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Computers, and black Jack Simulations

The use of computers for simulations is a very cool aspect of computers. In fact, I owe much of my carrier to this concept. I purchased my first AppleII computer to actually run black jack simulations. I wanted to play blackjack in casinos for a living, but needed to run some tests and numbers first (I wanted to know what I could win per hour, and the books did not give me good numbers in this regards).

Back then, to get any amount of computing power cost real dollars, and the concept of having my own computer that was fast enough to actually simulate the game of black jack was a most magical concept.

Even when at university we got a grant from the computing faculty to use the mainframe for the study of blackjack. We got tons of extra time on the mainframe, and did not have to pay for it.  We also started using “teams” (comprised of other CS students) to play the game at casinos.

Having played a good deal of blackjack in the Casino’s I started to ask the question:

If a person can learn to play black jack, and win, why do not more people not do this?

I mean, why not play blackjack?

It turns out that playing blackjack well does not really  require a special memory (practice takes care of this). However, it does requite that you are a person of logic, and also a very process originated type of person. In other words, you don’t believe in superstitions, or luck when it comes to cards. Of course, most casino’s are filled with people who are chasing luck, and not a process.

Basically, if you have the smarts, discipline and mind to focus on a project such as playing blackjack, then you are no doubt a educated Engineer, software developer, dentist,  doctor etc. In other words, those people that have the ability to win at blackjack are NOT playing blackjack because they already have a good career.  The discipline and focus you need to GET GOOD ENOUGH at this game is the same process you need to do well at school. If you did well at school and went on to get a university education then you had the means (money), and the smarts and resources to build one self a good career.

In fact, if you are involved in computing, then you likely started your own business and were able to put it all together. Or, if you are willing to put that kind of effort into playing blackjack, you might as well become the top selling real estate guy in your city. In other words, the abilities here requited are not easy, but starting your own SUCCESSFUL real estate companies is not easy either.

So, if you can be successful at blackjack, then you already likely are that kind of person who had the ability to get educated and likely already has a good job! (or started a good business etc.).

So, very few people play blackjack for a living, since the abilities needed means you already are doing something else more interesting or more lucrative!

However, for some of those that can’t find work in computing..perahps playing blackjack for a living is a possibility!

For a really interesting article in Wired on this subject, check out:

http://www.wired.com/wired/archive/10.09/vegas.html

It is wonderful article, since virtually the whole concept of card counting and computer simulations made the above possible. I studied and tried many of the concepts explained in the above a good number of years before those guys!

I wonder if there is any other business or venture that can benefit from simulations?

Albert D. Kallal
Edmonton, Alberta Canada
kallal@msn.com
http://www.attcanada.net/~kallal.msn

Albert D. kallal
Friday, September 05, 2003

Well, in additoin to your reasons, here's a few others not to make blackjack a career:
- smoke filled rooms
- noisy rooms
- surrounded by mostly drunk, innumerate morons who think gambling makes sense.

Other than that it sounds like fun :)

sgf
Friday, September 05, 2003

Interestingly enough, the founder of my previous employeer, Blair Hull, started out as a professional blackjack player.  He turned the knowledge/strategy of blackjack into an options tading philosophy and built one of the largest privately held trading companies - Hull Trading.

Unfortunately for the company, Blair (and the other partners) sold it to Goldman Sachs. GS sliced & diced it so that there are very few shards of the original left.

Old article about Blair/the company:
http://www.fastcompany.com/magazine/08/hullbenchmark.html

Blair's campaign website (note - no mention of Blackjack!):
http://www.blairhull.com/

RocketJeff
Friday, September 05, 2003

Also you need a large bankroll (e.g. $100K) to make any money out of the 1 or 2 percent edge you can get from card counting, and the standard deviation is so huge you have to not mind when you lose $10K in a single day (or hour) and just keep playing.

Add to that the stress from having to disguise yourself every day and duck and weave around pit bosses who are watching you closely and eager to kick you out for card counting... no thanks!

Rick
Friday, September 05, 2003

Just to add to Rick’s comment (which was correct - if the casino even _thinks_ you're card counting, they immediately ban you from playing [as do all the other casinos]).

I heard Blair talk about his blackjack days several times. The most elaborate card counters work as a team - the actual counter always makes small bets and doesn't try to win (or use the knowledge he’s gathering). If the table turns favorable, he signals to the other guys who come over and make large bets. These guys don't look like they're counting - they try to look like the average tourist (drink in hand, talks a lot, etc). Once the table goes cold (or they win enough) they leave.

