Fortune X Poker

Quote: Mental

It would be nice to confirm that the game can go directly from 12x to 2x on the next hand. If there was one dead hand after every bonus round, our models would be far off the mark.

Quote: Wizard

So, a rather crude spreadsheet YH gave me shows a feature probability of 11%. I questioned him on the difference with the 10.96% on the game screen and he said something to the effect of 10.96% being an average over multiple different games.

Quote: Mental

I did a new calculation with a substantially simpler game of my own creation. I designed the game to have a cutoff value for the last decision point that is exactly neutral to the EV for a vulture (NV). In other words, a vulture could get the same absolute EV by accepting the penultimate multiplier or by playing at 1X and then taking the forced multiplier on the next game.

I then calculate the RTP of the game via a billion game MC simulation two times with just one difference.

(A) I set the last cutoff point just below this neutral value, NV. This terminates the bonus round whenever the 'deal' comes up with the EV equal to the NV.

(B) I set the last cutoff point just above NV. This extends the bonus round whenever the 'deal' comes up with the EV equal to the NV. Bonus rounds are thus less frequent.

Anyone care to guess how the RTPs compare? Is the RTP higher under scenario (A) than under scenario (B), lower, or the same?
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Quote: Wizard

... I love everybody there [VideoPoker.com] and thank them very much for all their cooperation through the years ...

Quote: Wizard

I just updated my Fortune X page to show detailed calculations, that arrive at the 99.01% RTP. I fully agree if the feature probability is 11.00%, then that is what the rules screens should say. However, I hope I won't get anybody upset if I say in public they can be a little lazy in getting their rule screens correct at VideoPoker.com. The third page in their rules seems to be for a totally different game. However, I love everybody there and thank them very much for all their cooperation through the years and hope if they read this they will file it under peer review.
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Quote: Mental

I did a new calculation with a substantially simpler game of my own creation. I designed the game to have a cutoff value for the last decision point that is exactly neutral to the EV for a vulture (NV). In other words, a vulture could get the same absolute EV by accepting the penultimate multiplier or by playing at 1X and then taking the forced multiplier on the next game.

I then calculate the RTP of the game via a billion game MC simulation two times with just one difference.

(A) I set the last cutoff point just below this neutral value, NV. This terminates the bonus round whenever the 'deal' comes up with the EV equal to the NV.

(B) I set the last cutoff point just above NV. This extends the bonus round whenever the 'deal' comes up with the EV equal to the NV. Bonus rounds are thus less frequent.

Anyone care to guess how the RTPs compare? Is the RTP higher under scenario (A) than under scenario (B), lower, or the same?
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Quote: drrock

Getting back to Mental's question, so long as the space between different cutoffs contain no hands having EVs between them, there will be no difference in EV by moving the cutoff. However, if you move an optimal cutoff up or down, you cannot get a better than optimal result, and you might get a worse one.

Quote: gordonm888

Illustration of the point being discussed

Given
I am at the 8X multiplier level and have been dealt a hand of AQJ suited that has an EV of 1.411656 – above the Wizard’s threshold level of 1.41109914. I have to decide whether to accept or reject the 8x multiplier.

In this scenario, we will consider the value of the next two dealt hands no matter the decision on multipliers. This is equivalent to the situation in a simulation of a set number of trials, as if a player has made a decision to play a certain number of hands no matter what decisions are made.



So, when considering the subsequent dealt hand for both options, rejecting the 8x multiplier has more value for AQJ(suited) even though AQJs has an EV value above the Wizard’s threshold. I believe this has been Mental's point, and what drrock may be trying to test with his simulation.
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Quote: Mission146

I'm confused; if the player accepts the 8x multiplier, why is the assumption that he must play a hand after that one? While the player probably will, in most cases, the vulture would not.
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Quote: Mental

You may have missed my point: The NV EV is the EV after the deal where a vulture gets the same EV for accepting or declining the penultimate multiplier. Either choice is optimal for EV. Declining leads to a different variance, but the same expected monetary gain. The question is if the RTPs are the same for playing the game through many cycles.

