Please clear this up for the non-gamblers of consequence. If I purchase a lottery ticket or place a quarter on a cake wheel and win, I don't get my money back. What happens to my original bet on a roulette wheel or the race track when I win? Why?
Whether or not you get your original bet back depends on what you are betting on. In most casino table games you do get your original bet back if you win something. However with the lottery, horse races, slot machines, and cake wheels you do not get your money back if you win. The rule of thumb is that if you need to activate a machine to make the bet then the bet itself is gone. The cake wheel is obviously an exception to this rule but that game is for the benefit of fun and charity (I won a cake at a fair in Fresno years ago by the way).
Losing the original bet is not necessarily bad. If they returned it they would just depress the paybacks to recoup the money. If the odds are expressed as "to one," then you get your original bet back if you win. If they are expressed as "for one," then you don't.
In sports betting against the spread, it seems to me that the winner pays the 10% commission, not the loser. What am I missing?
You can look at it both ways. For example, suppose you bet $11 to win $10 on a game.
Loser pays: One could say it is an even-money bet, with a $1 refundable fee if you win. So, only the losers end up paying the fee.
Winner pays: A fair even-money bet would win $11 for an $11 bet. However, if the bet wins, the winner gets only $10. The missing dollar could be viewed as a commission or fee.
Personally, I view it as both pay in the form of a 4.54% house edge, assuming a 50% probability of winning.
I do not like to gamble. Any look at the math will show that I am bound to lose, the exception being blackjack
, if I could do the complex count. However, I do enjoy the casino atmosphere. So, the question is how do I stretch my money to lose as slowly as possible?
I seem to recall an article that stated that by placing two bets on the same roll in craps that I could cut down the house odds to the minimum. I'm not going to win big, but I won't lose big either. I know it doesn't sound very exciting, but I'm a boring guy. I figure my wife and I can side up the table, separately, and in effect cancel out each other, one will win big, the other will lose big. If we each bring a big enough roll, maybe we can stretch it out into a couple of hours of play. I think the bets were Pass & Don't Pass.
Fred, betting on opposite sides of the same bet is no fun. If you and your wife talk to each other, then you will look like fools by betting against each other. If you pretend to not know each other it will take away from the fun.
Your priority to minimize potential losses, yet still play, is not unusual. Personally, I would find a place with low minimums where you can feel comfortable with and play a low volatility game with a low house edge. Two slow games with lots of pushes are pai gow poker and pai gow (tiles). You may not have the patience to learn tiles, so my perscription is to take up pai gow poker
Do you believe that "wishful thinking" on behalf of the players can affect the outcome of a game. Note that I'm not concerned with the SIZE of the effect, just your philosophical opinion. Also, do you think that the manner in which a player tosses the dice in craps can cause a bias (good or bad) in the outcome. As always, your site is AWESOME.
Thanks for the kind words. No, I don't think that wishful thinking helps in the casino, all other things being equal.
The question on the dice influence is a hotly debated topic. Personally, I'm very skeptical. As I review this reply in 2013 I still have yet to see convincing evidence anybody can influence enough to have an advantage.
I’ve been a dealer in Vancouver, Canada for over six years. Having dealt all casino games (except craps) I’ve decided that a person’s best betting strategy would be to bet all their bankroll on a one-shot bet, preferably in bacarrat, and on Banker. My decision is based on the observation that the longer a player gambles, the more likely that the odds will get the best of them, and the more likely it is that they will lose everything. The one-shot method may not be entertaining, but surely it’d be more profitable (or I should say, "less damaging")
You’re absolutely right. The fewer bets actually made the better the odds are of actually winning. The expected loss is also a function of the total amount bet. If the player keeps circulating money back and forth between himself and the dealer the house edge will gradually grind the player down. However maximizing the odds of winning should not be the only objective to gambling. Having fun is also important. Plopping an entire bankroll on the table in one bet may not be as fun as playing it out gradually. It may also may have a greater chance at a large loss depending on how much play is involved in the alternative. If one really wants to cut down the house edge the best thing to do is put your money in the machine marked "change."
Some casinos offer "comps" for different levels of action. I was wondering if there was a way of approximating how much I would have to wager to earn these comps.
Your comp offers will depend on the product of your average bet, time played, hands per hour, house edge, and some "comp" constant, which is usually 33% to 40%. I indicate what one Vegas Strip casinos assumes for house edge and hands per hour in my house edge
Are all junket companies the same? The Grand Casino (a Park Place Entertainment holding) referred us to the Casino Connections junket firm based on our play. If we want to visit other casinos and let them pick up the airfare, et. al., how do we find the junket firm that serves the venue we want to visit and contact/convince them to take us on?
This is getting outside my area of expertise. When I was in college I went on almost free junket flights from Santa Barbara to Reno and had to be a minimum $5 player only. However, I rarely see advertisements for junkets any longer by airplane. I would suggest calling the casino host of a casino you like and prearranging a deal. However, I think you need to be a black chip player to get free hotel and airfare reimbursed.
You take a lot of your time or space on your web site to explain that no system can beat the house edge. I'd like to know what drives you to continue to play casino games if you know that you will loose in the end?
The rare times that I play a negative expectation game it is for entertainment. pai gow (tiles)
is the only game I find enjoyable enough to play without an advantage.
I understand the calculations made to determine the house advantage for the various games. My question is most applicable to the games with a large payout (for example, a royal flush in Let It Ride). Would it not be more reasonable to calculate the "house advantage" with the royal flush excluded? While hitting the royal is possible, it is extremely unlikely for the average player. Would this modified house advantage be more applicable to the average player? Thanks for the great site.
You make a good point. In terms of what to expect in the short run then you should ignore the highest hands. I know video poker players sometimes disregard royal flushes when determining their short-term expectations. However, as a mathematical purist, I can't help but consider every possible outcome, regardless of how unlikely.
I have been mulling over some money management techniques, and wanted to thank you for some sound advice (more to the point, probability theory) and it seems that win probabilities are based roughly on your starting bankroll. For example, estimating the probability of winning $100 on a $200 buy-in without going broke first. This is very helpful, but without doing the math (shame on me, I know), I get the feeling win limits should be based more on your betting unit, i.e. $1, $5, $10, etc. Basically, the idea you will have smaller fluctuations over time using a smaller bet than a larger one. I guess my actual question is this: if I have a given bankroll, (say $100) and a stated win limit (say $50), what (if any) betting unit would give the greatest possibility of success? I am thinking too small of a bet would give lower my chances of getting much above the mean and too large would run the risk of bankruptcy. Any advice or comments?
