I was recently introduced to a rather interesting paradox, by a rather interesting book. I’ve been reading Sixty Days and Counting by Kim Stanley Robinson, and it introduced what it called the “three-box game.” I was intrigued by it, and after a little research I found it was officially known as Bertrand’s box paradox.

I’m not as eloquent as Robinson, so I’ll just quote the book for his wonderful explanation.

One test that had caught even Frank, despite his vigilance, was the three-box game. Three boxes, all closed, one ten-dollar bill hidden in one of them; the experimenter knows which. Subject chooses one box, at that point left closed. Experimenter opens one of the other two boxes, always an empty one. Subject then offered a chance to either stick with his first choice, or switch to the other closed box. Which should he do?

Frank had decided it didn’t matter; fifty-fifty either way. He thought it through.

But each box at the start had a one-third chance of being the one. When subject chooses one, the other two have two-thirds chance of being right. After experimenter opens one of those boxes, always empty, those two boxes still have two-thirds of a chance, now concentrated in the remaining unchosen box, while the subject’s original choice still had its original one-third chance. So one should always change one’s choice!

It boggles the mind, right? After having it explained I do have some grasp of why it happens that way, but I still felt the need to empirically test it. So I wrote a little PHP script that plays the three-box game with itself, and set it to run a hundred thousand times each way. And what do you know? This is one thought-puzzle that holds up to empirical testing!

Played 100000 games each:  
bet kept: 33268 wins  
bet changed: 66738 wins

If you’re so inclined, here’s the PHP I used:

function boxgame ($change_bet) {
  $boxes = array();
  $bet = rand(1,3);
  $coin = rand(1,3);
  $boxes[1] = 'empty';
  $boxes[2] = 'empty';
  $boxes[3] = 'empty';
  $boxes[$coin] = 'coin';

  //this is where the actual game is played
  if ($change_bet == true) {
    foreach ($boxes as $box => $content) {
      if ($box == $bet) {
      }elseif ($content == 'coin') {
      }else {

    foreach ($boxes as $box => $content) {
      if ($box != $bet) {
        $bet = $box;

  if ($boxes[$bet] == 'coin') {
    $result = 'win';
  }else {
    $result = 'lose';

$iterations = 100000;
$results = array();
$results['keep bet'] = array();
$results['change bet'] = array();

for ($i = 1; $i <= $iterations; $i++) {
  $results['keep bet'][boxgame(false)]++;
  $results['change bet'][boxgame(true)]++;

echo "Played $iterations games each:<br>bet kept: ".$results['keep bet']['win']." wins<br>bet changed: ".$results['change bet']['win']." wins";