At this level, there's a lot more to the strategy of casino blackjack then just mathematics.

RocketJeff
Friday, September 05, 2003

Did anybody ever read the book "The Eudaemonic Pie" [ http://www.amazon.com/exec/obidos/tg/detail/-/0595142362/qid=1062778236/sr=1-1/ref=sr_1_1/103-8533350-3572631?v=glance&s=books ]?  I stumbled over it in the library when I was an undergrad.  It's a fun story about hippie hackers who build computers in their shoes to win at roulette.

Brian
Friday, September 05, 2003

I should point out that it's nonfiction, though the description makes it sound otherwise.

Brian
Friday, September 05, 2003

The reason few people play Blackjack for a living is that casinos, and bookmakers, don't like a level playing field.

Al;bert's comment about card players going into Real Estate is interesting. in the 1980's and early 90's the US had the best Bridge team in the world. Most of the players were in the real estate business.

Stephen Jones
Friday, September 05, 2003

Bringing Down the House http://www.amazon.com/exec/obidos/tg/detail/-/0743249992/qid=1062782968 is a good, recent book about graduate students playing BlackJack for profit.

Ted

Ted Graham
Friday, September 05, 2003

Albert, a number of my friends are real life professional casino gamblers.

Blackjack is very hard to beat these days. You basically need a team approach as described in Bringing down the house. Counting cards with an 8 deck shoe is next to impossible, and even if you can figure it out, or if you target smaller casinos who only use a 4 deck shoe, if you are a lone white guy that keeps winning at blackjack, you will be marked as a counter, asked to leave, and put on a blacklist that is faxed around to other casinos. After the MIT team got exposed, the team approach is also very hard. I've heard that there are still guys making money playing black jack, but I have no idea what strategy they are using. 

You are underestimating what it takes to be a pro gambler. The discipline and focus it takes to consistently make six figures a year gambling is probably about 10 times the discipline and focus it takes to get a 6 figure middle manager job.

My friends that are pros are all poker players. Poker is a game where you are competing with other poker players, not against a house that is trying to weigh the odds so that you can have no edge whatsoever. So, it is still possible to make a go at it as a pro poker player. It does take an enormous amount of practice and discipline to get to the point where you can support yourself playing poker. In fact I would say that if you applied the amount of discipline and focus necessary to become a pro poker player to ANYTHING, you would be equivalently successful doing whatever you chose, thus you really have to convince yourself that poker is what you want to do with your life.

Most of the pro poker guys i know are convinced that poker was their calling in life (weird as that may sound). It isn't as though they just thought being a pro poker player would be cool or fun, they truly believe that is what they are meant to be doing.

rz
Friday, September 05, 2003

Regarding simulations, your actual question. :)

In the USA, TONS of federal funds are being allocated to medical research. This means everything from genomics to epidemiology, to medical record processing. I personally work on a couple projects where we make heavy use of monte carlo techniques to simulate disease outbreaks. 

Thus if you are opportunistic and into simulations, target the research arms of a few universities near you and figure out how much grant money they have allocated to "BIOTERRORISM DEFENSE" or anything in epidemiology. Biomedical departments often have some good biologists and good statisticians, but have no one capable of translating their ideas into code.  Thus if you are able to simulate an epidemic based on some data you are given from their emergency room databases, you can position yourself as a technical consultant on some of these research projects. In my experience, I've been able to charge fees equivalent to what I charge for boring corporate work, and the work is more interesting, and far more relaxed. good luck.

rz
Friday, September 05, 2003

Anybody want to form a cabal to 'solve' roulette using digital cameras and shoe processors? The technology is much better these days.

Knowledge Maker
Friday, September 05, 2003

rz basically nailed everything:

- You can't win at Blackjack without cheating
- You can't cheat at Blackjack without being caught
- You'll swing -- very wide -- so you need a huge bankroll

If you want to make money gambling legally, the first, last, and only game to play is poker. You absolutely must be first rate at reading people.