In your example, assuming a negative expectation game, the best bet size to maximize the probability of reaching your win goal is $50. In a positive expectation game the best bet size is as small as possible. The reason is that the more you play the more the house edge will grind you down, or the more you will grind the casino down if you have the edge.
I was wondering about house edge: The house always has the advantage of change but one thing is forgotten and is not to be taken lightly when doing the calculations: I choose when to stop. If I see I am losing I may cut my losses. If I see that I won enough (there is no such thing) I can decide to stop betting. The house has no such option. How does it affect the calculation?
In the long run this kind of money management will neither help you nor hurt you. By cutting your loses at a certain point and walking away you risk missing a comeback. By walking away with a modest win you risk not turning it into an even bigger win. Of course things could also get worse too. In general you can assume the past does not matter and every hand is a new beginning. The best way to improve your odds is to cut down the house edge as much as possible. I’m not against money management but it won’t effect the house edge.
I have seen several references to ongoing bonus abuses from people who reside in Denmark. Do you know more details about what is going on here? What are they doing, and how can we avoid being bonus abusers?
I don’t know all the details but there is a lot of bonus abuse going on by players from Denmark. The way to avoid being labeled a bonus abuser is to always play much more than required. I hate to give an exact figure but exceeding requirements by at least 100% is a good idea. Giving free play to casinos you get repeated bonuses from is also good camouflage. It doesn’t look like you are a good faith gambler if you only play during promotions. In general don’t be too greedy.
Please explain "playing perfectly".
Playing perfectly means the same thing as using an optimal strategy. In other words making perfect use of the information available to maximize the expected outcome of the bet.
As I have little experience of casinos and do not think I can memorize all the finer points of the information your site offers, what would you recommend I play on my forthcoming trip to Las Vegas?
I’d recommend craps or baccarat. In craps stick to the line bets and the odds. In baccarat bet on the banker every time.
I have one simple question. I know you are the Wizard of odds but I need help. I play Craps and Texas Hold’em in the casinos. I fell I could be an intimidating force (THUS RAISING MY ODDS OF WINNING) if I could just figure out how to shuffle my poker chips. I have practiced but just can’t get it. I was hoping you could point me in a direction to learn this. Thank you for your time.
You asked the right person! I’m quite good at shuffling poker chips. Unfortunately I don’t often get to show it off because when I do play with chips it is usually counting cards or hole card reading, and in either case I don’t want to look like a pro. Anyway, start with a stack of 10 chips. Then cut them in half, putting two 5 stick chips side by side. Think of the two stacks as an 8. Put the 8 at about a 45 degree angle to your plane of symmetry. Put your thumb on the bottom of the 8, your index finger where the two stacks come together, and the other 3 fingers at the top of the 8. All five fingers should be barely touching the table. Then using your index finger gently lift both stacks while the other four fingers gently push the stacks together. After your index finger is about a quarter inch from the bottom of the table quickly move it away and keep pushing the stacks together with your other four fingers. It takes practice. I would recommend getting 10 nice clay chips and practicing at home or work. During commercials or whenever you have a minute to spare you can work on it. Before you know it you you’ll be riffling chips like a pro and casting fear in your fellow poker players.
Your Money Management page says:
"For those who sometimes lose too much and later regret their actions some self-constraints may be in order. I would suggest setting a specific loss point in these cases, for example $200. Personally I don’t set such limits on myself. If I’ve lost too much it won’t be fun any more and I’ll step away for that reason."
But for you, what does "too much" mean? On every other web page of your wonderful site, you warn against using gut feelings. But when it comes to losing, you say you stop when it doesn’t feel good. Especially with video poker, I set a bankroll size, and I stop when I lose that. Losing always sucks, whether it’s 1 credit or 300 credits.
You are right that I am not specific about money management. Unlike other gambling writers I do not put a lot of emphasis on how much to bet or when to walk away. For the recreational gambler there is no method of money management that can either add to or take from the house edge over the long run, so why dwell on it? So while I’m very specific on how to play your cards how much to bet is up to you. There should be some room for free will in gambling anyway. However I do say that if you’ve lost so much that gambling isn’t for fun any more then it is time to walk away.
I read your review of casinos and want to try Club Dice. Regarding the terms on their website, they mention bonus will be deduct from the account once you make the cash-in even you fulfill the wager requirement. I wonder then what is the meaning of this bonus?
This is what is called a "sticky bonus" and is not confined to Internet casinos. When I went to Germany I had to pay 20 Duetchmarks to get in. However if purchased 100 Duetchmarks in chips they gave me a free 20 in sticky chips. A sticky bonus or sticky chip is one that can never be cashed out. However if you bet it and win the winnings are real money. They way to get rid of sticky chips is to keep betting them until you lose them. In the case of Internet casinos you will need to keep cashing out winnings and always leave the sticky bonus in your account, then go back and play again.
Recently at the Miami Beach casino I did a 100% promotion for a $1000 deposit. Eventually I lost $1650 and tried to cash out my last $350. They declined it, stating in effect that the $1000 bonus was sticky. So I played more and kept cashing out when my balance was significantly above $1000. In the end I cashed out 7 or 8 times for a total of $4700.
In the long run sticky bonuses just force the player to bet more. The expected value of a sticky bonus is close to 100% less 2 times the house edge in even money based games.
Could you provide some details about the mechanics of "buying in", for example, at a crap table? Do dealers take cash only? If so, and I wish to make a substantial buy-in, does this mean I must arrive at the casino carrying nothing but cash (a risky action in LV)? What about travelers’ checks? Can I make advance arrangements for credit with the casino Cashier’s Office? If so, how does that work exactly? Get specific, please.
I remember my first trip to a casino I didn’t know how to actually get chips to play with and purchased them at the cage and walked them to the blackjack table. The proper way to buy in at the table is simply to lay your cash on the table and at the appropriate time the dealer will exchange the cash for chips. However if you wish to play for amounts too uncomfortable or unsafe to carry in cash you can wire funds to the casino in advance. Then all you have to do when you get there is ask the pit boss for chips and he will have you sign something, stating you are buying chips against your cash account. To get off topic a bit I think it is time the United States Treasury should start making $500 bills, making it easier to carry large amount of money. A 500 Euro note already exists, which is worth $598 U.S. dollars at the time of this writing.
