Rothstein Betting System
Poll
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| | 4 votes (44.44%) | ||
| | 2 votes (22.22%) | ||
| | 3 votes (33.33%) | ||
| | 1 vote (11.11%) | ||
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| | 1 vote (11.11%) | ||
| | 1 vote (11.11%) | ||
| | 4 votes (44.44%) |
9 members have voted
January 26th, 2024 at 1:36:23 PM
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It seems I get asked a lot to make more videos about betting systems. It's one of my least favorite topics when it comes to gambling, but I should balance that with what the public wants to know. Hopefully it also drives home that point that all betting systems are equally worthless and keeps people from paying money for them when there are many available for free.
I learned about the Rothstein in the craps chapter of Scarne's New Complete Guide to Gambling. It works like the Martingale, except the player always bets double the last bet plus one. Note the "plus one." So the betting progression goes 1, 3, 7, 15, 31, 63, etc.
With the Martingale, the player nets one unit for every win. With the Rothstein, the player nets one unit for every bet. So, while the player can win more/faster with the Rothstein, he can't last as many bets for any given bankroll. More reward equals more risk.
I just posted all the details of the Rothstein over at WoO. I welcome all comments are corrections. In particular, there was a lot of copying and pasting, so please let me know if anything wasn't updated properly.
The question for the poll is would you play the Rothstein? Multiple votes allowed.
Image source: Parade.com.
I learned about the Rothstein in the craps chapter of Scarne's New Complete Guide to Gambling. It works like the Martingale, except the player always bets double the last bet plus one. Note the "plus one." So the betting progression goes 1, 3, 7, 15, 31, 63, etc.
With the Martingale, the player nets one unit for every win. With the Rothstein, the player nets one unit for every bet. So, while the player can win more/faster with the Rothstein, he can't last as many bets for any given bankroll. More reward equals more risk.
I just posted all the details of the Rothstein over at WoO. I welcome all comments are corrections. In particular, there was a lot of copying and pasting, so please let me know if anything wasn't updated properly.
The question for the poll is would you play the Rothstein? Multiple votes allowed.
Image source: Parade.com.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
January 27th, 2024 at 6:03:04 AM
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I might be mistaken, but it seems like the "Rules" on the WoO might be not completely correct.
Shouldn't step 5. be something like:
If last bet made loses, then bet two times the last units plus an extra unit the next time and go back to step 4.
Also shouldn't the final note say that, instead of odd numbers, the bet sizes will be consecutive powers of two minus one: 1, 3, 7, 15, 31, 63, etc.?
In step 6. there is one redundant "you".
Shouldn't step 5. be something like:
If last bet made loses, then bet two times the last units plus an extra unit the next time and go back to step 4.
Also shouldn't the final note say that, instead of odd numbers, the bet sizes will be consecutive powers of two minus one: 1, 3, 7, 15, 31, 63, etc.?
units bet accum. loss possible win
1 1 1 - 0 = 1
3 4 3 - 1 = 2
7 11 7 - 4 = 3
15 26 15 - 11 = 4
31 57 31 - 26 = 5
63 120 63 - 57 = 6
In step 6. there is one redundant "you".
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
January 28th, 2024 at 5:39:41 AM
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It also seems, that in the game analyses the description and values of "Expected total bet" and "Expected net win" are exchanged:
Both "Five Step Analysis" and "Ten Step Analysis" of
* Craps Don't Pass Bet
* Single-Zero Roulette Even Money Bets
* Double-Zero Roulette Even Money Bets
* Player Bet in Baccarat
Both "Five Step Analysis" and "Ten Step Analysis" of
* Craps Don't Pass Bet
* Single-Zero Roulette Even Money Bets
* Double-Zero Roulette Even Money Bets
* Player Bet in Baccarat
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
February 1st, 2024 at 3:14:32 PM
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Thank you for those corrections, Thomas.
Here is my new video on the Rothstein.
Direct: https://www.youtube.com/watch?v=QzACmc4Ynjo
Here is my new video on the Rothstein.
Direct: https://www.youtube.com/watch?v=QzACmc4Ynjo
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
February 1st, 2024 at 4:02:38 PM
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Thanks for covering this system, Wiz. This is my betting strategy of choice on Baccarat.
I do a slight modification to keep it a bit more affordable. The first four steps remain 1-3-7-15, but after the fourth loss, I enter "Purgatory". In Purgatory, the levels are 7-15-31, and you must win five times without going broke in order to escape Purgatory and return to the one-unit bet.
This modification gives you seven levels with a 79-unit bankroll instead of a 247-unit bankroll.
Of course I do not kid myself into thinking this strategy would ever beat the house edge, however I do fairly well with it.
The key to success with it, ironically, is NOT to grind out a small win each time. The fatal flaw with Martingale has always been that the one losing session will wipe out ten or more winning sessions. Instead, one must strive to win at least 79 units each winning session. (Actually not that hard to do, that's roughly one shoe.) That way a loss will never cost you more than one win.
On the other side of the coin, it's imperative that you walk away after losing the seventh level, returning any money you have left to your bankroll. That way a loss will always be less than a win.
Again, it'll never beat the house edge, but for a short-term system, it ain't bad.
I do a slight modification to keep it a bit more affordable. The first four steps remain 1-3-7-15, but after the fourth loss, I enter "Purgatory". In Purgatory, the levels are 7-15-31, and you must win five times without going broke in order to escape Purgatory and return to the one-unit bet.
This modification gives you seven levels with a 79-unit bankroll instead of a 247-unit bankroll.
Of course I do not kid myself into thinking this strategy would ever beat the house edge, however I do fairly well with it.
The key to success with it, ironically, is NOT to grind out a small win each time. The fatal flaw with Martingale has always been that the one losing session will wipe out ten or more winning sessions. Instead, one must strive to win at least 79 units each winning session. (Actually not that hard to do, that's roughly one shoe.) That way a loss will never cost you more than one win.
On the other side of the coin, it's imperative that you walk away after losing the seventh level, returning any money you have left to your bankroll. That way a loss will always be less than a win.
Again, it'll never beat the house edge, but for a short-term system, it ain't bad.
Casinos are not your friends, they want your money. But so does Disneyland.
And there is no chance in hell that you will go to Disneyland and come back with more money than you went with.
- AxelWolf and Mickeycrimm
February 2nd, 2024 at 5:36:07 AM
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A thought experiment.
Imagine, a Casino offered Rothstein to valued high roller customers as a double-zero roulette side bet, operated by the dealer. For example, it might be called "The Rothstein Five" and be played on "red".
The player would buy in for 57 units (cheques) and then enjoy the excitement of the coming - up to five - rolls, without having to care about the individual betting.
The dealer would put the player's buy-in in special side-bet-spot and bet from there 1 unit on the first roll, 3 units on the second roll, etc. (7, 15, 31), until either a roll wins or all five rolls are lost.
In case of the loss, the player loses all 57 units.
In case of a win the player is paid the current wager plus the same amount, which is the win of the even money bet, plus any units left over from the buy-in which haven't been wagered, yet.
After losing or winning, the player is free to wager another 57 units on "The Rothstein Five".
That wouldn't be a sucker bet, I guess.
The procedure of "The Rothstein Five" might quite easily cause dealer errors, because the dealer has to keep track of the number of rolls already played and has to figure out how many units to bet on the next roll. Also, handling the 57 cheques might be inconvenient.
Fortunately, the procedure can be simplified using the property of the Rothstein which, in case of success, lets the player win one unit for each of the already played rolls:
The player buys in e.g. for $5700 (a handy high roller example), using any smallest number of cheques of any denominations the player whishes. The unit in this case is $100.
The dealer takes five $100 cheques from the dealer's rack and puts them on a side spot besides the buy-in side-bet spot.
Before each roll the dealer moves exactly one of these $100 cheques over to the player's buy-in.
If any of the five rolls wins, the player is paid the buy-in plus the cheques that have been added to it. (Remaining cheques on the side go back to the dealer's rack.)