I've watched a lot of pro poker, and almost without fail, the guys who win are what you'd call "loose" players. They flout the conventional statistical wisdom (almost all start by reading The Super System, which is an excellent intro book, and then move on from there). You seem them playing more hands, playing statistically bad hands, and they end up winning. They aren't playing the cards, they're playing their opponents. They're so good, their instinct makes it seem like they have X-ray vision. Those are the people who live to play, and love the game. You have to be at that level to make real money at it.

A typical large tournment will cost between $5k and $25k to enter, and have at least 100 players in it (I think the last World Series of Poker had almost 900 entrants). You're typically going to have to take at least 5th or 6th place to replace the money you spent entering the tournament. You have to be able to play poker for days straight in a tournament; some will last as long as a week, playing very very long days.

Brad Wilson (dotnetguy.techieswithcats.com)
Friday, September 05, 2003

You can win at Blackjack by counting the cards; it's perfectly legal, though they will ban you for it.

The point about the top class poker tournaments is that even many professionals don't enter expecting to make money. They get the experience of playing against top class opposition, and then does them in good stead when they recoup their losses by playing against palookas.

And I must mention one of my favourite Bridge stories. In a London club where a fair number of young wannabee Zia Mahmouds hung out and make a precarious living, there was a millionaire businessman who owned a couple of garment factories, and went to the Bridge club four or five times a week, leaving losses of tens of thousands of pounds a year on the table. One night the professionals were walking to the all night bus stop at four or five ion a chilly winter  morning, overcoats tightly buttoned up, when the businessmen drove past them in his Rolls Royce, and gave them a friendly wave from the driver's seat.

"Ah, there goes the loser", one of the professional card players announced, through chattering teeth.

Stephen Jones
Friday, September 05, 2003

"You can win at Blackjack by counting the cards; it's perfectly legal, though they will ban you for it."

Right. I was very careful to call it "cheating", meaning a violation of the rules, rather than "illegal", meaning a violation of a criminal law. Good points about poker tournaments. I mean, you can always play in cash games, but you have to play big enough stakes to get the idiots out of the game, and to make the winnings meaningful (because, for instance, most cash Texas Hold 'Em games will be Limit, rather than No Limit).

Brad Wilson (dotnetguy.techieswithcats.com)
Friday, September 05, 2003

As others have pointed out, even with a provably optimal style of play, the casino still has an edge in blackjack unless you count cards.  Doing this will eventually get you banned from the casino.  Winning at blackjack is more about tricking the casino than beating the game.

Poker is different.  Tournament poker is (mostly) a game of skill.  Luck is still involved of course, as it is with any endeavor.

Mike McNertney
Friday, September 05, 2003

There is no rule in blackjack that says you can't count the cards.

The casinos aren't banning you for cheating, they're banning you for playing well and winning.

And cheating (such as having a button hole TV camera attached to a lithium iron battery and a transmitter that sends the vidoe stream of the outside to where it can be slowed down and the individual cards noted) is a criminal offense in Nevada punishable by ten years jail.

Stephen Jones
Friday, September 05, 2003

Knowlege Maker,

You will probably have to include lidar (laser radar. Look it up in google) as one of your technologies. There are some new eyesafe ones that might be small and light enough to bring in a casino.

pdq
Friday, September 05, 2003

Counting cards was great the 60s and 70s with the likes of Ken Uston and Ed Thorpe but then casinos made it harder and you had to have teams which still isn't a cash cow. People that were successful or could be successful in that arena moved on to similar roles like, in the case of Hull, trading. By the time the general public hears about where these people are, that arena will have become unprofitable.

Tom Vu
Friday, September 05, 2003

All the pro BJ discussion is at bj21.com by the way...

The big things nowadays are team play like somebody said above, and alerting services for when some new riverboat casino opens up in Podunk, MO with local yokel new dealers who expose the hole card when they pull 'em out of the shoe if you sit in the right spot. Everybody swoops in the first weekend and cleans 'em out.

Louie the Chair
Friday, September 05, 2003

Casinos do indeed ban players they suspect of counting cards.  Here in NV the casinos can kick out anyone for any reason.  B.J. is the only game where the odds are actually in the player's favor, and the casinos ban you if you are good. 

That's why even though I live one mile from a casino, I never drop a dime in a machine or spend a chip at a table?  The truth of the matter is I fully support the casinos because they pay my income tax.  And I thank all the tourists that come to visit our great state, for helping me with my mortgage payment.