Great site! I’m a devoted fan who only bets on games with a small house edge.
I was surprised to find on the Nevada Gaming Control Board’s website, that the statewide casino win percentage for baccarat in 2003 was 19.62% and for mini baccarat, the casinos kept 13.81%. Why such a difference if the two games have the same house edge? By comparison, nickel slots (considered to have a lousy house edge) kept only 7.89% statewide! Why would slot machines (with a high house edge) keep less money than table games (with a low house edge)?
Thanks for the kind words. You are far from the only person to be confused about this. The reason is you are comparing the house advantage to the hold. The house advantage is the percentage you will lose on average of each dollar bet. The hold is the ratio of money the casino wins to chips purchased. This is going to be much higher than the house edge because in table games players circulate through the same chips for a while. So that baccarat figure is saying that of all the money dropped in the box in baccarat the casino won 19.62% and gave the players back the other 80.38%. Meanwhile the nickel slot figure is saying that of the total amount bet the casino kept 7.89% and gave players back 92.11%. To make a long answer short you are comparing apples and oranges.
First, great site. During a recent visit to Harrah’s, they gave me an option of either $100 match play or $50 in slot play. In your opinion with which is the best to take. (I took the match play). Also, for the match play would it be better to play all $100 on one hand, or multiple smaller hands (10 x $10 hands). Thanks
Thanks for the compliment. I recommend taking the match play. I’m sure the $100 in slot play was on specially designated machines. From anecdotal evidence I believe these free play slots are extremely stingy, set to pay back about 25%. That match play is worth about 48 cents on the dollar. I recommend betting in on the don’t pass in craps. The reason I favor that over blackjack is that blackjack has a lower probability of winning, thus reducing the value of the match play. For further explanation please see my October 30 2001 column
I work in a casino and have actually 86’ed people for various reasons. Where does the term actually come from?
According to Cecil Adams
the term originates from restaurants and soda fountains of the 1920s. He says it started out meaning to be out of something and then became an expression to drive off a customer.
p.s. In December 2004 another reader wrote with another explanation. According to inspirationline.com the term originates from a restaurant named Chumley’s at 86 Bedford Street in Greenwich Village, New York City. It started as a warning to leave the building because the police were coming and evolved to mean to get rid of something.
Hello, I just wanted to say that I love your site, and I have a question. I just got back from a trip to Vegas and noticed that some people bet way too high for their available bankroll, for example they will join a $5 minimum roulette table with $20. This greatly increases the house edge, because if that person loses the first 4 games, they have lost the entire bankroll and can no longer play. Using the standard deviation of a $5 game, can you calculate the minimum bankroll needed, so that say 95% of the time you can cover natural losing streaks? You did something similar on your betting systems page when you put a cap on the max bet for a Martingale better. How does the house edge change when a regular bettor only starts with $20, or $40, or whatever?
Thanks for kind words. The house edge is always the same in any game given the same rules and skill level of the player. The bankroll and betting strategy do not matter. Even if I sat down at a $5 game with $5 with the goal of winning $1,000,000 the house edge would still be the same. Although my probability of succeeding is low my worst casino scenario is nothing compared to the best case scenario.
But it is so much fun.
If I flat bet until I either win $100 or lose $1000 what is my probability of reaching both goals?
Ignoring the house edge the probability of achieving the winning goal is the loss marker divided by the sum of the loss marker and the winning marker. In this case 1000/(1000+100) = 1000/1100 = 90.91%. However the house edge will reduce the probability according to the house edge of the game played and the amount bet per hand, the smaller the bet the lower the probability of achieving the winning goal.
Dear Wizard, You advised a previous poster how one can learn to shuffle poker chips (Aug 25, 2003
). Becuase of your instruction, my shuffling abilities give me a strong edge (my poker skill, on the other hand, is another story...). I just want to give a few extra tips...
- Try starting with a stack of 6 chips instead of 10. Even those lacking any dexterity (like me) will have an easier time getting the feel of it.
- It is much harder to practice on a hard table. One cannot as easily get one’s fingers underneath the stack as on a felt table. If you don’t have access to a casino table (other than in the casino itself), practice on something soft, such as a mousepad or even a folded newspaper.
- Warm-up your hands before shuffling (especially if you are new at it). Shuffling puts your fingers in an odd position. It takes a while for them to get use to it.
- Learn to use both hands. It becomes much tougher when using your weaker hand, but it makes you look twice as intimidating when you show your fellow players that you are good with either hand.
Thanks again for the great site!
Thanks, I never thought to try with my left hand.
When are you going to do something on bad beat jackpots?
I get asked about bad beat jackpots about once a month. When I have the time I plan to add a section to my site about it. My hesitation is I’ll get asked about every bad beat jackpot in every poker room in the whole world.
I play a weekly social poker game. We have a guy who insists that dealing 2 or 3 or 5 straight cards to each player at a time is equally as random as dealing one card to each player. I assume that if a deck has been shuffled 6 or 7 times (depending on who you listen to) then he would be correct. But, it you have just finished a hand and shuffle only a couple times, dealing cards in groups or clumps like this would not be random. What do you say?
I agree with you. If the cards are well shuffled then it doesn’t matter. However if they are poorly shuffled then I think the dealer should deal the cards one at a time so any clumps are broken up among the various players.
Was reading through your site and browsing the section on tipping and had a few comments to share. I’ve always found it best to place the tip on top of my bet and have the dealer’s "playing with me" until I lose. Usually you are tipping when you are winning and if you get on a long winning streak a $1 tip can turn into much more for the dealers. Just make sure the dealer knows that the extra chip on top of your own bet is for them! Thanks for all the great information on all the games!
I’ve seen this happen before and I agree that some dealers like it. However in my opinion most don’t care because tips are pooled and shared among all the dealers. In 18 years of playing blackjack I have only once seen a dealer ask a player to do this.
[Bluejay adds: I always ask dealers which method they prefer, because some have a distinct preference. Some like the chip riding on top while others hate it. I like giving dealers the option, because just by asking I establish a small bond with them by showing that I’m considering their feelings.]
Why do people gamble at all? Especially in a casino, where they know they are supposed to lose? Can you see this happening in any other business? What is the psychology behind this? Something for nothing? No, they do not keep what they win, most just lose more back later, that can’t be it. Socializing? No, you can do that in a restaurant, that can’t be it, and the list goes on. So what really is the problem? I work in a casino, and see it everyday, and almost the same people, everyday. In many cases they do not seem to be having any fun, that cannot be the goal. So, what, in your opinion, is it that is so addicting about losing money, and not using the sense you have been given?