If the fifth roll loses, and as a result no more cheques on the side are available, then all the money goes to the dealer's rack. The player lost.
The question is whether it would be worthwhile for the casino to earn ¢84 (eighty-four cents) of every $100 wagered, in the long run.
If a VIP billionaire, who won't even blink when wagering some $200,000 on a single hand of Blackjack, demanded to play "The Rothstein Ten" and put the required $203,600 on the table, then again the unit would be $100.
The dealer now would have to put ten $100 cheques on the side and otherwise follow the same procedure:
Add one $100 cheque to the player's buy-in before each roll.
Repeat until either a roll wins or until ten rolls are lost and therefore no more $100 cheques are left on the side.
If a roll wins, pay the buy-in plus the cheques that have been added to it. Put any cheques, remaining on the side, back into the rack.
If the tenth roll loses then all the money goes to the rack.
Drawback: The casino will only earn ¢6 (six cents) of every $100 wagered, in the long run.
A strict casino management might insist on only offering the approved, and implemented, games and procedures, and might refuse to play "The Rothstein Ten". Instead they might offer the VIP billionaire the established "Rothstein Five" for units of $3,572. 57 of these units sum up to $203,604, basically the amount the VIP wishes to wager, anyway.
The high rolling player might go ballistc about this bad offer, beacause in the long run he would lose nearly 14 times as much as with "The Rothstein Ten".
Imagine, a Casino offered Rothstein to valued high roller customers as a double-zero roulette side bet, operated by the dealer. For example, it might be called "The Rothstein Five" and be played on "red".
The player would buy in for 57 units (cheques) and then enjoy the excitement of the coming - up to five - rolls, without having to care about the individual betting.
The dealer would put the player's buy-in in special side-bet-spot and bet from there 1 unit on the first roll, 3 units on the second roll, etc. (7, 15, 31), until either a roll wins or all five rolls are lost.
In case of the loss, the player loses all 57 units.
In case of a win the player is paid the current wager plus the same amount, which is the win of the even money bet, plus any units left over from the buy-in which haven't been wagered, yet.
After losing or winning, the player is free to wager another 57 units on "The Rothstein Five".
That wouldn't be a sucker bet, I guess.
= = = = = = = = = = = = = = =
The procedure of "The Rothstein Five" might quite easily cause dealer errors, because the dealer has to keep track of the number of rolls already played and has to figure out how many units to bet on the next roll. Also, handling the 57 cheques might be inconvenient.
Fortunately, the procedure can be simplified using the property of the Rothstein which, in case of success, lets the player win one unit for each of the already played rolls:
The player buys in e.g. for $5700 (a handy high roller example), using any smallest number of cheques of any denominations the player whishes. The unit in this case is $100.
The dealer takes five $100 cheques from the dealer's rack and puts them on a side spot besides the buy-in side-bet spot.
Before each roll the dealer moves exactly one of these $100 cheques over to the player's buy-in.
If any of the five rolls wins, the player is paid the buy-in plus the cheques that have been added to it. (Remaining cheques on the side go back to the dealer's rack.)
If the fifth roll loses, and as a result no more cheques on the side are available, then all the money goes to the dealer's rack. The player lost.
The question is whether it would be worthwhile for the casino to earn ¢84 (eighty-four cents) of every $100 wagered, in the long run.
Roll Probability Payout Expected Pay
--------------------------------------------
1 0.473684 5800 2,747.3684
2 0.249307 5900 1,470.9141
3 0.131214 6000 787.2868
4 0.069060 6100 421.2675
5 0.036347 6200 225.3545
(5 0.040386 0 0.0000)
============================================
total 1.000000 5652.1913
Expected loss per game:
5700.0000 - 5652.1913 = 47.8087
Expected loss per $100:
47.8087 / 57 = 0.838749
= = = = = = = = = = = = = = =
If a VIP billionaire, who won't even blink when wagering some $200,000 on a single hand of Blackjack, demanded to play "The Rothstein Ten" and put the required $203,600 on the table, then again the unit would be $100.
The dealer now would have to put ten $100 cheques on the side and otherwise follow the same procedure:
Add one $100 cheque to the player's buy-in before each roll.
Repeat until either a roll wins or until ten rolls are lost and therefore no more $100 cheques are left on the side.
If a roll wins, pay the buy-in plus the cheques that have been added to it. Put any cheques, remaining on the side, back into the rack.
If the tenth roll loses then all the money goes to the rack.
Drawback: The casino will only earn ¢6 (six cents) of every $100 wagered, in the long run.
Roll Probability Payout Expected Pay
---------------------------------------------
1 0.473684 203,700 96,489.4737
2 0.249307 203,800 50,808.8643
3 0.131214 203,900 26,754.6290
4 0.069060 204,000 14,088.2897
5 0.036347 204,100 7,418.5241
6 0.019130 204,200 3,906.3994
7 0.010069 204,300 2,057.0065
8 0.005299 204,400 1,083.1649
9 0.002789 204,500 570.3657
10 0.001468 204,600 300.3393
(10 0.001631 0 0.0000)
=============================================
total 1.000000 203,477.0565
Expected loss per game:
203,600 - 203,477.0565 = 122.9435
Expected loss per $100:
122.9435 / 2036 = 0.060385
= = = = = = = = = = = = = = =
A strict casino management might insist on only offering the approved, and implemented, games and procedures, and might refuse to play "The Rothstein Ten". Instead they might offer the VIP billionaire the established "Rothstein Five" for units of $3,572. 57 of these units sum up to $203,604, basically the amount the VIP wishes to wager, anyway.
The high rolling player might go ballistc about this bad offer, beacause in the long run he would lose nearly 14 times as much as with "The Rothstein Ten".
Roll Probability Payout Expected Pay
---------------------------------------------
1 0.473684 207,176 98,136.0000
2 0.249307 210,748 52,541.0526
3 0.131214 214,320 28,121.8837
4 0.069060 217,892 15,047.6746
5 0.036347 221,464 8,049.6620
(5 0.040386 0 0.0000)
=============================================
total 1.000000 201,896.2729
Expected loss per game:
203,604 - 201,896.2729 = 1,707.7271
Expected loss per $100:
1,707.7271 / 2,036.04 = 0.838749
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
March 28th, 2024 at 12:15:05 PM
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I'd like to ask you what you think is the amount wagered in the game of "The Rothstein Five" (as described above).
Three suggested scenarios have the same course of action with regards to the roulette outcomes. Only the modalities of handling the money differs in some way.
1) The player alone bets 1, 3, 7, 15, and 31 units according to the rules.
The player either loses any of these amounts or, on success, gets added the same amount to the bet for the even money win.
The player eventually either loses 57 units (the sum of the five amounts) or, on success, receives back the units most recently bet plus the same amount for the even money win and still keeps the units not bet so far. Remark: These three components add up to the same value as the original 57 units plus as many units as rolls played.
2) The player buys in for 57 units and is able to watch the dealer betting 1, 3, 7, 15, and 31 units on behalf of the player, according to the rules.
The player observes the dealer either losing any of these amounts or, on success, adding the same amount to the bet for the even money win.
The player eventually either loses all 57 units or, on success, receives back the units most recently bet plus the same amount for the even money win, as well as the units that haven't been bet so far. Remark: These three components add up to the same value as the original 57 units plus as many units as rolls played.
3) The player buys in for 57 units and is able to watch the dealer add one unit for each roll played.
The player eventually either loses all 57 units or, on success, receives back the original 57 units plus as many units as rolls played.
As you might guess, my answer is 57 units as the wager for "The Rothstein Five".
Would you agree?
Three suggested scenarios have the same course of action with regards to the roulette outcomes. Only the modalities of handling the money differs in some way.
1) The player alone bets 1, 3, 7, 15, and 31 units according to the rules.
The player either loses any of these amounts or, on success, gets added the same amount to the bet for the even money win.
