Multidimensional Conway's Game of Life


Addressing Conway’s Game of Life with a different approach for a multidimensional infinite space.

So… the Advent of Code is challenging us with a multi-dimensional generalization of Conway’s Game of Life, to be evaluated on an infinite space with $n$ dimensions. Cool!

This got me thinking… I can use a different representation and only keep track of the cells that are alive, without keeping the whole matrix. Especially now that it’s not a matrix any more, and it’s infinite in size. Ouch.

So, I settled for this:

  • coordinates are merged together in a single key string, like 1 5 -2 that stands for $x = 1$, $y = 5$, and $z = -2$;
  • the whole field is just a list of keys, representing the coordinates of alive cells.

Evolving from a state to the next one means calculating another list of active keys. This is done with the following function:

sub conway_3d_tick ($state) {
   my %previously_active;
   my %count_for;
   for my $cell ($state->@*) {
      my ($x, $y, $z) = split m{\s+}mxs, $cell;
      for my $xd (-1 .. 1) {
         my $X = $x + $xd;
         for my $yd (-1 .. 1) {
            my $Y = $y + $yd;
            for my $zd (-1 .. 1) {
               my $Z = $z + $zd;
               my $key = "$X $Y $Z";
               if ($xd == 0 && $yd == 0 && $zd == 0) {
                  $previously_active{$key} = 1;
               else {
   my @active;
   while (my ($key, $count) = each %count_for) {
      if ($previously_active{$key}) {
         push @active, $key if $count == 2 || $count == 3;
      else {
         push @active, $key if $count == 3;
   return \@active;

The first loop (for) iterates over all cells that are alive in the input state, and does the following:

  • increments the count of neighbors for surrounding cells;
  • marks the specific cell as previously active.

The second loop goes through all cells that have alive neighbors and checks for the count, to establish if the cell will contain an active element in the next round. If this is the case, the key is recorded in an array that is eventually produced as output.

Generalizing to 4 dimensions in the “second half” of the puzzle is trivial with copy-paste and adaptations. I know, I know… copy and paste code is not a good practice. But but… it’s very effective if you are in a hurry and you don’t care about future maintenance!

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