People generally gamble either for entertainment or because it’s a compulsion, so let’s look at each.
I think those who like gambling find it exciting and a safe way to get an adrenaline rush, much like riding a roller coaster. For the knowledgeable gambler the entertainment can actually be cheap. Although gambling feels like a job to me now I played recreational basic strategy blackjack for about a year before I went onto card counting. Playing $5 a hand under Atlantic City rules at a full table the expected loss per hour is only 2 cents per hand or about $1.20 per hour. That isn’t much to pay for the entertainment and free drinks. So those who play the better games and play them well could certainly make an argument that it is a small price to pay for entertainment.
Some people, like you, don’t see what is entertaining about gambling at all. That makes sense, since not everyone likes every form of entertainment. Just because some people like baseball doesn’t mean everyone will.
As for compulsive gambling, psychologists say compulsive gamblers fall into two groups: those who do it to it for the action and those who do it to escape reality. The action seekers tend to be men and gravitate towards the table games. The escapists tend to be women and gravitate towards slots and video poker. So that is my two cents. Keep in mind the only psychology I have studied was one semester in high school, 20 years ago (hard to believe it has been that long).
I'd like to how much does a casino poker chips weigh, and better yet do you know where is the best place you can purchase poker chips that feel and sound (when you drop them) are as close to the real deal as possible?
The standard is 11.5 grams. Casino-quality chips are made of a clay composite. Most poker chips sets are the same weight but the material is not as high quality and feels more like plastic. If you really want the best you could go to a casino and purchase a large quantity of $1 craps/poker chips from the cage at face value. If the casino changes the style, or goes the way of the dodo bird completely, the chips should go up in value. However for most recreational purposes there are always lots of sets available on eBay for about $50 for a 500-chip set. If you do get generic chips I would recommend true Paulson chips (there are many imitations), which are the same quality as casino chips. However Paulson no longer makes generic chips so the price will be significantly higher. If the price pushes $1 a chip, which it often does, I would just get actual casino chips instead.
By what method do casinos pay you when you cash out? For example, if you were to win say $10,000-$15,000 playing Roulette or Black Jack can you get that money in a cashiers check, money order, etc.? As one certainly doesn’t want to be walking around with or driving back to Canada with a bunch of cash on them!
I believe the policy at most casinos is that for large transactions you can have the funds any way you want. Before you consider laundering money by turning cash into checks be aware that casinos ask for a Social Security number and make a record of any transaction involving $10,000 or more.
If I make a maximum bet, can I still make a bet for the dealer?
Here in Vegas, yes you can.
Casino "comps" are supposed to be based on average bet x hours played x house edge (or something close to this). Why do they record your buy-in, your re-buys and the amount you color out? None of these should affect your comps. Is it truly theoretical loss or does actual loss(or win) come into play?
P.S. Your site is terrific and thanks for restarting the "Ask the Wizard" feature.
Thanks for the kind words. I asked a pit boss this question and he first agreed that comps are generally based on the product of average bet, hours played, and the house edge. The reason that buy-ins are recorded are to adhere to government regulations. There is different paperwork that must be filled out when buy-ins reach the $3,000 and $10,000 levels.
I’m getting married in October. Since my fiancée and I are inveterate gamblers, we’re going to honeymoon in Antigua at a resort with a casino. Are there any particular differences between gambling in the states and the Caribbean? Also, do you believe gambling to be an aphrodisiac?
Speaking only for the Bahamas and Curacao the gambling is similar to the U.S.. Don’t expect to find any great rules. The blackjack is played with six decks and the dealer hits a soft 17. Gambling is a good aphrodisiac if you win.
During a 4-hour layover in Vegas, what’s my best strategy to double a $2000 bankroll? what game, large or small bets, etc.?
First I would take a taxi to the Hard Rock, the closest major casino to the airport. I’m not sure how much odds the Hard Rock allows in craps
but I would guess 3-4-5. If that is the case then bet 1/7 of your bankroll on the don’t pass bet, or $275 to round down. If a point is established then lay the maximum on the odds, or $1650. If you win you’ll be a lot closer to your goal, the amount will depend on the point. Win or lose bet the lesser of 1/7 of your bankroll and 1/7 of how far you are from $4000. If you get close to either extreme just get it over with and bet everything if you’re low, or whatever you need to close the gap on $4000 if you’re high, and forget about the odds. Four hours should be enough time. However don’t dilly dally. The lines at security can get pretty bad. If your outbound flight is in terminal C be sure to ask an agent about the secret entrance.
If an immediate family member is a compulsive gambler, owes everyone money, and can’t function in the real world, is there a "blacklist" or other device on which to place his name so he will be banned from using internet gambling sites?
There should be but I have never heard of such a list. Even if there is such a list I think he would have to put his name on himself. The more respectable Internet casinos honor their own lists, and I have heard of loss refunds if the gambler proves he is getting treatment.
What are the odds a deck could be shuffled back into starting order, with either a totally random shuffling method or with a perfect rifiling shuffle, and how many times would it take?
The probability of a random shuffle resulting in starting order is 1 in 52!, or 1 in 8.06582*1067. If you did a perfect shuffle, in which last card was the first to come down, thus remaining last, it would only take 8 shuffles to be back to the starting order. If the 26th card was the first two come down then it would take 72 shuffles to back to the starting order.
Dear Wizard, you go to great lengths to explain the odds and why they are always in the casino’s favor. This being the case, why would anyone gamble?
Because it is so much fun!
Hi, fantastic site, keep up the good work :) I’m trying to work out risk of ruin statistics for any given combination of number of betting units and number of hands to play. Currently if the pair of values don’t appear in your RoR [Risk of Ruin] table in the blackjack appendix then I have no accurate percentage chance of ruin. Would you publish the formula for us please? Many thanks.
Thanks. I explain how I calculate the risk of ruin in video poker in my video poker appendix 1
. However for games where the bet amount is not always the same the calculations get very messy and computer simulations are necessary.
If I have a bet with another person, which is witnessed by three others and we shake on the bet. Is it by law a legal bet and must be paid?
No. Illegal debts are not enforceable. I’m not a lawyer but as I understand it unless the law specifically allows for gambling on whatever you bet on, and somebody had a license to take the bet, then the bet would have no legal protection. That is why I list "Honor thy gambling debts" as the first of my Ten Commandments of Gambling
. A true gentleman honors all his debts, but especially ones based on honor, like a bet made only verbally or with a handshake.
