A couple years ago, I came up with a card counting scheme to beat the game Catch-a-Wave. This uses the same High-low card counting method that is used for card-counting in blackjack, but with one important difference: blackjack card counting works in such a way that the odds are in favor of the players only when there is a positive count, but Catch-a-Wave counting favors the players with EITHER a high or a low count. I will go into this later, but first, a little background on Catch-a-Wave.
Up to 6 players sit at a table that is very similar to a blackjack table. Each player is dealt 1 card face up, and the dealer also is dealt one card face up.
The basic idea of this game is that you guess whether the next card is going to be higher or lower than the one you have showing. If you guess right, you can keep going. If you guess wrong you lose. You can stop at any time, and stay with the cards you have. The basic idea is that you want to end up with more cards than the dealer ends up with. The players all go first, then the dealer goes afterwards.
When it is your turn, you can either hit or stay. If you decide to "hit", you choose either "High" or "Low", meaning that you are guessing that the next card will be higher or lower than the card you are currently showing. If you guess correctly, you then have the same options as before: you can either stay or you can hit again and guess High or Low. If you ever guess incorrectly you lose your bet immediately regardless of what the dealer does. You also bust if the next card dealt is the same rank as the one currently showing. (ie. if you have a 4, and the next card dealt is also a 4). In fact, you do not have to take any hits at all, and you can just stay with one card.
After all the players have had their turn, it is the dealer's turn to go. The dealer then does essentially the same thing, but with some important differences:
1. The dealer must take at least one hit. (Players do not have to take any hits if they choose.)
2. The dealer must continue to hit until he gets a card between 5-10 (including 5 and 10), or until he busts.
3. If the dealer must hit, and his card is lower than 8, he goes "High"; if his card is 8 or higher, he goes "Low".
If you bust, you lose your bet immediately.
If the dealer busts, he pays out even odds to all players who did not bust (ie. who still have cards on the table).
If the dealer gets a card between 5-10 (a "stop card"), he stops, and then compares the number of cards he has with the number of cards each player has. If the dealer has more cards, then the house wins and you lose your bet. If you have more cards than the dealer, the house pays odds that are equal to the difference between the number of cards you and the dealer have. For example: if the dealer has 2 cards and you have 5 cards, the house pays out 3 to 1 odds, etc.
Catching a Wave: if you guess correctly six times (ending up with 7 cards showing), this is called "Catching a Wave", and it pays 6 to 1 odds, regardless of what the dealer does. (In a sense, this is analagous to getting a "Blackjack", although it is more rare, and it pays better.)
If you are still confused about the basic rules, here is a link to the Wizard of Odds page which describes the same stuff.
Catch-a-Wave was invented at Foxwoods, and it is advertised as having the best odds of any game in the casino. This is actually true, but only if you know how to play by the optimal strategy. Most players have no idea how to play this game, and lose a lot of money at it. Often times, people just continue to hit, trying to "Catch-a-Wave" each time. This is not a good strategy at all.
One thing that many average CAW players rarely take into consideration is the fact that the dealer's card determines your strategy as much as your own card does. For example, if the dealer has an 8 he has a more than 50% chance of busting, so you should only hit if you have a very high card (King or Ace) or a very low card (2 or 3). Whenever I am playing in this situation, and I stay with a queen, the whole table erupts in snickering and people saying "What? You can't stay with a queen!", etc, etc.
The reason that house has an advantage in this game is that the dealer goes last, after all the players have had their turns. What often happens is that half the players at a table will bust before the dealer even has his turn. So we must kep this in mind as we go on to the
My ex-housemate Pete Galea wrote a computer program which determined the optimal strategy for playing Catch-a-Wave. I wont go into the details of how he did this (because I don't know anything about the program itself), but in short, he came up with a strategy that supported Foxwoods' claim that the game has good odds. I don't remember exactly what these odds are, but I think it is somewhere around 99 cents payout per dollar bet.
The strategy is based on two variables, the dealer's card, and your card.