Quote: Mission146

I'm confused; if the player accepts the 8x multiplier, why is the assumption that he must play a hand after that one? While the player probably will, in most cases, the vulture would not.
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Quote: drrock

Quote: Mental

You may have missed my point: The NV EV is the EV after the deal where a vulture gets the same EV for accepting or declining the penultimate multiplier. Either choice is optimal for EV. Declining leads to a different variance, but the same expected monetary gain. The question is if the RTPs are the same for playing the game through many cycles.

Hey, Mental. Interesting brain-twister. It took me several readings to fully understand what you were trying to say here. I certainly did not know the particulars of the game that you had in mind with my first response, but before addressing the exact game that you are now outlining, I will first repeat what I said in my original response to you:

"However, if you have a cutoff that is optimal in a game like Fortune X, and you either increase the cutoff by a bit to induce more declines of the current multiplier or decrease the cutoff by a bit to get more acceptances of the current multiplier, two things can happen. The one thing that cannot happen is that the return increases. If the return increases, your cutoff was not optimal, whether you have designed it to be neutral or not."

This still stands for the game that you have clarified here.

The key here is not whether the optimal cutoff is also neutral for a vulture. Whether you are above or below optimal is what is important, not whether or not the cutoff is also vulture-neutral.

A side note: As you might have seen in my previous response to Mission and Gordon, the vulture's optimal cutoffs, whether knife-edge neutral or not, actually will in practice be different than the optimal cutoffs for maximizing EV with continuous play. It is possible given that the step function like the one you outlined for me of the possible hand EVs is such that the cutoffs for both vulture and continuous play could be equal, and they will probably be fairly close, but they are not guaranteed to be equal.

The calculation for an optimal vulture cutoff considers only the hands up to the end of the current bonus round. The continuous-play optimal cutoff considers also the hands after the bonus round, the most important aspect of this is how soon the next bonus round can start.

The game that you designed is an excellent example for what is above. I get that the game you designed has a cutoff of a dealer card of 2 allows a vulture the same EV whether he accepts a 20x and stops or passes on the 20x, invests one more bet and takes a shot at 36x.

And I can see that you did a billion simulations for various cutoffs. Unfortunately, the analytical EVs on this game are so close that you will not be able to determine via simulation whether for continuous play (which is what a billion-rep simulation is imitating), which cutoff is better than the other, whether there is a small difference between EVs or a difference of zero. And if there is a difference, which cutoff produces the larger EV.

On your game, for continuous play, it turns out the optimal EV occurs when you accept 20x ONLY when the dealer's card is 1. When you do that, the game EV is 0.9996540. When you have the policy of accepting 20x also on dealer cards of 2 (as well as 1), the game EV drops ever so slightly to 0.9996528. I could include a few more decimal places since Excel generally keeps 14 or 15, but it would be tough to expect a simulation to be able to tell the difference between these two EVs. I could also supply exact numbers corresponding to what your simulations were estimating, but that will have to wait, at least for now.

If you think of the acceptance percentage for a particular multiplier as a step function, you will see that with your game, the only percentages that make sense are 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, or 100%. Sure you can come up with something like 16% where you would accept when the dealer has a 1 and 60% of the time when he has a 2, but you will be incredibly lucky for that to be optimal. Of course, you could also be technically correct by saying you will accept the 20x multiplier when the dealer's card is less than or equal to 1.6, but that is only because there is no possibility of a dealer card being in the open interval (1,2).
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Quote: Mental

Quote: Mission146

I'm confused; if the player accepts the 8x multiplier, why is the assumption that he must play a hand after that one? While the player probably will, in most cases, the vulture would not.
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I am confused as to why you are even interested in the vulturing RTP. It is big enough in any case. I make roughly 0% of my income by vulturing games. I make it by finding long plays with RTP + bennies > 100%. We are trying to determine the long term RTP. I am trying to figure out if a neutral EV vulture multiplier should be generally be accepted by a long term player or not.
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Quote: Mental