If you play long enough you will lose.  Most people don't seem to be able to understand this.  May believe the longer they play the better chance they have of hitting it big.  Crazy logic, but ...  well you know...

christopher baus
Friday, September 05, 2003

"If you play long enough you will lose.  Most people don't seem to be able to understand this."

I don't think that people fail to understand this. I'd say that people who gamble recreationally (as opposed to obsessively) probably overwhemlingly believe one (or both) of these things:

1. They consider the money played as the cost of entertainment;

2. They know that, with some things, there's an off chance of winning big, but they aren't counting on it.

Brad Wilson (dotnetguy.techieswithcats.com)
Friday, September 05, 2003

A very interesting article somewhat related to this:

http://www.nytimes.com/2001/06/03/magazine/03DAHLMAN.html

It's about a guy who has "hacked" the racetrack system and made a relatively consistent, predictable living (and a good one at that).  (NY Times seems to have a bug in that you can access their pay articles through direct links, BTW).

But I must agree with everyone else, in that this requires just as much effort as a "legit" job.  Some people just like this kind of stuff, just like you and I like software.

And while a lot of gamblers could use some basic math skills, the best ones seem to have an edge that goes above and beyond what calculations can give you.  They seem to have a hypersensitity to patterns... they can achieve the same or better results without ever sitting down with a computer, but rather by playing millions of games.

Also I read some article in the NY Times as well talking about a guy who recently won a big poker tournament.  He actually honed his skills online (where you can play hands much faster).  So apparently he says that there is nothing about reading someone's "poker face", which obviously you can't do on the internet -- it's about reading the bets.  The money talks.

Andy
Friday, September 05, 2003

>>Counting cards with an 8 deck shoe is next to impossible, and even if you can figure it out,

Actually, playing with 2, 4, 6 or 8 decks is NO DIFFERENT in terms of difficulty. That is a very common myth.

In fact, there was even errors in that Wired article. Such as making one mistake is a really bad thing. Making one mistake is not such a big deal at all.  One hand/ one mistake is not going to make, or break the situation in one night. (you certainly don't want to make mistakes, but ONE mistake is NOT make or break situation).

However, while playing BJ against more decks does not increase difficulty of play at all, it does lower your odds by the fact that you get LESS betting opportunities to make money.

In fact, if the casinos just went to a 10 deck shoe and shuffled up when they reach 5 decks each time, it would be VERY difficult to make money.

Albert D. Kallal
Edmonton, Alberta Canada
kallal@msn.com
http://www.attcanada.net/~kallal.msn

Albert D. kallal
Saturday, September 06, 2003

This quote from the article Andy mentions is irrestible;

---" Dahlman puts it a little differently. "I'm a MOTO player," he says. "Master of the Obvious. It's just that more things are obvious to me than to most people." ------

Stephen Jones
Saturday, September 06, 2003

Albert, they could just do single-deck and shuffle after every hand, too... but while they're shuffling, people aren't playing the game so the casinos are giving up profit, AND a lot of players won't play any more because they mistakenly think they can count and have an edge (when they can't,  or make too many errors).

Louie the Chair
Saturday, September 06, 2003

Simulations aren't very interesting. Neural nets that learn how to play by playing against themselves are interesting.

Check out the book Blondie24.

The newest thing in Blackjack is an automated system that tracks the players, their bets and their cards. Guess what happens to players who consistently raise their bets when the count is favorable?

Oh, and team stuff won't work in a lot of places, Casinos simply implement a rule that you can't enter a game in the middle of the shoe.

Clutch Cargo
Sunday, September 07, 2003

>>>The newest thing in Blackjack is an automated system that tracks the players, their bets and their cards. Guess what happens to players who consistently raise their bets when the count is favorable?

Gosh, can you like homer Simpson go duh?? How stupid do you think we are? You think that a casino is even stupider?  Casinos have always watched tables like a hawk.

The whole point of that article was about how steps were taken to not tip off the casino. Fact is the card counters and betters DID NOT CHANGE the amount they bet. This went a long way to fooling the casinos.

Also, if you been in casinos, you don’t see clean cut University type students sitting at a table with $1000 of chips.  The fact that these guys dressed the gamblers part and did things like use people of Asian decent etc really did go a long to keep the casinos off guard.

>>>Oh, and team stuff won't work in a lot of places, Casinos simply implement a rule that you can't enter a game in the middle of the shoe.