First it must be understood that certain cards are better to have while others are worse. For example, the best card for you to have is either a 2 or an Ace. You should always hit with a 2, 3, K, or A, because your chance of busting is quite small. On the other hand, 8 is the worst card for you to have. Your chance of busting with an 8 is greater than 50%. This is because 8 is exactly in the middle of the 13 card ranks in the deck, and also because you bust if you are dealt is an 8 (i.e pushing equals a bust). So in short, if you get an 8, you should stay more often because you are afraid of busting.
The dealer's card is an important and often overlooked factor in determining your strategy. If the dealer has an 8, there is a probability that he will bust on his first hit, so you don't have to hit as often, and you still have a good chance of winning. (This is the same concept as "staying" in Blackjack when the dealer has a 16.) On the other hand, if the dealer has a card at either extremity of the deck, say a 2 or 3, or a K or A, he has a smaller chance of busting so you must hit more often.
Pete came up with a series of "arrays" (one array for each dealer card) that tells you if you should Hit or Stay. Each array composed of 6 numbers, representing the six possible hits you may have to make on your way to "Catching a Wave".
Here are the arrays. These should be memorized. (NOTE: Some dealer cards share the same array).
Dealer Card Array 2, A 0 1 3 5 5 3 3, K 0 1 5 5 5 3 4, Q 0 3 5 5 5 1 5 3 5 5 5 3 1 6 7 5 7 5 3 1 7 9 7 7 5 1 0 8 9 9 7 5 1 0 9 7 7 7 5 1 0 10 5 5 7 5 3 1 J 1 5 5 7 5 1
Each number in the array, represents a "range" of cards, which is centered on the middle card which is 8. If your card falls within this range, then you stay... otherwise you should hit. For example if the number in the array is 3, this represents the 3 cards centered on and around the number 8... in other words 7, 8, 9. If you have 7, 8, or 9 you should Stay. If you have any other card, you should Hit. The number in the array will always be an odd number.
If the number in the array is 5, you stay with: 6, 7, 8, 9, 10.
If the number in the array is 3, you stay with: 7, 8, 9.
If the number in the array is 1, you stay with: 8. In other words, you hit with any card, except for 8.
If the number in the array is 0, you hit no matter what.
This will become more clear as you memorize the arrays.
You should also memorize the range which is represented by the numbers in the arrays. For example the number "7" means you stay with any card 5 through J. If you memorize these, you will be able to play faster and more intuitively.
The dealer has a 2 and you have an 8. This is not a good start. Look at the array for a dealer card of 2: 0 1 3 5 5 3. The first number tells you what to do with your first card, whether you should hit or stay. 0 means you hit no matter what. So you take a hit and you guess "high". Yopu are dealt a Queen. OK, now look back at the array: 0 1 3 5 5 3. The second number is 1, meaning you stay with only one card, the 8. You don't have an 8 so you hit and you obviously guess "low" because you have a Queen. Your next card is a 10. OK, now what? Look at the array. The third number is 3, representing the 3 cards you should stay with: 7, 8, 9. You have a 10, so you have to hit and you guess "Low" again. The dealer gives you a 7. Nice! OK, the next number in the array is 5. That means you stay with 6, 7, 8, 9, or 10. You have a 7, so you stay. You now have 5 cards on the table, and a good chance of winning even odds or better. Now the dealer goes, and deals himself a 5, which means he must stop. Now the dealer has 2 cards and you have 5 cards, so the house pays 3-to-1 odds on your bet.
Using these arrays as described above will give you a good payout for Catch-a-Wave. But to actually beat the game you have to count cards.
Because Catch-a-Wave is a rather simple game in which you guess if the next card is "high" or "low", I immediately began to think that it would be ideally suited to card counting. The counting strategy which I devised has been proven to work in a computer simulation of the game, and I will give the results of that later.
First off, I decided to use the same basic "High/Low" count that is used in blackjack: cards 2-6 are assigned a value of 1; cards 10-A are assigned a value of -1; and cards 7-9 are assigned a value of zero. As you see cards come out of the deck you keep adding the values to keep a "running count". To get the "true count" you must divide the running count by the estimated number of decks remaining in the shoe. A high count means that there are a lot of high cards left in the deck, and a low count means that there are a lot of low cards left. (Please see other websites for a more complete description of how to keep a high/low count.)