I was asking whether long-term RTP changes when you move the cutoff through a neutral vulture EV (Wiz says vulture indifferent EV). If the RTP changes, is the optimum cutoff above or below this special EV? In a VP game, you will probably never find such a lucky neutral EV. That is why I invented a simple game where this is a special neutral EV value. I failed spectacularly in creating a game where there is a clear difference moving the cutoff through the vulture indifferent EV. I thought for sure there would be a clear answer one way or the other.
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Quote: Mission146

Quote: Mental

Quote: Mission146

I'm confused; if the player accepts the 8x multiplier, why is the assumption that he must play a hand after that one? While the player probably will, in most cases, the vulture would not.
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I am confused as to why you are even interested in the vulturing RTP. It is big enough in any case. I make roughly 0% of my income by vulturing games. I make it by finding long plays with RTP + bennies > 100%. We are trying to determine the long term RTP. I am trying to figure out if a neutral EV vulture multiplier should be generally be accepted by a long term player or not.
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Fair enough! In any case, this wouldn’t be my game of choice for long-term play; the variance must be obscene.
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Quote: drrock

And I can see that you did a billion simulations for various cutoffs. Unfortunately, the analytical EVs on this game are so close that you will not be able to determine via simulation whether for continuous play (which is what a billion-rep simulation is imitating), which cutoff is better than the other, whether there is a small difference between EVs or a difference of zero. And if there is a difference, which cutoff produces the larger EV.

On your game, for continuous play, it turns out the optimal EV occurs when you accept 20x ONLY when the dealer's card is 1. When you do that, the game EV is 0.9996540. When you have the policy of accepting 20x also on dealer cards of 2 (as well as 1), the game EV drops ever so slightly to 0.9996528. I could include a few more decimal places since Excel generally keeps 14 or 15, but it would be tough to expect a simulation to be able to tell the difference between these two EVs.

Quote: drrock

I think that the Wizard's calculation methods were very close if not equivalent to calculating vulture optimal cutoffs.
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Quote: Wizard

Quote: drrock

I think that the Wizard's calculation methods were very close if not equivalent to calculating vulture optimal cutoffs.
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Yes, my cutoffs are exactly what a vulture should play.

I recognize a long-term player, trying to maximize RTP, should have slightly higher cut-offs, but that's just not how I analyze casino games.
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Quote: DRich

Thank you all, this has been one of the best threads on this site for quite a while.
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Quote: drrock

Quote: Mental

I was asking whether long-term RTP changes when you move the cutoff through a neutral vulture EV (Wiz says vulture indifferent EV). If the RTP changes, is the optimum cutoff above or below this special EV? In a VP game, you will probably never find such a lucky neutral EV. That is why I invented a simple game where this is a special neutral EV value. I failed spectacularly in creating a game where there is a clear difference moving the cutoff through the vulture indifferent EV. I thought for sure there would be a clear answer one way or the other.
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Hmmmm....

I am not 100% certain, but what we have seen here thus far is that continuous-play optimal cutoffs have been more exclusive than vulture optimal cutoffs, again whether these happen to be your definition of vulture-neutral or not. By "more exclusive," I mean the multiplier acceptance probability will be lower (or, in the cases where there is no difference, at least not higher).

And I can say that is what happened in the game you invented as well, though the EVs were so, so close.

I think that the Wizard's calculation methods were very close if not equivalent to calculating vulture optimal cutoffs. I know that is what I used as initial values for cutoffs to feed into the iterative method that I used to determine what I thought should be continuous-play optimal cutoffs.

Those are the only examples that I have explored.

Other things to ponder: Is it possible to have two different consecutive hands on our spectrum that using either as a cutoff will imply the same EV? I don't know. Could slight differences in your game accomplished this? Again, I don't know. With marketable video poker games, I think it is unlikely but not 100% impossible I suppose.

