So here's a puzzle:
Last night we went to the Melbourne Crown Casino, which is a labyrinth of neon madness the likes of which mankind was not meant to comprehend.
In the midst of the roulette, blackjack, pontoon and far-too-many poker machines, there is a little game that the Crown likes to call "Sic-Bo". The rules of Sic-Bo are simple - three six-sided dice are rolled, and you bet on what the result will be.
There are all sorts of complex bets you can make, but relevantly to our problem the bottom area of the betting table allows you to bet on the results of single dice. The numbers one through six are shown, and you can place your chips on any of these numbers.
If you bet on a number, and that number appears on any of the dice, the game pays 1 to 1. (That is, your $5 bet wins you a further $5.) If the number comes up on two dice, it pays 2 to 1 ($10 for your $5). And if all three dice show the number you bet on, it pays 12 to 1 ($5 yields $60).
This, I have to say, initially confused us. Those are good odds. I believe the phrase "money farm" was used. As we saw it, this was guaranteed returns, such that everything we knew and understood about casinos was turned upside down.
Here's how our reasoning went. If you place a $5 bet on each and every one of the six numbers, you couldn't lose money. In a case where all three of the dice show different numbers, you'd lose on half the numbers and win 1 to 1 on the others, thereby breaking even. Where you got a double and a single, that was the same total win as singles, so you were even. And on in the 1 in 36 occasions on which triples came up, you pocketed a profit. Rinse and repeat; increase your stake to make the money come in faster. There is no dice result on which the house doesn't pay out; cocked dice results in bets off but no loss to the player.
Using this logic we started doing all sorts of interesting maths that suggested we should immediately quit our jobs and just work the Sic-Bo tables 24/7, happily raking in buckets of cash while the other players looked on in an intoxicating mixture of awe and admiration.
So I have two questions. The first, which we have now solved, is in the case where we bet evenly on all six numbers, how does the house make its money? And the second question, which we haven't finished the numbers on yet, is does it make a difference if you instead just bet continually on a single number?