Magic squares of odd order

Magic squares of odd order is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

A magic square is a   NxN   square matrix whose numbers (usually integers) consist of consecutive numbers arranged so that the sum of each row and column, and both long (main) diagonals are equal to the same sum (which is called the magic number or magic constant).

The numbers are usually (but not always) the 1^st   N²   positive integers.

A magic square whose rows and columns add up to a magic number   but   whose main diagonals do not, is known as a semimagic square.

task requirements

For any odd   N,   generate a magic square with the integers   1 ──► N²   and show the results.

Optionally, show the magic number.

Use an   N   of   5   for the first example.

Also see

• Wiki entry: Method_for_constructing_a_magic_square_of_odd_order
• MathWorld (TM) entry: Magic_square

C

<lang c>#include <stdio.h>

1. include <stdlib.h>

int main(int argc, char **argv) {

1. define M(x) ((x + n - 1) % n)

       int i, j, k, n, *m;
       char fmt[16];

       if (argc < 2 || (n = atoi(argv[1])) <= 0 || !(n&1))
               fprintf(stderr, "forcing size %d\n", n = 5);

       m = calloc(n*n, sizeof(*m));

       i = 0, j = n/2;
       for (k = 1; k <= n*n; k++) {
               m[i*n + j] = k;
               if (m[M(i)*n + M(j)])
                       i = (i+1) % n;
               else
                       i = M(i), j = M(j);
       }

       for (i = 2, j = 1; j <= n*n; ++i, j *= 10);
       sprintf(fmt, "%%%dd", i);

       for (i = 0; i < n; i++) {
               for (j = 0; j < n; j++)
                       printf(fmt, m[i*n + j]);
               putchar('\n');
       }

       return 0;

}</lang>

Output:

% ./a.out
forcing size 5
  15   8   1  24  17
  16  14   7   5  23
  22  20  13   6   4
   3  21  19  12  10
   9   2  25  18  11

Perl 6

<lang perl6>sub MAIN ($n = 5) {

   note "Sorry, must be an odd integer." and exit if $n %% 2;

   my ($x, $y, $i, @sq) = $n/2, 0, 1;

   @sq[($i % $n ?? $y-- !! $y++) % $n][($x % $n ?? $x++ !! $x) % $n] = $i++ for ^($n**2);

   say $_>>.fmt("%{$i.chars}d ") for @sq;

   say "\nThe magic number is ", [+] @sq[0].list;

}</lang>

Output:

Default, No parameter:

24  15   1  17   8 
14   5  16   7  23 
 4  20   6  22  13 
19  10  21  12   3 
 9  25  11   2  18 

The magic number is 65

With a parameter of 19

106  297  127  318  148  339  169  360  190    1  192   22  213   43  234   64  255   85  276 
296  126  317  147  338  168  359  189   19  191   21  212   42  233   63  254   84  275  105 
125  316  146  337  167  358  188   18  209   20  211   41  232   62  253   83  274  104  295 
315  145  336  166  357  187   17  208   38  210   40  231   61  252   82  273  103  294  124 
144  335  165  356  186   16  207   37  228   39  230   60  251   81  272  102  293  123  314 
334  164  355  185   15  206   36  227   57  229   59  250   80  271  101  292  122  313  143 
163  354  184   14  205   35  226   56  247   58  249   79  270  100  291  121  312  142  333 
353  183   13  204   34  225   55  246   76  248   78  269   99  290  120  311  141  332  162 
182   12  203   33  224   54  245   75  266   77  268   98  289  119  310  140  331  161  352 
 11  202   32  223   53  244   74  265   95  267   97  288  118  309  139  330  160  351  181 
201   31  222   52  243   73  264   94  285   96  287  117  308  138  329  159  350  180   10 
 30  221   51  242   72  263   93  284  114  286  116  307  137  328  158  349  179    9  200 
220   50  241   71  262   92  283  113  304  115  306  136  327  157  348  178    8  199   29 
 49  240   70  261   91  282  112  303  133  305  135  326  156  347  177    7  198   28  219 
239   69  260   90  281  111  302  132  323  134  325  155  346  176    6  197   27  218   48 
 68  259   89  280  110  301  131  322  152  324  154  345  175    5  196   26  217   47  238 
258   88  279  109  300  130  321  151  342  153  344  174    4  195   25  216   46  237   67 
 87  278  108  299  129  320  150  341  171  343  173    3  194   24  215   45  236   66  257 
277  107  298  128  319  149  340  170  361  172    2  193   23  214   44  235   65  256   86 