The player eventually either loses 57 units (the sum of the five amounts) or, on success, receives back the units most recently bet plus the same amount for the even money win and still keeps the units not bet so far. Remark: These three components add up to the same value as the original 57 units plus as many units as rolls played.
2) The player buys in for 57 units and is able to watch the dealer betting 1, 3, 7, 15, and 31 units on behalf of the player, according to the rules.
The player observes the dealer either losing any of these amounts or, on success, adding the same amount to the bet for the even money win.
The player eventually either loses all 57 units or, on success, receives back the units most recently bet plus the same amount for the even money win, as well as the units that haven't been bet so far. Remark: These three components add up to the same value as the original 57 units plus as many units as rolls played.
3) The player buys in for 57 units and is able to watch the dealer add one unit for each roll played.
The player eventually either loses all 57 units or, on success, receives back the original 57 units plus as many units as rolls played.
As you might guess, my answer is 57 units as the wager for "The Rothstein Five".
Would you agree?
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
April 6th, 2024 at 2:39:14 PM
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I'd like to summarize the information I have gathered so far about the Rothstein, showing
"The Rothstein Five" as a game of its own has the following parameters:
In the case of an eventual winning roll, this accumulated loss is completely refunded by the gain. This gain in addition includes the winning amount which the player receives. As long as the player keeps winning Rothsteins, the player will reuse the same betting units over and over again for each successive Rothstein.
Only in the case of an eventual losing roll, in the fifth step of "The Rothstein Five", the player actually loses 57 units to the casino and will have to fetch another 57 units from the wallet in order to play another "Rothstein Five".
The establlished method of analysis results in these parameters:
This means that a player will suffer from the same long term loss as a player playing "The Rothstein Five", by betting 9.083655 units on the first roll together with the Rothstein player and then waiting for the Rothstein player to resolve, before betting another 9.083655 units.
However, this also means that the Rothstein player will enjoy more rolls than the flat betting player: approx. 2 rolls on average and, due to variance, in some cases up to five rolls.
On a per roll basis the parameters are
This means that, if the flat betting player doesn't want to wait for the Rothstein player to resolve, the player may instead bet 4.483700 units each roll, together with the Rothstein player's rolls, and will not lose any more money than before.
Remark: The expected value has to be multiplied by the average number of Rothstein rolls to be comparable with Rothstein as well as with the 9.083655 units flat betting.
NOTE: The expected values, or more precisely the expected returns, in all four cases (Rothstein observed, Rothstein, flat, flat roll) are identical.
Conclusion
As demonstrated, "The Rothstein Five" is a game of its own. It has its own amount wagered and its own house edge. The amount wagered is not instantly obvious when observing the player's action. Only being able to spot the actual amount lost, reveals the size of the wager. The consequence of this "veiled" wager is that the house edge of "The Rothstein Five" is much smaller than the house edge of the base game, upon which Rothstein is brought to life.
In the simulation provided, four different courses of action, connected to "The Rothstein Five", are compared:
The casino, however, would very probably not offer "The Rothstein Five" to their patrons due to the horribly low house edge.
Players, on the other hand, can employ "The Rothstein Five" on their own and make use of the low house edge as long as they are willing to afford an amount of 57 units per game.
Outlook
Rothstein is one example among the class of finite betting sequences with negative progression, also called progression on losses. The method of analysis used here also applies to other sequences within this class, for example to the Martingale, which supposedly is the most famous among them.
It is straight forward to apply this method of analysis as well to the complementary class, the finite betting sequences with positive progression, also called progression on winnings. The probably most famous example here might be the Parlay, also called Paroli. With positive progression, the amount wagered remains the initial bet throughout the sequence but, as the downside, the house edge rises rapidly. This higher house edge makes games of this class more appealing to casinos. Examples are: Double Action Roulette, Double Ball Roulette, 2G$ side bet (any two greens in a row).
The Excel simulation.
I am posting the contents of my Excel sheet as a HEX encoded dump in two localized versions:
Please see here the instructions how to rebuild the content in your own Excel sheet.
Columns A through E contain the results of the simulated game parameters and their theoretical counterparts.
For easy comparison the categories in column F may be used by switching on the filter for the columns A through F.
The value in cell B2 is preset to 5 for "The Rothstein Five" and may be replaced by a different value out of 1 through 10 to simulate Rothsteins of that length.
- that "The Rothstein Five" is a game of its own, with its own house edge,
- how the results of the established method of analysis fit in,
- what it would mean on a per roll basis,
"The Rothstein Five" as a game of its own has the following parameters:
wager: 57 units
EXPECTED VALUE: -0.478087 units
house edge: 0.008387
In the simulation, see the entries labeled "Rothstein" and "Rothstein observed",
together with the values labeled "intermediate", which also include the plausibility check.
together with the values labeled "intermediate", which also include the plausibility check.
Why are the 57 units the amount wagered in "The Rothstein Five"?
The reason why the 57 units are the wager of "The Rothstein Five" is that 57 units is the only amount the player ever loses playing "The Rothstein Five" and that no wager is needed for a sequence eventually won in Rothstein.Why would the player lose exactly 57 units
while one can watch the player lose different amounts at the table while playing Rothstein?
The player, betting individual rolls, accumulates individual losses until Rothstein is resolved.while one can watch the player lose different amounts at the table while playing Rothstein?
In the case of an eventual winning roll, this accumulated loss is completely refunded by the gain. This gain in addition includes the winning amount which the player receives. As long as the player keeps winning Rothsteins, the player will reuse the same betting units over and over again for each successive Rothstein.
Only in the case of an eventual losing roll, in the fifth step of "The Rothstein Five", the player actually loses 57 units to the casino and will have to fetch another 57 units from the wallet in order to play another "Rothstein Five".
In which universe would a player be able to win money in a gambling game without any wager?
A few examples that explain, why a winning hand does not need a wager:This situation of winning without a wager happen millions of times every day in the world.
The dealer in a casino, as the opponent player to a patron player, and being supposed to win over the patron for the casino, never puts any wager on the table before the roll.
In the case of the dealer winning, i.e. the patron losing, this is no problem. The dealer's wager is neither needed nor would any be tolerated by the casino.
In the opposite case of the dealer losing and the patron winning, however, the patron has to trust the dealer paying the win, even though this wasn't obviously indicated by a dealer's wager before the roll. While the casino distrusts the patrons, patrons on the contrary have to trust the dealer, representing the casino.
The dealer in a casino, as the opponent player to a patron player, and being supposed to win over the patron for the casino, never puts any wager on the table before the roll.
In the case of the dealer winning, i.e. the patron losing, this is no problem. The dealer's wager is neither needed nor would any be tolerated by the casino.
In the opposite case of the dealer losing and the patron winning, however, the patron has to trust the dealer paying the win, even though this wasn't obviously indicated by a dealer's wager before the roll. While the casino distrusts the patrons, patrons on the contrary have to trust the dealer, representing the casino.
In a one-on-one situation, where both parties trust each other, no money has to be put on the table, before the bet is resolved. Each side is assured to receive the agreed upon wager from the other player in case of winning. In particular, the winning party of both parties doesn't need any wager because it only receives from but never has to give to the opposing losing party.
As shown above, a wager put on the table by a player, is returned back to the player together with the winning amount, if the player wins the roll. The player may now bet this same wager on a successive roll without having to fetch any fresh money from the wallet. Looking back to the previous winning roll, the player could have used a chip without monetary value to indicated the player's decision in the game. No wager was needed to win this previous roll.
As an example of such an equation, E(X)=p*W-q*L is the expected value for Bernoulli games, with
p: the probabillity of winning,
q: the probability of losing,
W: the amount won in the winning case,
L: the amount lost in the losing case.
From the equation it is clear that, in the losing case, a player has to pay the amount L.
For the winning case, nothing is stated about the amount the player should wager. Only the amount W, received by the player, is given in the equation. In the winning case no wager is needed.
As a consquence, the amount L defines the amount wagered, because it is the only amount that is transferred from the player to the casino.
p: the probabillity of winning,
q: the probability of losing,
W: the amount won in the winning case,
L: the amount lost in the losing case.