As I started to use this counting technique, some things became immediately obvious. For example, if you have a positive count and you must hit with an 8, it is obvious that you would guess "High" in this scenario, because there are more high cards left in the shoe. In general, your odds of winning increase as the count gets further away from zero, either positive or negative.
Consider the following scenario: Say you have a count of +8, so there are a lot of high cards in the deck. The dealer thus has a higher probability of dealing himself a high card. Let's say the dealer gets a Jack. At this point, the dealer's chance of busting is significantly greater than if he had the same card with a count of 0.
The count also allows you to alter your strategy to maximize your chances of winning.
Consider the array of 13 card ranks in a deck: (2 3 4 5 6 7 8 9 10 J Q K A) When the count is Zero, "8" is clearly the worst card for you to have, because it is exactly in the middle of this array, and you have the highest probability of busting. However, when the count is either high or low, this will not always be the case. For example, when the count is +8, the middle card in the array of remaining cards is assumed to be 9. Likewise, when the count is -8, the middle card in the array is assumed to be 7.
Therefore, when you have a high count (+8 for example), and the dealer has a 9, you know that the dealer now has the card with the greatest chance of busting. So you alter your strategy accordingly.
As mentioned before, we gradually found that the player seems to get an advantage when the count becomes either high or low. This assumption was verified by a computer simulation of the game which was written by Dave Craft. The worst scenario for a player, statistically speaking, is when the count is at or near zero.
It is important to note that this represents a significant advantage over Blackjack card counting, because the advantage goes to the player with BOTH a positive and a negative count. In Blackjack, only a positive count is good for the player. Therefore, we would expect to be able to increase our bet amount twice as often as in Blackjack counting.
The computer simulation was tested with a variety of strategies. In general, the odds seem to go to the player once the count gets to about +/-3 or so. When the count is greater, say +/-8, the player's advantage is quite significant.
When you have a count that is skewed either in a positive or a negative direction you should alter your strategy. For example, if the count is around +8 or higher, you should alter your strategy in 2 ways.
1. First, look at the dealers card, then subtract one from this card and use the resultant array.
For example: if the count is +8, and the dealer's card is 9, use the array for 8 (as if his card was 8). Or if the count is +8, and the dealer's card is K, use the array for Q.
The reason for this should be obvious. If the dealer has a 9, this is actually now a worse card (for the dealer) than 8 , so you play as if the dealer actually had an 8.
2. The second way you change your strategy is that you change the card that the range (of cards you stay on) is centered on. For example, if the count is high (+8 or so), you ADD 1 to get the card you center the array on, in this case 9.
So using these 2 strategy alterations together, if you have a +8 count you subtract one from the dealer's card to get the array, then you center the ranges in the array on 9 (instead of 8).
With a high (absolute value) negative count (-8 or lower) you would do the exact opposite: add 1 to the dealer's card and use that array; then center your ranges on 7.
Dave's computer simulation yielded interesting results, as follows:
1. Positive counts were generally better for the player than negative counts, but negative counts were also in the player's favour.
2. When to increase your bet: generally the odds become in your favor at a count of +3 and -4. At this point you should increase your bet.
3. As the count increases either in a positive or a negative direction, your odds also improve. For example a +8 count is very good. A -9 count is also very good.
Here is a sample of the type of results we got from the computer simulation using the card counting strategy alterations and the following bet strategy:
If the count is between -4 and +3, bet the minimum ($2). If the count is between +3 and +6 or between -4 and -8, bet $30 (a medium sized bet). If the count is greater than or equal to 6, or less than or equal to -8, bet $100 (the maximum bet in this simulation). We shifted the arrays as described above when the count was greater than 8 or less than -8.
Playing 5 million games using the above strategy returned $802,570 profit. If the maximum bet was increased, say to $500, I am sure the return would have been much better.