Wednesday, February 3, 2010

Fox 240

Now this asks little more than geometry.
Note that the random process described in the question is discrete.
Ladybugs "appear" in next vertices at the same time.

5 comments:

  1. B - is my answer.

    ReplyDelete
  2. Such a simple problem, and the correct answer isn't even listed as an option!

    F) 3*sqrt(2)/4

    Hint: Checkerboard

    -DM

    ReplyDelete
  3. Probability of each distance case:

    * Distance 0: 25%
    * Distance 1: 0%
    * Distance sqrt(2): 75%
    * Distance sqrt(3): 0%

    Average distance = 3*sqrt(2)/4 = approx. 1.061

    ReplyDelete
  4. The Perl source code of my simulation program is shown below.

    #!/usr/local/bin/perl

    $N = 1000;
    $f = 100/$N;
    $sum = 0;
    %sums = (0, 0, 0, 0);
    ($x1, $y1, $z1) = (0, 0, 1);
    ($x2, $y2, $z2) = (1, 1, 1);
    %vectors = (
    "0,0,0", [[1,0,0],[0,1,0],[0,0,1]],
    "0,0,1", [[1,0,0],[0,1,0],[0,0,-1]],
    "0,1,0", [[1,0,0],[0,-1,0],[0,0,1]],
    "0,1,1", [[1,0,0],[0,-1,0],[0,0,-1]],
    "1,0,0", [[-1,0,0],[0,1,0],[0,0,1]],
    "1,0,1", [[-1,0,0],[0,1,0],[0,0,-1]],
    "1,1,0", [[-1,0,0],[0,-1,0],[0,0,1]],
    "1,1,1", [[-1,0,0],[0,-1,0],[0,0,-1]]);

    MAIN: {
     my $d;
      for ($i=0; $i<$N; $i++) {
       $d = abs($x1-$x2)+abs($y1-$y2)+abs($z1-$z2);
       $sums[$d]++;
       $sum += sqrt(($x1-$x2)**2 + ($y1-$y2)**2 + ($z1-$z2)**2);
       ($x1,$y1,$z1) = &get_next($x1,$y1,$z1);
       ($x2,$y2,$z2) = &get_next($x2,$y2,$z2);
      }
      printf("Average distance: %.2f\n", $sum/$N);
      printf("(0: %.0f%) (1: %.0f%) (sqrt(2): %.0f%) (sqrt(3): %.0f%)\n",
       $sums[0]*$f, $sums[1]*$f, $sums[2]*$f, $sums[3]*$f);
    }
    sub get_next() {
     my ($x,$y,$z) = @_;
     my ($dx,$dy,$dz);
     my $index = int(rand(3));
     my $key;

     $key = sprintf("%d,%d,%d", $x, $y, $z);
     $x += $vectors{$key}->[0][$index];
     $y += $vectors{$key}->[1][$index];
     $z += $vectors{$key}->[2][$index];

     return($x,$y,$z);
    }

    ReplyDelete
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