Even if we figure out that games that we may define as substantially similar to Fortune X always work in a particular manner, there may be some (even minor) alteration that causes the behavior of the relationship to differ from what we see here.
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Quote: DRich

Thank you all, this has been one of the best threads on this site for quite a while.
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Quote: Mental

If you use a neutral EV cutoff point for your bonus round, you are basically saying "Let me get back to playing the base game as quickly as I can. After all, the base game has an RTP of < 50%. I want the highest possible percentage of my games to be base games, not the lucrative bonus games. Let me terminate this bonus round."

I would go in a different direction.
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Quote: Mental

Quote: DRich

Thank you all, this has been one of the best threads on this site for quite a while.
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Don't worry. Your normal programming of dice control, astrology, gamblers fallacy, roulette systems, and unsupported claims of gambling prowess will resume after this brief interruption.
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Quote: Mental

Hands in the bonus round that are not multiplied are worse losers than games outside the bonus round.
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Quote: drrock

Quote: Mental

Hands in the bonus round that are not multiplied are worse losers than games outside the bonus round.
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I am thinking it is as simple as "optimal cutoff points are higher than indifference points between two states (or multipliers) when the indifference point has an EV above the game EV," and "optimal cutoff points are lower than indifference points when the indifference point has an EV below the game EV."

Other things equal, in our search for optimal EV, if we increase the frequency of states with EVs above game EV, we raise game EV, and vice versa. EV's above 1 are what makes things vulturable if that is what you are interested in, but that is not particularly germane to this post.

The indifference point could be exactly equal to the optimal cutoff, but only in the knife-edge case if the indifference point's EV was exactly equal to the game's EV (a virtually zero-probability situation without some clever mathematical manipulation).

To the statement quoted, it is true of course, but only on average and not when considering the decision around an indifference point. All the indifference points in these examples have EVs above 1 (and above the game EV); so all such hands are winners.

And, thus, also in this example, all the optimal cutoffs are higher than the indifference points.
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Quote: drrock

Quote: Mental

Hands in the bonus round that are not multiplied are worse losers than games outside the bonus round.
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I am thinking it is as simple as "optimal cutoff points are higher than indifference points between two states (or multipliers) when the indifference point has an EV above the game EV," and "optimal cutoff points are lower than indifference points when the indifference point has an EV below the game EV."

Other things equal, in our search for optimal EV, if we increase the frequency of states with EVs above game EV, we raise game EV, and vice versa. EV's above 1 are what makes things vulturable if that is what you are interested in, but that is not particularly germane to this post.

The indifference point could be exactly equal to the optimal cutoff, but only in the knife-edge case if the indifference point's EV was exactly equal to the game's EV (a virtually zero-probability situation without some clever mathematical manipulation).

To the statement quoted, it is true of course, but only on average and not when considering the decision around an indifference point. All the indifference points in these examples have EVs above 1 (and above the game EV); so all such hands are winners.

And, thus, also in this example, all the optimal cutoffs are higher than the indifference points.
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double m = 8, M = 12, B = 0.9898078220, Y = 12.92240, C = 23.46054;
double Q = q+1.0;
double y = m*Q*(1.0-q);
y += q*q;
y *= E;
y /= 2;
y += M*B*q;
// so, y = (m*Q*(1.0-q)*E/2) + (q*q*E/2) + (M*B*q);
double c = 2*Q;
double R = (Y+y)/(C+c);

q: 0.6721  E:  2.11  I:  1.4112  O: 1.4197  R:   97.054
q: 0.6405  E:  2.21  I:  1.4112  O: 1.4170  R:   97.985
q: 0.6116  E:  2.31  I:  1.4114  O: 1.4142  R:   98.969
q: 0.5851  E:  2.41  I:  1.4114  O: 1.4114  R:  100.000
q: 0.5606  E:  2.51  I:  1.4114  O: 1.4084  R:  101.072
q: 0.5379  E:  2.61  I:  1.4114  O: 1.4052  R:  102.181
q: 0.5201  E:  2.71  I:  1.4115  O: 1.4107  R:  103.323

Quote:

An interesting mental exercise, no pun originally intended when I started this sentence.