From the equation it is clear that, in the losing case, a player has to pay the amount L.
For the winning case, nothing is stated about the amount the player should wager. Only the amount W, received by the player, is given in the equation. In the winning case no wager is needed.
As a consquence, the amount L defines the amount wagered, because it is the only amount that is transferred from the player to the casino.
The establlished method of analysis results in these parameters:
wager: 9.083655 units
EXPECTED VALUE: -0.478087 units
house edge: 0.052632
In the simulation, see the entries labeled "flat".
This means that a player will suffer from the same long term loss as a player playing "The Rothstein Five", by betting 9.083655 units on the first roll together with the Rothstein player and then waiting for the Rothstein player to resolve, before betting another 9.083655 units.
However, this also means that the Rothstein player will enjoy more rolls than the flat betting player: approx. 2 rolls on average and, due to variance, in some cases up to five rolls.
On a per roll basis the parameters are
wager: 4.483700 units
expected value: -0.235993 units
house edge: 0.052632
average # rolls for Rothstein: 2.025852
EXPECTED VALUE for 2.025852 rolls: -0.478087 units
In the simulation, see the entries labeled "flat rolls".
This means that, if the flat betting player doesn't want to wait for the Rothstein player to resolve, the player may instead bet 4.483700 units each roll, together with the Rothstein player's rolls, and will not lose any more money than before.
Remark: The expected value has to be multiplied by the average number of Rothstein rolls to be comparable with Rothstein as well as with the 9.083655 units flat betting.
NOTE: The expected values, or more precisely the expected returns, in all four cases (Rothstein observed, Rothstein, flat, flat roll) are identical.
Conclusion
As demonstrated, "The Rothstein Five" is a game of its own. It has its own amount wagered and its own house edge. The amount wagered is not instantly obvious when observing the player's action. Only being able to spot the actual amount lost, reveals the size of the wager. The consequence of this "veiled" wager is that the house edge of "The Rothstein Five" is much smaller than the house edge of the base game, upon which Rothstein is brought to life.
In the simulation provided, four different courses of action, connected to "The Rothstein Five", are compared:
- An observer, watching the individual action of the Rothstein player.
- The Rothstein player, knowing about the winning cases, the losing cases and therefore about the actual wager of the Rothstein sequence.
- A flat betting player, wagering the average of the bets of the Rothstein player, on a single roll for each Rothstein.
- A flat betting player, wagering the adjusted average of the bets of the Rothstein player, on each individual roll together with the Rothstein player.
The casino, however, would very probably not offer "The Rothstein Five" to their patrons due to the horribly low house edge.
Players, on the other hand, can employ "The Rothstein Five" on their own and make use of the low house edge as long as they are willing to afford an amount of 57 units per game.
Outlook
Rothstein is one example among the class of finite betting sequences with negative progression, also called progression on losses. The method of analysis used here also applies to other sequences within this class, for example to the Martingale, which supposedly is the most famous among them.
It is straight forward to apply this method of analysis as well to the complementary class, the finite betting sequences with positive progression, also called progression on winnings. The probably most famous example here might be the Parlay, also called Paroli. With positive progression, the amount wagered remains the initial bet throughout the sequence but, as the downside, the house edge rises rapidly. This higher house edge makes games of this class more appealing to casinos. Examples are: Double Action Roulette, Double Ball Roulette, 2G$ side bet (any two greens in a row).
The Excel simulation.
I am posting the contents of my Excel sheet as a HEX encoded dump in two localized versions:
- Decimal comma and semicolon-separated formula parameters. (Used and tested by me.)
- Decimal point and comma-separated formula parameters. (Sorry, I'm not able to test it; please report any problems to me.)
3d 22 54 48 45 20 52 4f 54 48 53 54 45 49 4e 20 22 26 56 4c 4f 4f 4b 55 50 28 42 32 3b 41 43 3a 41 44 3b 32 3b 30 29 09 64 65 66 69 6e 65 64 09 68 65 75 72 69 73 74 69 63 73 09 74 68 65 6f 72 65 74 69 63 61 6c 09 64 65 76 69 61 74 69 6f 6e 09 66 69 6c 74 65 72 20 74 6f 20 63 6f 6d 70 61 72 65 09 09 6e 75 6d 62 65 72 09 63 68 61 6e 63 65 09 73 74 65 70 28 52 6f 74 68 73 74 65 69 6e 29 09 62 65 74 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 77 69 6e 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 6c 6f 73 65 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 69 6e 74 65 72 6d 65 64 69 61 74 65 28 6c 6f 73 73 20 61 63 63 75 6d 75 6c 61 74 65 64 29 09 69 6e 74 65 72 6d 65 64 69 61 74 65 28 72 65 66 75 6e 64 65 64 29 09 69 6e 74 65 72 6d 65 64 69 61 74 65 28 70 6c 61 75 73 69 62 69 6c 69 74 79 20 63 68 65 63 6b 29 09 62 65 74 28 52 6f 74 68 73 74 65 69 6e 29 09 77 69 6e 28 52 6f 74 68 73 74 65 69 6e 29 09 6c 6f 73 65 28 52 6f 74 68 73 74 65 69 6e 29 09 62 65 74 28 66 6c 61 74 29 09 77 69 6e 28 66 6c 61 74 29 09 6c 6f 73 65 28 66 6c 61 74 29 09 62 65 74 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 77 69 6e 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 6c 6f 73 65 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 67 72 6f 75 70 69 6e 67 09 66 69 6e 09 09 09 0a 6d 61 78 69 6d 75 6d 20 23 20 73 74 65 70 73 09 35 09 09 09 09 63 6f 6d 6d 6f 6e 09 09 3d 49 4e 54 28 52 41 4e 44 28 29 2a 33 38 29 2d 31 09 3d 49 4e 54 28 28 48 32 2b 31 37 29 2f 31 38 29 09 31 09 3d 32 5e 4a 32 2d 31 09 3d 28 49 32 3d 31 29 2a 4b 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 4b 32 09 3d 49 46 28 4a 32 3d 31 3b 4d 32 3b 4e 31 2b 4d 32 29 09 3d 28 4c 32 3e 2d 4e 32 29 2a 2d 4e 32 09 3d 4c 32 2d 4f 32 2d 52 32 09 3d 28 4a 32 3d 31 29 2a 24 44 24 33 09 3d 28 49 32 3d 31 29 2a 4a 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 28 4a 32 3d 24 42 24 32 29 2a 24 44 24 33 09 3d 28 4a 32 3d 31 29 2a 24 44 24 31 30 09 3d 28 49 32 3d 31 29 2a 54 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 54 32 09 3d 44 31 30 2f 44 37 09 3d 28 49 32 3d 31 29 2a 57 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 57 32 09 30 09 09 09 31 09 28 4e 29 4f 4e 45 0a 6d 61 78 69 6d 75 6d 20 77 61 67 65 72 28 52 6f 74 68 73 74 65 69 6e 29 09 09 09 3d 31 2b 28 32 3c 3d 42 32 29 2a 33 2b 28 33 3c 3d 42 32 29 2a 37 2b 28 34 3c 3d 42 32 29 2a 31 35 2b 28 35 3c 3d 42 32 29 2a 33 31 2b 28 36 3c 3d 42 32 29 2a 36 33 2b 28 37 3c 3d 42 32 29 2a 31 32 37 2b 28 38 3c 3d 42 32 29 2a 32 35 35 2b 28 39 3c 3d 42 32 29 2a 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 31 30 32 33 09 09 63 6f 6d 6d 6f 6e 09 09 3d 49 4e 54 28 52 41 4e 44 28 29 2a 33 38 29 2d 31 09 3d 49 4e 54 28 28 48 33 2b 31 37 29 2f 31 38 29 09 3d 49 46 28 49 32 3d 31 3b 31 3b 49 46 28 4a 32 3d 24 42 24 32 3b 31 3b 4a 32 2b 31 29 29 09 3d 32 5e 4a 33 2d 31 09 3d 28 49 33 3d 31 29 2a 4b 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 4b 33 09 3d 49 46 28 4a 33 3d 31 3b 4d 33 3b 4e 32 2b 4d 33 29 09 3d 28 4c 33 3e 2d 4e 33 29 2a 2d 4e 33 09 3d 4c 33 2d 4f 33 2d 52 33 09 3d 28 4a 33 3d 31 29 2a 24 44 24 33 09 3d 28 49 33 3d 31 29 2a 4a 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 28 4a 33 3d 24 42 24 32 29 2a 24 44 24 33 09 3d 28 4a 33 3d 31 29 2a 24 