The magic number is 3439

REXX

<lang rexx>/*REXX program generates and displays true magic squares (for odd N). */ parse arg N .; if N== then N=5 /*matrix size ¬given? Use default*/ w=length(N*N); r=2; c=(n+1)%2-1 /*define initial row and column. */ @.=. /* [↓] uses the Siamese method.*/

   do j=1  for n*n;   br=r==N & c==N; r=r-1;  c=c+1   /*BR=bottom right*/
   if r<1 & c>N then do;  r=r+2;  c=c-1;    end       /*R under, C over*/
   if r<1       then r=n; if r>n  then r=1; if c>n then c=1  /*overflow*/
   if @.r.c\==. then do; r=r+2; c=c-1; if br then do; r=N; c=c+1; end;end
   @.r.c=j                            /*assign #───►square matrix cell.*/
   end   /*j*/                        /* [↑]  can handle even N matrix.*/
                                      /* [↓]  displays (aligned) matrix*/
      do   r=1  for N;  _=            /*display 1 matrix row at a time.*/
        do c=1  for N;  _=_ right(@.r.c, w);  end  /*c*/    /*build row*/
      say substr(_,2)                 /*row has an extra leading blank.*/
      end   /*c*/                     /* [↑]   also right-justified #s.*/

say /*might as well show a blank line*/ if N//2 then say 'The magic number (or magic constant is): ' N*(n*n+1)%2

                                      /*stick a fork in it, we're done.*/</lang>

output   using the default input of   5:

17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

The magic number  (or magic constant is):  65

output   using the default input of   3:

8 1 6
3 5 7
4 9 2

The magic number  (or magic constant is):  15

output   using the input of:   19:

192 213 234 255 276 297 318 339 360   1  22  43  64  85 106 127 148 169 190
212 233 254 275 296 317 338 359  19  21  42  63  84 105 126 147 168 189 191
232 253 274 295 316 337 358  18  20  41  62  83 104 125 146 167 188 209 211
252 273 294 315 336 357  17  38  40  61  82 103 124 145 166 187 208 210 231
272 293 314 335 356  16  37  39  60  81 102 123 144 165 186 207 228 230 251
292 313 334 355  15  36  57  59  80 101 122 143 164 185 206 227 229 250 271
312 333 354  14  35  56  58  79 100 121 142 163 184 205 226 247 249 270 291
332 353  13  34  55  76  78  99 120 141 162 183 204 225 246 248 269 290 311
352  12  33  54  75  77  98 119 140 161 182 203 224 245 266 268 289 310 331
 11  32  53  74  95  97 118 139 160 181 202 223 244 265 267 288 309 330 351
 31  52  73  94  96 117 138 159 180 201 222 243 264 285 287 308 329 350  10
 51  72  93 114 116 137 158 179 200 221 242 263 284 286 307 328 349   9  30
 71  92 113 115 136 157 178 199 220 241 262 283 304 306 327 348   8  29  50
 91 112 133 135 156 177 198 219 240 261 282 303 305 326 347   7  28  49  70
111 132 134 155 176 197 218 239 260 281 302 323 325 346   6  27  48  69  90
131 152 154 175 196 217 238 259 280 301 322 324 345   5  26  47  68  89 110
151 153 174 195 216 237 258 279 300 321 342 344   4  25  46  67  88 109 130
171 173 194 215 236 257 278 299 320 341 343   3  24  45  66  87 108 129 150
172 193 214 235 256 277 298 319 340 361   2  23  44  65  86 107 128 149 170

The magic number  (or magic constant is):  3439