I'm afraid it is a bit more complicated than this in order to confirm or refute the general statement here.

Quote:

I think the earlier assertion still holds:

"I am thinking it is as simple as 'optimal cutoff points are higher than indifference points between two states (or multipliers) when the indifference point has an EV above the game EV,' and 'optimal cutoff points are lower than indifference points when the indifference point has an EV below the game EV.'" [along with the later clarification: "it was the average of the two hands that needed to be above the game EV for this assertion to hold"]

Quote:

"Game EV" is determined by all 6 multipliers, their frequencies, and their respective expected values for each multiplier state. Game EV relates to all hands played with all the possible multipliers, not just the two states for which an indifference point is being calculated.

In Mental's example above, game EV is based only on hands involving the 8x and 12x multipliers, leaving out the other four. That is likely why there was an equivalence found between the parameters labeled O and I.

Quote:

Mental: "For this discussion, I assume that the fraction of 2x triggers is 0.11 and the 2x, 3x, and 5x multipliers are frozen at 8.901087, 4.889740, and 2.567263."

Unfortunately, in any actual game of this type, the 2x, 3x, and 5x cutoffs are dependent upon the 8x cutoff, and its resulting probability and the average payout when 8x is selected. So, when conditions change, the other cutoffs are also subject to change as well, which, until convergence, go into determining the overall Game EV. Even when the probability of the 2x trigger, which has nothing to do with a player decision, changed from 10.96% to 11%, all the cutoff values would change. I haven't previously reported on that, because my calculations, completed prior to the beginning of this thread, all involved a 2x probability of 10.96%, as shown, until at least a few days ago, on videopoker.com's help screen. Here it turned out not to affect any actual multiplier acceptance probabilities since the 11% cutoffs remained in the same "gaps" in video poker dealt hand EV distributions as the 10.96% cutoffs. However, it does clearly affect the game EV, because now there are a higher percentage of hands with non-1x multipliers. Once that 11% was accepted as the actual probability of a 2x appearing, I was able to reconcile the earlier 98.8784% EV that I calculated with YH's stated 99.01%.

Quote:

This may sound circular, which is why it requires iteration, in general, to find the optimal cutoffs. When calculating cutoffs for an actual game, we get calculations for frequencies and expected values for each multiplier; but these, in turn, allow us to recalculate cutoffs. When we get the same cutoffs that we started with in an iteration, we are finished.

Quote:

I can say that I am not certain if I accurately put together all the nuances in the game that Mental described. On first reading, I was anticipating that a multiplier uniformly distributed from 0 to 2 would be applied to the dealt EV, which would require a different calculation, but would have an interesting distribution that was continuous and an expected value EQUAL to the dealt hand EV. I think there would still be considerable calculation involved in the analysis if that were the case, but in this case, the probabilities of accepting the 8x multiplier would be similar to what we see in the actual Fortune X game.

With the genie and the description given,

"Imagine a 'genie' shows up after an 8x game is dealt. The genie promises to produce an exact EV after the draw no matter which cards you hold. The genie picks a random number r such that 0<r<1. The genie displays an EV=rE to the player so he can make a decision. Clearly, 0<EV<E",

it looks like the decision point perhaps is being made after the draw on an 8x hand? Although I can't tell for certain, perhaps the letter E means the dealt hand EV, and we are actually still making the decision prior to the draw on the basis of E. Either way, E seems to be about 1.0 higher than our original cutoff point, which means that 8x will be selected less often. If so, this also means that the relative frequencies of 8x and 12x will change, as well as the respective EVs when 8x and 12x are selected. If that happens, that will change the overall game EV, which will potentially change the other cutoffs, etc., etc.

Quote:

At this point, it looks like all parties have come to agreement (or agreed to disagree) as to the final resolution of how to calculate the actual Fortune X's EV; e.g., it looks like the Wizard got what he wanted and published his evaluation of Fortune X. I likely don't see all the ramifications of this additional issue but who knows how all of us increasing our vast knowledge will pay benefits in the future. ;) Cheers!