Lots are mentioned several times in the bible, most famously for using lots to decide who went home with Jesus’ robe. What exactly are lots and were they used for gambling?
My webmaster Michael Bluejay insisted on providing his own answer first:
Actually, drawing lots wasn’t their first idea for deciding who got Jesus’ robe. First they were going to draw a horse, but they didn’t have the right color crayons. So they each decided to draw whatever they wanted, and one drew Carey while another drew Barrymore. But the judging became an apples-to-oranges kind of comparison so they settled on drawing lots. Of course, everyone likes lots because lots means plentiful. You always see signs that say "Lots for Sale", but you never see a sign that says "Only a Little Bit for Sale". When you think about it, that place "Big Lots" is kind of redundant. It’s like saying "Abundant Abundance". But if companies can get away with saying "Pizza Pizza" (or agar agar) then I guess there’s no problem. Anyway, to answer your question, lots were first used for gambling in Cow Bingo. You know, it’s the game where a cow is placed on a lot marked off in a grid and people bet on which grid square the cow will poop in. The most famous use of lots in gambling is their role as the first part of the LOTtery.
(groan) Now that that’s over with, I asked my friend and bible expert, Tom R. the "Watchman on the Wall", about this. He quoted various bible dictionaries. The bottom line is that lots were not used for gambling but to choose a name randomly. This was accomplished by writing one name each on pieces of wood or stone, putting them in a bottle, and shaking just one out.
I had a blackjack and the dealer paid me. Then at the end of the deal he turned his cards over and he discovered he had a blackjack. He forgot to check his hand before paying me. The supervisors wanted me to return the winnings. I refused. Seems I remember these scenario from a book on blackjack. Was I correct?
I say you should have returned the winnings. I have never seen this addressed in any book. However, is a book really required? That is what you have a conscience for. You were asked to make things right, it is the right thing to do so.
Love your site! I just ran across your detailed list of Las Vegas Blackjack tables and their edges, so I was wondering: Suppose that there two BJ tables, one with an edge of 0.2% with a table minimum of $10 and 0.4% with a table minimum of $5. Both tables have the same $0.02 loss per hand. Is there any advantage to choose one table over the other?
Thanks. To answer this question you first have to ask yourself why you are gambling in the first place. If you are trying to lose as little as possible then you shouldn’t play at all. However, if you are playing for the fun of gambling then I would choose the $10 game with the 0.2% edge. The expected loss will be the same but you’ll get more of a fix with the larger wagers.
Playing blackjack, at what average dollar bet can I Expect:
- Free Food & Beverage
- Free Lodging
- One of those high roller suites
- Free golf at Wynn
- A new car
- Free airfare.
The basic formula for comps is that the casino will give you back a percentage of your theoretical loss. That percentage can vary by game, the higher the house edge the higher the percentage. I asked a former Vegas casino manager and he said the comp rebate is about 15%. Other pertinent pieces of the equation are 60 hands per hour in blackjack with an average house edge of 1%. So the value of comps you could expect would be (average bet) × (hours played) × 60 × 1% × 15%. Let’s assume 16 hours of play. You can then back out the average bet required. Let’s assume food and beverage has a value of $500. Then the average bet required would be 500/(16*60*0.01*0.15) = $347. A free room might be worth $1,000, so an average bet of $694 would be required. There is a whole spectrum of suites, roughly ranging in value from $1,000 to $10,000 a day, so an average bet of $1,389 to $13,889 would be required. Free golf might be worth $500, so back to $347 for that. I’ve heard of Vegas casinos comping shopping sprees at the Fashion Show mall, but they don’t sell cars there. If we assume $2,000 for airfare then $1,389. At high levels of play this may also be subject to skill level, the better you are the less you will get. They also might have some sympathy and give you more than you are entitled if you had a really bad run of luck. For rooms, you will have more bargaining power if you ask for one during a slow time when they have vacancies anyway.
Hey Wizard - I love the site! My wife and I just returned from Vegas, and I really noticed the proliferation of 6:5 Blackjack games all up and down the strip. Thanks to your site, and the numerous articles that you wrote or linked to, we were able to avoid these games entirely. I did notice, however, that most of these tables seemed to be close to capacity, and I’m wondering how long you think it will take before the ’blissfully ignorant masses’ begin to hear about how insidious this small rule change is? At first glance, it seems like the ’pitch game’ that my wife and I like so much (and could only find downtown in the past) and I think a lot of people just don’t realize the edge they’re giving up. If we hadn’t read your site, we might have thought the casinos were responding to customer demand for pitch games. Keep up the great work.
Thanks. The ratio of 6 to 5 games will only go up, in my prediction. You could ask the same question about lots of games. Why do people play 8/5 jacks or better video poker when a 9/6 game is across the aisle? Why do people play double zero roulette when the same casino has single zero? Why do people bet more than the minimum on the pass line, but then don’t take full odds in craps? Granted, sometimes game limits are an issue but you see people betting large amounts foolishly all the time. My own guess is that it is easier to believe in luck and superstition than to do a little reading about the odds.
"What luck for rulers, that men do not think." - Adolph Hitler
When I first got a casino player card to rate my play, I had two cards for the account and shared one with my mom so we could get a higher rating. Eventually she got her own card and now when we play side by side, spending the same amount of time and money, she will rate higher returns in bonus cash. She also rates higher returns at other casinos even though we never shared a card at them. Do casinos share player ratings? And why after a number of years of playing individually am I still not receiving the same player rating as my mom when we basically play the same way? The casinos give me no answers on how they rate players. Also, a friend of ours goes to the casino maybe once or twice a year, spends about the same as we do per visit (yet we go 6 to 8 times a year, spending more), and she gets even higher bonus cash than both of us combined! Can you explain this and advise how to correct it?
I wish I had a good answer for you. Usually cash back or free play is a percentage of points earned. However, it doesn’t sound like this is the case with your casino. Mailers are often a mystery. Here in Vegas, professional gamblers often exchange information about play vs. mailer, to try to determine the least amount of play required to get the best mailer. It also sometimes has to do with the average play per trip. So often it can hurt you to play just a little bit per day. Tournaments and comps can get even more mysterious. Yes, casinos do indeed share information with other casinos in the same company, which sometimes results in getting offers from casinos you rarely or never play at. It also sometimes helps to play hard-to-get. If you play on a regular basis, the casino’s rating formula may peg you as somebody who will play regardless of incentive. However, an intermittent player may need more enticing to get through the door. If the casino told you everything about how it rewards players, managers would be concerned that you may discover the system and begin playing as little as possible to get the greatest reward.