44 24 31 30 09 3d 28 49 33 3d 31 29 2a 54 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 54 33 09 3d 57 32 09 3d 28 49 33 3d 31 29 2a 57 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 57 33 09 3d 49 46 28 4a 33 3d 31 3b 4d 4f 44 28 5a 32 2b 31 3b 32 29 3b 5a 32 29 09 09 09 32 09 54 57 4f 0a 20 09 09 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 33 09 54 48 52 45 45 0a 23 20 72 6f 6c 6c 73 20 70 6c 61 79 65 64 09 09 3d 43 4f 55 4e 54 49 46 28 57 3a 57 3b 22 3e 30 22 29 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 34 09 46 4f 55 52 0a 23 20 52 6f 74 68 73 74 65 69 6e 73 20 70 6c 61 79 65 64 09 09 3d 43 4f 55 4e 54 49 46 28 4a 3a 4a 3b 22 3d 31 22 29 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 35 09 46 49 56 45 0a 61 76 65 72 61 67 65 20 23 20 72 6f 6c 6c 73 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 35 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 31 30 3c 3d 42 32 29 2a 31 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 37 2d 44 37 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 36 09 53 49 58 0a 20 09 09 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 37 09 53 45 56 45 4e 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 53 55 4d 28 4b 3a 4b 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 38 09 45 49 47 48 54 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 39 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 33 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 37 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 35 2b 32 30 2f 33 38 2a 28 31 35 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 31 2b 32 30 2f 33 38 2a 28 33 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 36 33 2b 32 30 2f 33 38 2a 28 36 33 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 32 37 2b 32 30 2f 33 38 2a 28 31 32 37 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 32 35 35 2b 32 30 2f 33 38 2a 28 32 35 35 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 35 31 31 2b 32 30 2f 33 38 2a 28 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 31 30 32 33 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 31 30 2d 44 31 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 39 09 4e 49 4e 45 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 31 30 09 54 45 4e 0a 74 6f 74 61 6c 20 67 61 69 6e 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 53 55 4d 28 4c 3a 4c 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 53 55 4d 28 4d 3a 4d 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 31 32 2b 43 31 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 31 34 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 2d 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 2d 33 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 2d 37 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 35 2b 32 30 2f 33 38 2a 28 2d 31 35 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 31 2b 32 30 2f 33 38 2a 28 2d 33 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 36 33 2b 32 30 2f 33 38 2a 28 2d 36 33 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 32 37 2b 32 30 2f 33 38 2a 28 2d 31 32 37 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 32 35 35 2b 32 30 2f 33 38 2a 28 2d 32 35 35 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 35 31 31 2b 32 30 2f 33 38 2a 28 2d 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 30 32 33 2b 32 30 2f 33 38 2a 2d 31 30 32 33 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 31 35 2d 44 31 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 31 35 2f 2d 43 31 30 09 3d 44 31 35 2f 2d 44 31 30 09 3d 43 31 36 2d 44 31 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 31 32 2b 43 31 33 29 2f 2d 43 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 53 55 4d 28 51 3a 51 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 31 39 2f 43 36 09 3d 44 33 09 3d 43 32 30 2d 44 32 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 67 61 69 6e 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 53 55 4d 28 52 3a 52 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 53 55 4d 28 53 3a 53 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 32 32 2b 43 32 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 32 34 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 28 31 3d 42 32 29 2a 28 2d 31 29 2b 28 32 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 32 2b 32 30 2f 33 38 2a 28 28 32 3d 42 32 29 2a 28 2d 34 29 2b 28 33 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 28 33 3d 42 32 29 2a 28 2d 31 31 29 2b 28 34 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 34 2b 32 30 2f 33 38 2a 28 28 34 3d 42 32 29 2a 28 2d 32 36 29 2b 28 35 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 35 2b 32 30 2f 33 38 2a 28 28 35 3d 42 32 29 2a 28 2d 35 37 29 2b 28 36 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 36 2b 32 30 2f 33 38 2a 28 28 36 3d 42 32 29 2a 28 2d 31 32 30 29 2b 28 37 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 28 37 3d 42 32 29 2a 28 2d 32 34 37 29 2b 28 38 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 38 2b 32 30 2f 33 38 2a 28 28 38 3d 42 32 29 2a 28 2d 35 30 32 29 2b 28 39 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 39 2b 32 30 2f 33 38 2a 28 28 39 3d 42 32 29 2a 28 2d 31 30 31 33 29 2b 28 31 30 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 31 30 2b 32 30 2f 33 38 2a 28 28 31 30 3d 42 32 29 2a 28 2d 32 30 33 36 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 32 35 2d 44 32 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 32 35 2f 2d 43 32 30 09 3d 44 32 35 2f 2d 44 33 09 3d 43 32 36 2d 44 32 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 32 32 2b 43 32 33 29 2f 2d 43 31 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 66 6c 61 74 29 09 09 3d 53 55 4d 28 54 3a 54 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 66 6c 61 74 29 09 09 3d 43 32 39 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 33 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 37 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 35 2b 32 30 2f 33 38 2a 28 31 35 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 31 2b 32 30 2f 33 38 2a 28 33 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 36 33 2b 32 30 2f 33 38 2a 28 36 33 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 32 37 2b 32 30 2f 33 38 2a 28 31 32 37 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 32 35 35 2b 32 30 2f 33 38 2a 28 32 35 35 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 35 31 31 2b 32 30 2f 33 38 2a 28 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 31 30 32 33 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 33 30 2d 44 33 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 67 61 69 6e 28 66 6c 61 74 29 09 09 3d 53 55 4d 28 55 3a 55 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 66 6c 61 74 29 09 09 3d 53 55 4d 28 56 3a 56 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 66 6c 61 74 29 09 09 3d 43 33 32 2b 43 33 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 66 6c 61 74 29 09 09 3d 43 33 34 2f 43 36 09 3d 44 31 30 2a 28 31 38 2f 33 38 2d 32 30 2f 33 38 29 09 3d 43 33 35 2d 44 33 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 66 6c 61 74 29 09 09 3d 43 33 35 2f 2d 43 33 30 09 3d 44 33 35 2f 2d 44 33 30 09 3d 43 33 36 2d 44 33 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 33 32 2b 43 33 33 29 2f 2d 43 32 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 53 55 4d 28 57 3a 57 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 33 39 2f 43 35 09 3d 44 33 30 2f 44 37 09 3d 43 34 30 2d 44 34 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 67 61 69 6e 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 53 55 4d 28 58 3a 58 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 53 55 4d 28 59 3a 59 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 34 32 2b 43 34 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 34 34 2f 43 35 09 3d 44 34 30 2a 28 31 38 2f 33 38 2d 32 30 2f 33 38 29 09 3d 43 34 35 2d 44 34 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 34 35 2f 2d 43 34 30 09 3d 44 34 35 2f 2d 44 34 30 09 3d 43 34 36 2d 44 34 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 34 32 2b 43 34 33 29 2f 2d 43 33 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09
