I think some of the problem for those who don't understand the solution is they don't understand the game. The game is not a variation of the sock drawer; it is not some sort of card game; it is not flipping a coin; it is not a game with infinite choices or chances, nor one where your choices are affected by any other player. And you probably will only have one opportunity to play the game for real. But they analyze the Hall game as if it were something it isn't.

The game is simple:
2 separate chances to pick a prize from 3 choices; the second chance is given only after one choice is eliminated. To make the game more exciting, the contestant is given the option of changing choices before a remaining door is opened. One opportunity to play.

       * While it is unknown at first which door contains the prize, the written explanations cause us to think, perhaps sub-consciously, that they do. So the analyze the problem as if they do.
       * They don't understand what the game really is and what it consists of. It is one person (not multiple people in sequence), choosing for a prize from 3 doors.
       * But that is only the set-up round. The host knows where the prize is, and the door you chose. For suspense and the entertainment value of the show, the Host will never show the prize or your choice immediately; he will only show the non-prize from the remaining door (either of two doors if your choice happened to contain the prize.).
       * The Host will not immediately open one of the remaining doors to determine the game's outcome. If he did, you would always only have 1 chance in 3 of winning. Pure luck, and not a very exciting game.
       * So there is a second round to the game. The host now offers you the opportunity to stay with your original choice, or switch to the remaining door. Stay/switch, switch/stay--it seems like you now have a fifty-fifty chance of getting the prize. This would be true if you had no other information, but you do have other information--the elimination of one door. That information results in your actually having 2 out of 3 chances of winning the prize if you follow the math. The math is the math is the math, and it dictates switching in order to give you the better chance of getting the prize.

No strategy offers a sure thing, as has been often pointed out. But which would you rather have, 1 chance out of 2 (50%), or 2 chances out if 3 (67%)? You don't have to understand the math, but you have act as if you do.