Do you have a good rule for getting up from the blackjack table for a non-card counter playing basic strategy? Obviously we’d all like to quit while we’re ahead, but how far ahead. And is there a time to quit when you’re behind?
For recreational gambling, my rule is to get up when you’re not having fun any longer.
Consider a hypothetical game based on the roll of a die. If die lands on 1, the player loses $1 and the game ends. If the die lands on anything else, the player’s wins $1. At this point the player may let it ride, or quit. The player may keep playing, doubling each bet, until he loses or quits. What is the best strategy?
Speaking only in terms of maximizing expected value, the player should play forever. While the probability is 1 that the player will eventually lose, at any given decision point the expected value always favors going again. It seems like a paradox. The answer lies in the fact that some events have a probability of 1, but still may not happen. For example, if you threw a dart at a number line from 0 to 10, the probability of not hitting pi exactly is 1, but it still could happen.
However, for practical purposes, there is some stopping point. This is because the happiness money brings is not proportional to the amount. While it is commonly accepted that more money brings more happiness, the richer you get, the less happiness each additional dollar brings you.
I believe a good way to answer this question is to apply the Kelly Criterion to the problem. According to Kelly, the player should make every decision with the goal of maximizing the expected log of his bankroll after the wager. To cut to the end of this (I cut out a lot of math), the player should keep doubling until the wager amount exceeds 96.5948% of his total wealth. Wealth should be defined as the sum of the amount won plus whatever money the player had before he made the first wager. For example, if the player had $100,000 to start with, he should keep doubling up to 23 times, to a win of $4,194,304. At that point the player’s total wealth will be $4,294,304. He will be asked to wager 4,194,304/4,294,304 = 96.67% of his total wealth, which is greater than the 96.5948% stopping point, so he should quit.
Recently I visited Charles Town Races and Slots, betting on the Kentucky derby. A Hispanic guy had hit a good payout on a slot machine for $6,000 and seemed to be having some sort of ID problem. I was in the casino for about an hour. In passing him on the way out, still standing by the machine, he still seemed to be having a problem. My question is if he has no ID (for whatever reason) can he still get the payout? The casino is in the state of WV. Would the rules prohibit a person in the country illegally from betting or wining if he or she has bet?
I forwarded this one to Brian, who is a former gaming regulator, and currently a casino manager. Here is what he said,
The casino would not know that someone was in the country illegally. If he had a valid passport, the jackpot would be honored. The illegal may not know this, be scared or they may not have a valid ID to show. Whenever someone wins $1,200 or more, ID is required for tax purposes. If someone doesn’t have his ID, the jackpot would be held in the cage waiting for them to claim it. In most cases, the person has legitimately forgotten their ID; however, sometimes you run into a problem, such as a minor who was gaming. If he doesn’t claim it, the money has to be added back into revenue because the deduction (jackpot) was never paid or there are abandoned property rules that prevail. Also, like the U.S., most countries tax worldwide income. To that end, the U.S. has tax treaties with several countries to withhold or notify the respective governments of monies won in the U.S. so Uncle Sam always gets his cut.
How much do you tip if you win in a live casino poker tournament? I have been tipping 10% for wins under $3,000. Am I over tipping? How much do the big winners in the WSOP tip?
I think you’re over-tipping. I think a good range is 2% to 5%, the greater the win, the lower the percentage.
Do you think any individual player can beat the odds over the long run in California games that every seated player has the same opportunity to bank, once every round, assuming players have to pay $1 to the house every hand?
Yes and no. Those games usually have a banker advantage, so if you took it at every opportunity, you would have a long-term advantage. However, there are agreements among the casinos and banking organizations not to let ordinary players do this excessively, as if it were a business, as opposed to recreational gambling.
Are you familiar with "The Fundamental Formula of Gambling"? I would love to hear your thoughts on it, as it is never mentioned on your site. The formula is:
N = log(1 - DC)/ log(1 - p), where
DC = Degree of certainty that an event will appear
P = probability of the event
N = number of trials
That is just an obvious extension of the rule that log(ab
)=b×log(a). It is not worthy of any special term. I suppose the formula might be helpful in answering some questions about the probability of a succession of losses. For example, suppose a video poker player wants to know how many hands he would have to play, such that the probability of a royal drought is exactly 5%. The probability of a royal per hand in 9/6 Jacks or Better
, with optimal strategy, is 0.00002476. The degree of certainty that at least one royal will appear is 95%. So, the number of hands in a 5% royal drought would be log(1-.95)/log(1-0.00002476) = 120,989.
However, you don’t need to use that formula to solve that problem. It could be set up as:
.05 = (1-0.00002476) n
log(.05) = n × log(1-.00002476)
-1.301 = n × -0.000010753
n = 120,989
My question is about the history of California gaming law, especially craps and roulette. Do you know why California does not allow the dice or the ball and wheel to directly determine the outcome of their respective games? Or do you know where this law originated? What were California law makers thinking when they wrote this?
Sorry, I don’t know the history or reason behind that law. It was probably a misguided compromise between puritan and gambling interests. They were likely thinking the same kind of thing Mississippi lawmakers were when they only permitted non-Indian gaming on “riverboats.” We all saw the result of that brilliant idea after Hurricane Katrina. As I’ve been saying for years, my opinion is if you’re going to allow gambling, then drop the pretenses and allow it the whole way.
I need to buy an official felt. What kind do most casinos use? When I was looking for one, one of them was velveteen. Is there any difference?
To be honest with you, I don’t know much about it. Casino felts are generally much better quality than those for home use. For my home poker games, we put a home-use roulette felt on top of my dining room table. By the end of the evening, there are little green felt balls everywhere. In Macau, the felts are much smoother and harder than those in Vegas. I see you can buy casino felts on eBay for about $90.
I’m a maths teacher, and I wanted to use your site as part of an investigation into the Mathematics of Gambling. However, I am concerned that exposure to a discussion of gambling systems will encourage, rather than discourage, gambling. Do you have any recommendations for problem gamblers? Or ways to avoid becoming a problem gambler? Sorry this question isn’t simple and Mathematical :) I want to combine a knowledge of the maths with an ethical/political understanding of the issue. Australia has a big Pokie (slot machine) gambling problem.