3d 22 54 48 45 20 52 4f 54 48 53 54 45 49 4e 20 22 26 56 4c 4f 4f 4b 55 50 28 42 32 2c 41 43 3a 41 44 2c 32 2c 30 29 09 64 65 66 69 6e 65 64 09 68 65 75 72 69 73 74 69 63 73 09 74 68 65 6f 72 65 74 69 63 61 6c 09 64 65 76 69 61 74 69 6f 6e 09 66 69 6c 74 65 72 20 74 6f 20 63 6f 6d 70 61 72 65 09 09 6e 75 6d 62 65 72 09 63 68 61 6e 63 65 09 73 74 65 70 28 52 6f 74 68 73 74 65 69 6e 29 09 62 65 74 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 77 69 6e 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 6c 6f 73 65 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 69 6e 74 65 72 6d 65 64 69 61 74 65 28 6c 6f 73 73 20 61 63 63 75 6d 75 6c 61 74 65 64 29 09 69 6e 74 65 72 6d 65 64 69 61 74 65 28 72 65 66 75 6e 64 65 64 29 09 69 6e 74 65 72 6d 65 64 69 61 74 65 28 70 6c 61 75 73 69 62 69 6c 69 74 79 20 63 68 65 63 6b 29 09 62 65 74 28 52 6f 74 68 73 74 65 69 6e 29 09 77 69 6e 28 52 6f 74 68 73 74 65 69 6e 29 09 6c 6f 73 65 28 52 6f 74 68 73 74 65 69 6e 29 09 62 65 74 28 66 6c 61 74 29 09 77 69 6e 28 66 6c 61 74 29 09 6c 6f 73 65 28 66 6c 61 74 29 09 62 65 74 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 77 69 6e 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 6c 6f 73 65 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 67 72 6f 75 70 69 6e 67 09 66 69 6e 09 09 09 0a 6d 61 78 69 6d 75 6d 20 23 20 73 74 65 70 73 09 35 09 09 09 09 63 6f 6d 6d 6f 6e 09 09 3d 49 4e 54 28 52 41 4e 44 28 29 2a 33 38 29 2d 31 09 3d 49 4e 54 28 28 48 32 2b 31 37 29 2f 31 38 29 09 31 09 3d 32 5e 4a 32 2d 31 09 3d 28 49 32 3d 31 29 2a 4b 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 4b 32 09 3d 49 46 28 4a 32 3d 31 2c 4d 32 2c 4e 31 2b 4d 32 29 09 3d 28 4c 32 3e 2d 4e 32 29 2a 2d 4e 32 09 3d 4c 32 2d 4f 32 2d 52 32 09 3d 28 4a 32 3d 31 29 2a 24 44 24 33 09 3d 28 49 32 3d 31 29 2a 4a 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 28 4a 32 3d 24 42 24 32 29 2a 24 44 24 33 09 3d 28 4a 32 3d 31 29 2a 24 44 24 31 30 09 3d 28 49 32 3d 31 29 2a 54 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 54 32 09 3d 44 31 30 2f 44 37 09 3d 28 49 32 3d 31 29 2a 57 32 09 3d 28 28 49 32 3d 31 29 2d 31 29 2a 57 32 09 30 09 09 09 31 09 28 4e 29 4f 4e 45 0a 6d 61 78 69 6d 75 6d 20 77 61 67 65 72 28 52 6f 74 68 73 74 65 69 6e 29 09 09 09 3d 31 2b 28 32 3c 3d 42 32 29 2a 33 2b 28 33 3c 3d 42 32 29 2a 37 2b 28 34 3c 3d 42 32 29 2a 31 35 2b 28 35 3c 3d 42 32 29 2a 33 31 2b 28 36 3c 3d 42 32 29 2a 36 33 2b 28 37 3c 3d 42 32 29 2a 31 32 37 2b 28 38 3c 3d 42 32 29 2a 32 35 35 2b 28 39 3c 3d 42 32 29 2a 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 31 30 32 33 09 09 63 6f 6d 6d 6f 6e 09 09 3d 49 4e 54 28 52 41 4e 44 28 29 2a 33 38 29 2d 31 09 3d 49 4e 54 28 28 48 33 2b 31 37 29 2f 31 38 29 09 3d 49 46 28 49 32 3d 31 2c 31 2c 49 46 28 4a 32 3d 24 42 24 32 2c 31 2c 4a 32 2b 31 29 29 09 3d 32 5e 4a 33 2d 31 09 3d 28 49 33 3d 31 29 2a 4b 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 4b 33 09 3d 49 46 28 4a 33 3d 31 2c 4d 33 2c 4e 32 2b 4d 33 29 09 3d 28 4c 33 3e 2d 4e 33 29 2a 2d 4e 33 09 3d 4c 33 2d 4f 33 2d 52 33 09 3d 28 4a 33 3d 31 29 2a 24 44 24 33 09 3d 28 49 33 3d 31 29 2a 4a 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 28 4a 33 3d 24 42 24 32 29 2a 24 44 24 33 09 3d 28 4a 33 3d 31 29 2a 24 44 24 31 30 09 3d 28 49 33 3d 31 29 2a 54 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 54 33 09 3d 57 32 09 3d 28 49 33 3d 31 29 2a 57 33 09 3d 28 28 49 33 3d 31 29 2d 31 29 2a 57 33 09 3d 49 46 28 4a 33 3d 31 2c 4d 4f 44 28 5a 32 2b 31 2c 32 29 2c 5a 32 29 09 09 09 32 09 54 57 4f 0a 20 09 09 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 33 09 54 48 52 45 45 0a 23 20 72 6f 6c 6c 73 20 70 6c 61 79 65 64 09 09 3d 43 4f 55 4e 54 49 46 28 57 3a 57 2c 22 3e 30 22 29 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 34 09 46 4f 55 52 0a 23 20 52 6f 74 68 73 74 65 69 6e 73 20 70 6c 61 79 65 64 09 09 3d 43 4f 55 4e 54 49 46 28 4a 3a 4a 2c 22 3d 31 22 29 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 35 09 46 49 56 45 0a 61 76 65 72 61 67 65 20 23 20 72 6f 6c 6c 73 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 35 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 31 30 3c 3d 42 32 29 2a 31 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 37 2d 44 37 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 36 09 53 49 58 0a 20 09 09 09 09 09 63 6f 6d 6d 6f 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 37 09 53 45 56 45 4e 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 53 55 4d 28 4b 3a 4b 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 38 09 45 49 47 48 54 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 39 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 33 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 37 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 35 2b 32 30 2f 33 38 2a 28 31 35 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 31 2b 32 30 2f 33 38 2a 28 33 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 36 33 2b 32 30 2f 33 38 2a 28 36 33 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 32 37 2b 32 30 2f 33 38 2a 28 31 32 37 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 32 35 35 2b 32 30 2f 33 38 2a 28 32 35 35 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 35 31 31 2b 32 30 2f 33 38 2a 28 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 31 30 32 33 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 31 30 2d 44 31 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 39 09 4e 49 4e 45 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 31 30 09 54 45 4e 0a 74 6f 74 61 6c 20 67 61 69 6e 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 53 55 4d 28 4c 3a 4c 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 53 55 4d 28 4d 3a 4d 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 31 32 2b 43 31 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 31 34 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 2d 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 2d 33 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 2d 37 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 35 2b 32 30 2f 33 38 2a 28 2d 31 35 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 31 2b 32 30 2f 33 38 2a 28 2d 33 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 36 33 2b 32 30 2f 33 38 2a 28 2d 36 33 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 32 37 2b 32 30 2f 33 38 2a 28 2d 31 32 37 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 32 35 35 2b 32 30 2f 33 38 2a 28 2d 32 35 35 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 35 31 31 2b 32 30 2f 33 38 2a 28 2d 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 30 32 33 2b 32 30 2f 33 38 2a 2d 31 30 32 33 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 31 35 2d 44 31 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 52 6f 74 68 73 74 65 69 6e 20 6f 62 73 65 72 76 65 64 29 09 09 3d 43 31 35 2f 2d 43 31 30 09 3d 44 31 35 2f 2d 44 31 30 09 3d 43 31 36 2d 44 31 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 31 32 2b 43 31 33 29 2f 2d 43 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 53 55 4d 28 51 3a 51 