Yes, what huge hit the casino takes. A nice couple walking with a stack of chips from the roulette table can’t even stop at a few blackjack tables on the way out. They have  to sit and wait for the shoe to finish? Yea,...good luck there...those folks will just continue walking.  Same goes for people walking with winnings from the slots. Can’t park at the blackjack tables for few quick bets? Casinos are simply loathe to create a atmosphere where you can’t walk up to a table and place a few bets. A horrible solution at best. 

Good software is all about making things easy. Good casinos is all about making betting easy.

If you are telling me that Casinos are much better at watching players and tracking their winnings, you are kind revealing a secret like humans breath air. The fact that humans breath air, and casinos track what happens at each table should not be the slightest surprise to you.

Albert D. Kallal
Edmonton, Alberta Canada
kallal@msn.com
http://www.attcanada.net/~kallal.msn

Albert D. kallal
Sunday, September 07, 2003

Albert, can you a similar strategy be applied to bingo? ;)

Mike
Monday, September 08, 2003

>>to bingo?

Hum, gee, Mike that is a interesting question!

I don’t play bingo!, but the use of simulation software to shed some light on the that game would be most interesting.

Since there is not much choice in the way how you play bingo, there is not too much of skill issue here. You can’t really vary how you play the game.

However, deciding on how many cards to purchase at the start, and a few other possible issues certainly could be learned by simulating the game in a computer.

I suspect there is some room to change odds at purchase time of the cards, but I never have seen how the process works when you purchase your bingo cards. (ie, what is the best number of cards to purchase for a given return etc.).

For example, simulation software will certainly tell you what squares are the best to own in the game of Monopoly. (the orange guys with the $100 houses are about the best).

Writing of the simulation software for bingo would not be much effort at all but I don’t see much in the way of using the resulting information. (you never know!!).

I think I played bingo at birthday partly when I was about 6 years old. That is the last time I played, and I have NEVER played a game of bingo where I had to “pay” to play.

I don’t know that game at all, and of what I remember, I don’t think there is much you can vary in terms of play...

Albert D. Kallal
Edmonton, Alberta Canada
kallal@msn.com
http://www.attcanada.net/~kallal.msn

Albert D. kallal
Monday, September 08, 2003

>>Gosh, can you like homer Simpson go duh?? How stupid
>>do you think we are? You think that a casino is even
>>stupider?  Casinos have always watched tables like a
>>hawk.

I can’t even parse this paragraph. I’m pretty sure you don’t really want me to state how stupid I think you are. All I really get out of it is the impression that english is your second language.

The point I was making was that this new system is all computerized, the feed from multiple digital cameras is processed, the software recognizes the value of the cards, the composition of the deck (plus or minus for the player) and the color and number of chips that are bet. It flags card counters without any human intervention.

>The whole point of that article was about how steps were
>taken to not tip off the casino. Fact is the card counters
>and betters DID NOT CHANGE the amount they bet. This
>went a long way to fooling the casinos.

The 'Big Player' act described in the article dates back to Ken Uston in the 1970s. Do you think that fools anyone in the casinos after Uston’s book was published?

Also, if you read the book and not the tiny article, you’d know that they didn’t fool the casinos. They were caught repeatedly.

>Yes, what huge hit the casino takes. A nice couple
>walking with a stack of chips from the roulette table can’t
>even stop at a few blackjack tables on the way out.

Geeze. You do know that there’s more than one blackjack table in a casino don’t you? So there’s a good chance that one of the tables will always be at a point where players can enter.

I was at the Rio a few weeks ago and they had the no mid-deck entry rule in effect.

>If you are telling me that Casinos are much better at
>watching players and tracking their winnings, you are
>kind revealing a secret like humans breath air. The fact
>that humans breath air, and casinos track what happens
>at each table should not be the slightest surprise to you.

Again, parsing error at or near 'kind revealing'. Too many errors detected. Aborting processing. File closed.

Clutch Cargo
Monday, September 08, 2003

Clutch  if I was a bit harsh, my apologies.

There is SO MUCH myth I read everyday, I tend react quite harsh. I am sorry about my snap at you, it was un-called for.

My only point was that even 20 years ago, varying your bet in a Casio was all alarm bells back then. Today, you can’t even get close to doing that.