My philosophy is the world would be a better place if there was unfettered access to truthful information. In that spirit, I would have no compunction to discuss the topic. If everybody knew the truth about how slot machines work, and how expensive they are to play, there would be a lot fewer players -- recreational and compulsive.
I’m aware of Australia’s love of pokies. When I was at a gambling conference in Sydney, I had the pleasure of listening to your Nick Xenophon chastising the audience for making such an addictive product. Personally, I favor mandating that machines be labeled with the return percentage that they are expected to pay.
Here in the U.S., we would say "math teacher," or "knowledge of math," by the way.
What is 48/2(9+3)? A lot of forums are having a heated debate on this.
In evaluating mathematical expressions you use the following order of priority:
- Parenthesis (what is inside them)
- Multiplication and division (equal priority)
- Addition and subtraction (equal priority)
After this narrowing down, if you have terms of equal priority, then you go left to right. So, in this case we evaluate the parenthesis first, resulting in 48/2 × 12. Division and multiplication are equal in priority, so we do the division first, because it is furthest left. That gives us 24 × 12=288.
I have to go on record as saying that while I feel that 288 is the right answer, the syntax of the expression is terrible. It never hurts to put in extra parenthesis or brackets for clarification. I would have expressed this as (48/2) × (9+3).
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
The story goes that some people believed that Friday the 13th was unlucky because it seemed the 13th of the month occurred more often on Friday than on any other day of the week. Does the 13th really occur on Friday most often or not?
I have to admit my initial answer to this question, in my forum, was wrong. There are 14 different calendars only, seven for each day of the week the year starts on, by two for whether or not there is a leap day. I incorrectly thought that there would be a 2800-year balanced cycle. However, that is not the case.
Before going further, let's review the leap year rules:
- Years evenly divisible by four are leap years, except...
- Years evenly divisible by 100 are not leap years, except...
- Years evenly divisible by 400 are leap years.
If it were not for the third rule there would be a nice 700-year cycle. However, the 400-year rule breaks the 700-year balanced pattern, and starts it over from the starting point. So there is a 400-year cycle, but it isn't balanced. The following table shows how often the 13th of each month falls on each day of the week in a cycle.
13th of the Monthby Day of the Week
The average number in a cycle is 685.71. However, Friday exceeds the average at 688. So, every 400 years we get 2.29 extra Friday the 13ths.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
I have some match play chips. Unlike the usual case, the casino in question allows them to be used in any game. What bet would you recommend I use them on?
That is unusual. Said casino probably has no clue what they are doing. For the benefit of other readers, let me review what a match play chip is. These are chips that you match with real money when making a bet. If you win, you are paid on both, and your real money wager is returned. If you lose, you lose both. Nothing happens on a push.
So a match play chip may be used only once on a resolved bet. If the casino allows you to use it on any bet, the proper strategy is to put it on a long-shot bet. This is because the cost of not getting the match play back after a win is a lot less on a long-shot bet than an even-money wager.
The following table shows various bets in three different games and the expected number of units won. For the purposes of the table, it is assumed if the player gets a tie he keeps repeating the same bet until it is resolved. You can see the highest expected value is on a single-number bet in roulette at 87% of face value.
Match Play Expected Value
Please explain what an APR interest rate is.
APR stands for Annual Percentage Rate. The purpose of it is to equate an interest rate with possible points and compounded monthly to an APY (annual percentage yield), which is an interest rate with no points and compounded annually.
For those who don't know, when you take out a mortgage, the bank often charges a finance fee based on the amount of the mortgage. For each point, the borrower must pay 1% of the mortgage amount to the bank as an additional fee. Sometimes this fee is tacked on to the principal amount.
The APR interest rate is hypothetical. If the borrower negotiated with the lender to increase the interest rate, in exchange for no points, and compound interest annually, then the APR interest rate would result in exactly the same payment. Let's look at an example.
Suppose the borrower wants a loan of $250,000. The bank charges 5.625% interest, compounded monthly, with two points, based on a 30-year mortgage. What would be the APR? The finance fee is 2% of $250,000, which equals $5,000. The borrower then asks the bank to add that to the principal, for a loan of $255,000. I won't get into the monthly payment calculation, so take it on faith that it comes to $1,467.92.
Assuming there were no points, an interest were compounded annually, what interest rate would equate to the same monthly payment of $1,467.92 on a loan of $250,000? By trial and error I find an interest rate of 5.9635% and no points and compounded annually results in the same monthly payment of $1,467.92. So, a way to phrase this would be, "A 30-year fixed loan at 5.625% interest with two points has an APR of 5.9635%."
What was the first casino game granted a patent?
I don't know. What I think I can correctly say is the earliest casino game patent for a game played today is for Caribbean Stud Poker. There probably were other patents before it for games that didn't make it. The Caribbean Stud patent was filed on April 18, 1988 and issued on June 6, 1989. Patent number 4,836,553.
Not that you asked, but at that time casino game patents were valid for 17 years from date of issue, or 20 years from date of filing, whichever was more. In 1995 the term was extended to 20 years from date of filing. In the case of Caribbean Stud, the patent would have expired in 2008. However, I think it still has valid trademarks, meaning a casino could offer the game without paying royalties, but would have to think of another name that is not trademarked.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
How many hands need to be played of a casino game for the casino to be confident of showing a profit?
It depends on the game, naturally. The greater the house advantage and lesser the variance, the greater the chance is. It also depends on the level of confidence.
The following table shows how many bets are required, assuming the same bet amount, for the casino to be confident, at various levels of confidence, for some common games. For example, to have a 95% chance of showing a net profit on the Banker bet in baccarat, the casino would need to deal 20,791 hands.