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 31 39 2f 43 36 09 3d 44 33 09 3d 43 32 30 2d 44 32 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 67 61 69 6e 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 53 55 4d 28 52 3a 52 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 53 55 4d 28 53 3a 53 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 32 32 2b 43 32 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 32 34 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 28 31 3d 42 32 29 2a 28 2d 31 29 2b 28 32 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 32 2b 32 30 2f 33 38 2a 28 28 32 3d 42 32 29 2a 28 2d 34 29 2b 28 33 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 28 33 3d 42 32 29 2a 28 2d 31 31 29 2b 28 34 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 34 2b 32 30 2f 33 38 2a 28 28 34 3d 42 32 29 2a 28 2d 32 36 29 2b 28 35 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 35 2b 32 30 2f 33 38 2a 28 28 35 3d 42 32 29 2a 28 2d 35 37 29 2b 28 36 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 36 2b 32 30 2f 33 38 2a 28 28 36 3d 42 32 29 2a 28 2d 31 32 30 29 2b 28 37 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 28 37 3d 42 32 29 2a 28 2d 32 34 37 29 2b 28 38 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 38 2b 32 30 2f 33 38 2a 28 28 38 3d 42 32 29 2a 28 2d 35 30 32 29 2b 28 39 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 39 2b 32 30 2f 33 38 2a 28 28 39 3d 42 32 29 2a 28 2d 31 30 31 33 29 2b 28 31 30 3c 3d 42 32 29 2a 31 38 2f 33 38 2a 31 30 2b 32 30 2f 33 38 2a 28 28 31 30 3d 42 32 29 2a 28 2d 32 30 33 36 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 32 35 2d 44 32 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 52 6f 74 68 73 74 65 69 6e 29 09 09 3d 43 32 35 2f 2d 43 32 30 09 3d 44 32 35 2f 2d 44 33 09 3d 43 32 36 2d 44 32 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 32 32 2b 43 32 33 29 2f 2d 43 31 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 66 6c 61 74 29 09 09 3d 53 55 4d 28 54 3a 54 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 66 6c 61 74 29 09 09 3d 43 32 39 2f 43 36 09 3d 31 38 2f 33 38 2a 31 2b 32 30 2f 33 38 2a 28 31 2b 28 32 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 2b 32 30 2f 33 38 2a 28 33 2b 28 33 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 37 2b 32 30 2f 33 38 2a 28 37 2b 28 34 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 35 2b 32 30 2f 33 38 2a 28 31 35 2b 28 35 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 33 31 2b 32 30 2f 33 38 2a 28 33 31 2b 28 36 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 36 33 2b 32 30 2f 33 38 2a 28 36 33 2b 28 37 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 31 32 37 2b 32 30 2f 33 38 2a 28 31 32 37 2b 28 38 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 32 35 35 2b 32 30 2f 33 38 2a 28 32 35 35 2b 28 39 3c 3d 42 32 29 2a 28 31 38 2f 33 38 2a 35 31 31 2b 32 30 2f 33 38 2a 28 35 31 31 2b 28 31 30 3c 3d 42 32 29 2a 31 30 32 33 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 09 3d 43 33 30 2d 44 33 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 67 61 69 6e 28 66 6c 61 74 29 09 09 3d 53 55 4d 28 55 3a 55 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 66 6c 61 74 29 09 09 3d 53 55 4d 28 56 3a 56 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 66 6c 61 74 29 09 09 3d 43 33 32 2b 43 33 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 66 6c 61 74 29 09 09 3d 43 33 34 2f 43 36 09 3d 44 31 30 2a 28 31 38 2f 33 38 2d 32 30 2f 33 38 29 09 3d 43 33 35 2d 44 33 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 66 6c 61 74 29 09 09 3d 43 33 35 2f 2d 43 33 30 09 3d 44 33 35 2f 2d 44 33 30 09 3d 43 33 36 2d 44 33 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 33 32 2b 43 33 33 29 2f 2d 43 32 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 75 6e 69 74 73 20 62 65 74 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 53 55 4d 28 57 3a 57 29 09 09 09 62 65 74 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 61 76 65 72 61 67 65 20 77 61 67 65 72 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 33 39 2f 43 35 09 3d 44 33 30 2f 44 37 09 3d 43 34 30 2d 44 34 30 09 77 61 67 65 72 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 20 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 67 61 69 6e 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 53 55 4d 28 58 3a 58 29 09 09 09 67 61 69 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 6c 6f 73 73 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 53 55 4d 28 59 3a 59 29 09 09 09 6c 6f 73 73 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 74 6f 74 61 6c 20 72 65 74 75 72 6e 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 34 32 2b 43 34 33 09 09 09 72 65 74 75 72 6e 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 65 78 70 65 63 74 65 64 20 76 61 6c 75 65 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 34 34 2f 43 35 09 3d 44 34 30 2a 28 31 38 2f 33 38 2d 32 30 2f 33 38 29 09 3d 43 34 35 2d 44 34 35 09 65 76 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 68 6f 75 73 65 20 65 64 67 65 28 66 6c 61 74 20 72 6f 6c 6c 73 29 09 09 3d 43 34 35 2f 2d 43 34 30 09 3d 44 34 35 2f 2d 44 34 30 09 3d 43 34 36 2d 44 34 36 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 0a 5f 5f 5f 5f 5f 20 28 68 6f 75 73 65 20 65 64 67 65 2c 20 62 61 73 65 64 20 6f 6e 20 74 6f 74 61 6c 73 20 6f 6e 6c 79 29 09 09 3d 28 43 34 32 2b 43 34 33 29 2f 2d 43 33 39 09 09 09 68 65 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09
Please see here the instructions how to rebuild the content in your own Excel sheet.
- Open the spoiler then left-click and swipe over the first few characters, moving the mouse to the empty second line until the HEX content is being highlighted. Release mouse button.
HINT: It is a bit tricky to position the mouse pointer in the right spot in front of the first character. You know you found that spot if the HEX content becomes highlighted. - CTRL-C: copy the HEX content.
-
Choose any HEX to ASCII converter to retrieve the readable Excel content.
I chose https://www.utilities-online.info/hex-to-ascii applying the following steps:
a) Left-click into the left frame labeled "HEX".
b) CTRL-V: Paste the HEX content to the HEX frame.
c) Submit "Convert". The readable text is now displayed in the right frame.
d) Left-click into the right frame. -
Follow the final steps to fill in your Excel sheet:
CTRL-A: Select all text. - CTRL-C: Copy all text.
- Go to a fresh blank Excel sheet.
- Click cell A1. Only select it, do not open it for editing.
- CTRL-V: Paste all text to Excel. The cells are filled accordingly.
- Decide how many rolls you would like to simulate (e.g. 3800).
- CTRL-G: Go To dialog opens.
- In field "Reference" enter the target cell for column H. (e.g. H3801) REMARK: The +1 takes the column heading into account.
- Leave with RETURN or "OK".
- Set an "x" into the curent cell (H3801).
- CTRL-<up-key> gets you back to cell H3.
- CTRL-SHIFT-<right-key> selects cells H3 through Z3.
- CTRL-C copies this portion of the simulation.
- CTRL-SHIFT-<down-key> selects all relevant cells down to the chosen row (e.g. H3 through Z3801).
- RETURN pastes into all cells accordingly.
- Select row 1.
- Inside row 1 right-click and choose "Format Cells...".
- Select "Alignment" and set "Horizontal Alignment" to "Center"; then in "Orientation" left-click the top diamond.
- Leave with "OK".
- Left-click the top left corner of the spread sheet to select all cells. Remark: It is the square connecting row label "1" and column label "A".
- Move mouse pointer over the separating line between column label "A" and column label "B" until the pointer turns into the <-|-> sign and double-left-click.