And yes, those folks did get caught despite NOT varying the bets. I was only pointing out that varying bets has been a bad thing to do for the last 20 years, and should not come as a surprise to anyone at all!  Varying bets today is really a non issue, because you have not been able to do this for years! So, why bother to point out you can’t vary the bet when you have been able to do so for 20 years?

>Geeze. You do know that there’s more than one blackjack table in a casino don’t you? So there’s a good chance that one of the tables will always be at a point where players can enter.

Yes, but it still cuts down on the ability of people to roam, or stop and visit a cute girl at a table. It also stops that impulse type bets.

If drop in and see some friends at a table and and just one hand has been dealt, then I can ‘t even join my friends  at the table until the whole round is gone. With only 1, or 2 players at the table, I have stand around for quite a long time (especially with the number of decks they use). This wait time costs the casino a lot of money.

The issues of no entry during the shoe rules is not the best from the Casino’s point of view. However, as you mention, it would seem Casinos are starting to adopt this practice anyway! This just proves your point as to how difficult card counting has become.

Albert D. Kallal
Edmonton, Alberta Canada
kallal@msn.com
http://www.attcanada.net/~kallal.msn

Albert D. kallal
Monday, September 08, 2003

ALbert, I was just kidding about bingo.  Bingo is basically random, period.

Mike
Monday, September 08, 2003

[SLIGHT SPOILER WARNING]
I just read "Bringing Down the House" on Sunday.  The impression I got from it was that the team was sold out more than it was caught.  Someone sold info about them to the casino's security firm.  They would have been caught eventually, especially as survailance technology improved, but the impression the book gives is that they would have made a lot more money before that happened.

Brian
Tuesday, September 09, 2003

Hey, no problem Albert.

>Yes, but it still cuts down on the ability of people to roam, >or stop and visit a cute girl at a table. It also stops that >impulse type bets.

Oddly enough the one place I never saw people playing at the Rio was at the tables where the dealers are very cute women in bikinis. (Some sort of tie-in with the Rio's Bikini nightclub)

Clutch Cargo
Tuesday, September 09, 2003

"Fact is the card counters and betters DID NOT CHANGE the amount they bet."

If you don't change your bet to take advantage of a deck that's heavily in your favor, then you can't flip the odds over in your favor.

"The issues of no entry during the shoe rules is not the best from the Casino’s point of view. However, as you mention, it would seem Casinos are starting to adopt this practice anyway!"

Since you live in Canada, I'll have to defer to your knowledge of any Canadian casinos. I'm a fairly frequent visitor to Las Vegas, and I've never seen any casino, anywhere, implement a rule that says you can only join when beginning a new shoe.

Brad Wilson (dotnetguy.techieswithcats.com)
Tuesday, September 09, 2003

>>"Fact is the card counters and betters DID NOT CHANGE the amount they bet."

>If you don't change your bet to take advantage of a deck that's heavily in your favor, then you can't flip the odds over in your favor.

You are 100% correct. However, changing your bet gets your caught in a flash. So, what is done is that you have card counters at several tables. They give a VERY subtle (secret) single to a guy who rooms around playing the big better gambler type guy. He comes to the table and simply places the same big bet he has been playing at every other table in the place. So, the counters never change their bets, nor does this “rooming” guy. With a few counters, they can keep this big better guy quite busy. So, in fact, no one actually does change their bet in the casino eyes. This approach went a long way to not getting caught.

>>The issues of no entry during the shoe rules is not the best from the Casino’s point of view. However, as you mention, it would seem Casinos are starting to adopt this practice anyway!"

Well, I did not say that is happening, and I said I was quite surprised that casinos would adopt such a policy. However, one poster mentioned that in Rio they are starting to do this. I was simply saying, if the Casino’s are starting to do this, then so be it. (but it certainly is a surprise that they would adopt this policy, and I have not seen it myself).

Albert D. Kallal
Edmonton, Alberta Canada
kallal@msn.com
http://www.attcanada.net/~kallal.msn

Albert D. kallal
Wednesday, September 10, 2003

Anyone in London want to form a team and give this a go, count me in.

I am thinking of perfecting the skill, and then doing casinos in resorts like Sun City in south africa.

At least I know they don't have mafia style policing there, and even if I am caught, I get to keep my fingures!

Tapiwa
Thursday, September 11, 2003

that was meant to be fingers....

the figures I can live without. But then again, figures=digits=finger!!

I need to get out more.

Tapiwa
Thursday, September 11, 2003

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