Number of Hands for Net Profit at Various Confidence Levels
|Baccarat ||Banker ||1.06% ||0.93 ||12621 ||20791 ||29520 ||41588 |
|Baccarat ||Player ||1.24% ||0.95 ||9741 ||16046 ||22783 ||32097 |
|Baccarat ||Tie ||14.36% ||2.64 ||555 ||915 ||1299 ||1830 |
|Blackjack ||Strip rules ||0.28% ||1.15 ||277046 ||456388 ||648001 ||912912 |
|Casino War || ||2.88% ||2.24 ||9935 ||16367 ||23238 ||32739 |
|Craps ||Pass ||1.41% ||1.00 ||8259 ||13606 ||19318 ||27216 |
|Craps ||Don't Pass ||1.36% ||0.99 ||8631 ||14219 ||20188 ||28442 |
|Pai gow || ||1.50% ||0.75 ||4106 ||6764 ||9604 ||13530 |
|Pai gow poker || ||1.46% ||0.75 ||4334 ||7140 ||10137 ||14281 |
|Three Card Poker ||Ante ||3.37% ||1.64 ||3890 ||6407 ||9098 ||12817 |
|Three Card Poker ||Pairplus ||7.28% ||2.85 ||2517 ||4146 ||5887 ||8294 |
|Jacks or Better ||9/6 ||0.46% ||4.42 ||1589830 ||2649876 ||3782305 ||5348060 |
|Jacks or Better ||9/5 ||1.55% ||4.42 ||142476 ||243815 ||350929 ||500032 |
|Jacks or Better ||8/5 ||2.70% ||4.4 ||48094 ||84383 ||123102 ||176586 |
|Jacks or Better ||7/5 ||3.85% ||4.38 ||25100 ||43688 ||64109 ||92533 |
|Jacks or Better ||6/5 ||5.00% ||4.36 ||13754 ||26569 ||39923 ||58235 |
This is based on the Normal Distribution in all cases except for Jacks or Better
. That approximation becomes untrustworthy if the number of expected events of any one outcome is five or less. So, for video poker, I used the Poisson distribution for the royals and the Normal approximation otherwise.
For blackjack, the rules are: 6 decks, dealer stands on soft 17, double after split allowed, surrender allowed, re-splitting aces allowed.
This question is raised and discussed in my forum at Wizard of Vegas
There is a prize draw at the local casino every week where the prize draw winner is offered $5,000 cash or the option to pick a mystery bag containing one of the following prizes:
- $2,000 cash
- $4,000 non-negotiable chips
- $6,000 non-negotiable chips
- $8,000 cash
- $10,000 cash
What would you choose and why?
First, I assume the "non-negotiable chips" are the use-till-you-lose type. These have a value of about 98% of face value, depending on the rules of the games you can use them on.
That said, the average bag is worth $5,960. That is almost 20% more than the cash offer. Even looking at it as a utility of money problem, you should still take a bag, even if you have no other wealth.
This question was asked and discussed in my forum at Wizard of Vegas
Costa Rica sports book BookMaker.eu put out a wager on the number of false statement Trump would make in his televised address on the border wall. They set the over/under line at 3.5. 96% of the action was on the over and they lost $276,424. What would you have made a fair line on that bet?
According to BookMaker, the Washington Post, who keep a count political false statements in general, would be used as the source of number of lies. According to that source, Trump averaged 15 false statement per day during 2018. The next question to be answered in analyzing this bet is how much time does Trump spend making public statements a day? Between tweets, interviews, and off-the-cuff statements, 20 minutes seems like a reasonable estimate to me. A nice round number at least. Simple division gives us 15/20 = 0.75 false statements per minute, or one every 80 seconds.
The address was estimated to last six to eight minutes by the media before it started. Let's split the difference and go with seven minutes. Seven minutes at 0.75 false statements a minute gives us an estimated 5.25 false statements. So, I would have set the over/under at 5.5.
By the way, if we assume 5.25 to be the mean number of false statements, then the probability of three or less false statements is 23.17%, if we assume the total is distributed according to the Poisson distribution, which I think is a reasonable assumption.
By the way, in the end, the number of false statements was scored at six.
This question is asked and discussed in the very long thread on Trump at Wizard of Vegas, but discussion of this specific topic starts here.
What are the top five largest casinos in the United States?
As measured by square feet of gambling space, here they are. This comes as a surprise to me, as I've barely heard of the two Oklahoma casinos in the top five.
Top Five U.S. Casinos
|Winstar || Thackerville, OK || 519,000 |
| Mohegan Sun || Uncasville, Connecticut || 364,000 |
| Foxwoods || Mashantucket, CT || 344,000 |
| San Manuel || Highland CA || 220,000 |
| Riverwind || Norman OK || 216,000 |
What is the best game to play?
It depends on the rules of the game and how well you play it. Limiting the answer to popular games, assuming you play the optimal strategy and stick to all the best bets when given a choice, I’d narrow down the best games to the four in the following list. (The percentage shown is the element of risk of those games, which is the ratio of how much you can expect to lose to how much you bet, which I think is a proper measurement of the value of a game.)
- blackjack (six decks, dealer stands on soft 17, double after split allowed, surrender allowed, re-splitting aces allowed) — 0.25%
- craps (3-4-5x odds, laying the maximum odds allowed) — 0.27%
- video poker (9-6 jacks or better) — 0.46%
- Ultimate Texas Hold 'Em — 0.53%
What is your favorite game?
My answer would be whichever game has the lowest element of risk at whatever casino I'm in. However, the answer to the question about which game I find the most fun to play is pai gow (tiles). I dislike volatility and tiles offers a slow game with lots of pushes. It’s also a challenging game to understand and play well. I find that other players are generally smart people and pleasant to play with.
What do you think of my betting system?
All betting systems are equally worthless. Not only can't a betting system overcome the house edge, it can't even dent it. If a betting system makes gambling more fun, be my guest. Just don't delude yourself that it will help in the long run.
Which is your favorite casino in Las Vegas?
The casino that I feel offers the best odds and overall value is South Point
Casino (insert name here) is cheating. Can you please warn your readers about them? I know because (insert adjective-laden story about losing here).
This kind of accusation rarely comes with any evidence behind it other than adjectives. What rare times I get some actual numbers, the loss could easily be explained as ordinary bad luck. Nevertheless, I have exposed cases of cheating at Internet casinos several times, starting with such accusations. So if you suspect a casino is cheating, please follow the scientific method before writing to me; in other words, make a hypothesis about how the casino cheating, gather evidence to confirm or deny the hypothesis, and finally analyze the evidence. I’m happy to help with step 3.
Why are you such a Debbie Downer when it comes to gambling? You take all the fun out of it with your mathematical strategies, which take away my free will.
If you want to lose more by making mistakes, go ahead. I can only lead a horse to water. You don't have to drink it if you don't want to.
The following puzzle appeared in the March 6, 2021 New York Times.
The rules are pretty simple:
- Each row, column and region must have exactly two stars.
- No two stars may touch, not even diagonally.
Can you help with a solution?
This is called a Two not Touch puzzle. The button below shows my answer and solution.
Here is my solution (PDF).