    29    79
# [5,]   80   20   70   10   60  121   50  100   40    90    30
# [6,]   31   81   21   71   11   61  111   51  101    41    91
# [7,]   92   32   82   22   72    1   62  112   52   102    42
# [8,]   43   93   33   83   12   73    2   63  113    53   103
# [9,]  104   44   94   23   84   13   74    3   64   114    54
# [10,]   55  105   34   95   24   85   14   75    4    65   115
# [11,]  116   45  106   35   96   25   86   15   76     5    66
# 
# magic number = 671
# sum primary diag = 671
# sum secondary diag = 671
# sum row 1 = 671
# sum col 1 = 671
# sum row 2 = 671
# sum col 2 = 671
# sum row 3 = 671
# sum col 3 = 671
# sum row 4 = 671
# sum col 4 = 671
# sum row 5 = 671
# sum col 5 = 671
# sum row 6 = 671
# sum col 6 = 671
# sum row 7 = 671
# sum col 7 = 671
# sum row 8 = 671
# sum col 8 = 671
# sum row 9 = 671
# sum col 9 = 671
# sum row 10 = 671
# sum col 10 = 671
# sum row 11 = 671
# sum col 11 = 671