Columns H through AA hold all the individual components of the simulation. The columns can be grouped as follows:
Grouping the rows belonging to the same Rothstein sequence:
| H-J | roulette rolls, even money events, the Rothstein sequence |
| K-M | wager, gain, and loss, as perceived by an observer of the action |
| N-P | intermediate values cencerning the refund, including the plausibility check |
| Q-S | wager, gain, and loss of "The Rothstein Five" |
| T-V | wager, gain, and loss of flat betting as analyzed, using the established method |
| W-Y | wager, gain, and loss of flat betting on each roll |
| Z | assistance for highlighting Rothstein sequences (see below) |
Grouping the rows belonging to the same Rothstein sequence:
- Select cell H2.
- Press an hold CTRL-SHIFT, then press <down-key> once, then press <right-key> once, and release CTRL-SHIFT to select all simulation cells.
- In menu "Home" open the selection "Conditional Formatting".
- Select "New Rule...".
- In "Style:" choose "Classic"
- In the selection saying "Format only ..." choose "Use a formula to determine which cells to format".
- In the next field enter "=$Z2".
- In "Format with:" choose "Custom Format...".
- In the new dialog "Format Cells" choose selection "Fill".
- In "Background color:" replace "No Color" by a light grey tone.
- Leave "Format Cells" with "OK", leave "New Formatting Rule" with "OK".
Columns A through E contain the results of the simulated game parameters and their theoretical counterparts.
For easy comparison the categories in column F may be used by switching on the filter for the columns A through F.
The value in cell B2 is preset to 5 for "The Rothstein Five" and may be replaced by a different value out of 1 through 10 to simulate Rothsteins of that length.
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
April 9th, 2024 at 3:58:55 AM
permalink
If I’ve followed this correctly, you are converting a betting progression into a single side bet at the table (eg, 5 step Rothstein) and then showing that, if it’s a negative betting progression, the house edge for the side bet is lower than the betting progression (and vice versa for a positive betting progression). Is that right?
And also showing that the expected loss is identical in the side bet and progression?
This conclusion seems uncontroversial to me.
House edge is an imperfect indicator of wager “value” since its denominator is simply the initial wager amount. Expected loss is a better way to compare wagers.
And also showing that the expected loss is identical in the side bet and progression?
This conclusion seems uncontroversial to me.
House edge is an imperfect indicator of wager “value” since its denominator is simply the initial wager amount. Expected loss is a better way to compare wagers.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
April 9th, 2024 at 6:16:43 AM
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Nearly: The house edge for the side bet, which again is the progression, is lower than the house edge of the individual bet, upon which the progression is built. (For a positive progression it is higher than the house edge of the individual bet.)Quote: unJonIf I’ve followed this correctly, you are converting a betting progression into a single side bet at the table (eg, 5 step Rothstein) and then showing that, if it’s a negative betting progression, the house edge for the side bet is lower than the betting progression (and vice versa for a positive betting progression). Is that right?
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Exactly. The expected loss of the side bet is identical to that of the progression. (Again because the side bet is the progression.)Quote:[...]
And also showing that the expected loss is identical in the side bet and progression?
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Does this still hold, even with my remarks above?Quote:[...]
This conclusion seems uncontroversial to me.
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I agree: Expected loss is the amount the player actually loses over a certain period of play and therefore is best suited to compare games.Quote:[...]
House edge is an imperfect indicator of wager “value” since its denominator is simply the initial wager amount. Expected loss is a better way to compare wagers.
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Nevertheless, I would argue that wager and house edge may still be used to compare games, as long as only flat bet games are compared. In my analysis there are the obvious flat bet games of 9.083655 units and of 4.483700 units, both having a 0.052632 house edge. The 57 units side bet is a flat bet game as well, having a house edge of 0.008387.
These three shall not be compared to any sequence of individual progressive bets, because the "refunding effect" taking place within the progression, is not considered. By ignoring the refund in the cases of a winning roll (within the same sequence), amounts are added to the total amount wagered which are not wagers (because winning hands don't need a wager).
For a finite progression, it is always possible to calculate the equivalent flat bet game which then is comparable to other flat bet games.
"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
April 9th, 2024 at 7:36:11 AM
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We are in agreement. I think you could also use this analysis in sports betting where parlays are common to show that while the house edge in the parlay is higher than manually parlaying, the expected loss is the same.
Question for you is whether you calculate the Element of Risk for your Rothstein side bet as higher, lower or the same as the Element of Risk for manually going through a Rothstein progression. In other words does EoR fix the issue inherent in House Edge calculations for non-flat bet games.
Question for you is whether you calculate the Element of Risk for your Rothstein side bet as higher, lower or the same as the Element of Risk for manually going through a Rothstein progression. In other words does EoR fix the issue inherent in House Edge calculations for non-flat bet games.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
April 9th, 2024 at 10:14:39 AM
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I'm not familiar with sports betting, but as far as I understand the Wizard's explanation of parlay sports bets, they basically work like any parlay, e.g. in roulette.Quote: unJonWe are in agreement. I think you could also use this analysis in sports betting where parlays are common [...]
The sports betting explanation ...
Quote: Sports Betting[...]
PARLAYS
Parlay bets are a good and popular way to potentially win big with a small wager. The way they work is the bettor picks two or more events, which all must win or push to win the bet. One or more loss will cause the whole parlay bet to lose.
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link to Wizard of Odds
... might translate to roulette as follows:
PARLAYS
Parlay bets are a good and popular way to potentially win big with a small wager. The way
they work is the bettor picks a number of two or more successive rolls, which all must
win [...] to win the bet. One [...] loss will cause the whole parlay bet to lose.
My point is, that a parlay offered as single game is exactly the same as manually playing the parlay.Quote: unJon[...] to show that while the house edge in the parlay is higher than manually parlaying, the expected loss is the same. [...]
The expected loss is the same in both cases.
Furthermore the wager is the same in both cases. Although it seems that the manual player wagers different amounts, in the case of losing (within the same sequence), the player loses back all these intermediate "wins" togehter with the player's initial bet. Effectively the player only loses the initial wager. A parlay ends up to be another Bernoulli game: The player either loses the initial wager if the progression fails (because any of the individual bets fails) or the player wins the maximum amount in the only other case where the progression succeeds (because all the individual bets succeeded).
Summary: In positive progressions intermediate wins are lost back, which is the complement to negative progressions, where intermediate losses are refunded.
I guess you are referring to this definition:Quote: unJon[...] Question for you is whether you calculate the Element of Risk for your Rothstein side bet [...]
I hope I interpret this properly: Again the term "total amount bet" is used which, as I understand it, here refers to all the amounts that a player puts in the betting spot during a progressive sequence. As I explained, this sum of amounts is not the player's wager, because it does not consider the refunding effect in negative progressions (or the losing-back effect in positive progressions, respectively).Quote: Glossary[...]
ELEMENT OF RISK
The ratio of the expected player loss to the total amount bet. This is a good measure of comparing the value of one bet against another. For games in which there is no raising the element of risk will be the same as the house edge. However for games in which the player may increase his bet the element of risk will be lower. For example in Caribbean Stud Poker the house edge is 5.22%. However the average amount bet per hand is 2.0445 times the initial bet, so the element of risk would be 0.0522/2.0445 = 2.55%.
[...]link to Wizard of Odds
Maybe the Wizard can comment ...
Being offered the Rothstein as a side bet or playing it through manually is of no difference. Neither in Element Of Risk, if refunding is considered, nor in the House Edge. The established methods of analyzing non-flat bet games calculate the parameters for individual rolls, not those for the progression as a whole. The seemingly inherent issue in House Edge calculations can only be solved by taking the effects of refunding and losing-back into account. This then consequently results in the parameters of the Bernoulli flat bet game, having a House Edge (or an Element of Risk) which is smaller than the house edge of the individual rolls upon which the progression is built.Quote: unJon[...] as higher, lower or the same as the Element of Risk for manually going through a Rothstein progression. In other words does EoR fix the issue inherent in House Edge calculations for non-flat bet games.
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"When it comes to probability and statistics, intuition is a bad advisor. Don't speculate. Calculate." - a math textbook author (name not recalled)