Quote: gordonm888

Does anyone want to take a stab at the cutoffs for JOB VP? We would need to use a hit frequency for the bonus feature, I guess 11% is the obvious suspect. Alternatively, we might parameterize the JOB cutoffs as a function of feature hit frequency. I think it would be fun to see to what extent the cutoffs are different than DDB.

Obviously, we'll need a table with EVs and probabilities for JOB hand categories such as Mental kindly provided for DDB.
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Quote: gordonm888

Does anyone want to take a stab at the cutoffs for JOB VP? We would need to use a hit frequency for the bonus feature, I guess 11% is the obvious suspect. Alternatively, we might parameterize the JOB cutoffs as a function of feature hit frequency. I think it would be fun to see to what extent the cutoffs are different than DDB.

Obviously, we'll need a table with EVs and probabilities for JOB hand categories such as Mental kindly provided for DDB.
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Quote: drrock

Mental, I don’t claim to be a better mathematician. You’re demonstrating lots of tricks and techniques yourself. But who is better sometimes won’t get us to a better answer. And we can all make mistakes. These always end up being longer than I would like. If you don't have a lot of time now, reread your concluding paragraph and look at (5) below. See if this raises at least a doubt as to your conclusion.
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Quote: gordonm888

Does anyone want to take a stab at the cutoffs for JOB VP? We would need to use a hit frequency for the bonus feature, I guess 11% is the obvious suspect. Alternatively, we might parameterize the JOB cutoffs as a function of feature hit frequency. I think it would be fun to see to what extent the cutoffs are different than DDB.

Obviously, we'll need a table with EVs and probabilities for JOB hand categories such as Mental kindly provided for DDB.
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Quote: drrock

I say this is troubling because we saw that the 98.98% base game for DDB became 99.01% under Fortune X. And I understand a lot of games with novel features like multipliers have an EV that is designed to be equal to or just a touch above that of the base game. Clearly 97.61% is pretty underwhelming for a 99.54% base game.
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Quote: drrock

Once that coefficient changes, even if none of the other transition probabilities (or coefficients) change, the answer for all the p's in that system of equations changes. The game EV is p1 x EV1 + p2 x EV2 + p3 x EV3 + p5 x EV5 + p8 x EV8 + p12 x EV12; so if even one p changes, the game EV changes, other things equal. And you can’t really have only one p changing, since they must add to 1. I did not see what your estimates were for each of the 12 items above in your example. I know you say they are all combined in one figure, but unless you break them out into those 12 items, I don’t think I can find precisely where we differ. But please see if any of what follows helps.

When your genie game gets an EV of 100% by changing EV8. p1, p2, p3, and p5 will also change, which is why I was suggesting iteration even if EV2, EV3, and EV5 do not change. We don't have to worry about EV1 or EV12 because they will not change, but p1 and p12 will change.
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Quote: drrock

(6) If you need any more convincing, please try to calculate separately the 12 components of game EV above. I might be able to see how these compare, and if they balance out with what we think the transition probabilities are going from multiplier to multiplier.
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Quote: drrock

Quote: gordonm888

Does anyone want to take a stab at the cutoffs for JOB VP? We would need to use a hit frequency for the bonus feature, I guess 11% is the obvious suspect. Alternatively, we might parameterize the JOB cutoffs as a function of feature hit frequency. I think it would be fun to see to what extent the cutoffs are different than DDB.

Obviously, we'll need a table with EVs and probabilities for JOB hand categories such as Mental kindly provided for DDB.
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Mult	Wizard*	DrRock	Difference	LoHandEV	HiHandEV	InBetween
2x	8.514869	8.646949	0.132080	6.000000	9.000000	0
3x	4.739431	4.781851	0.042420	4.302498	6.000000	0
5x	2.546356	2.557272	0.010916	2.531915	2.595745	0
8x	1.420753	1.421622	0.000869	1.419056	1.421832	0

LoHandEV shows the EV of the hands that have the highest EV that are still below the Wizard's cutoff. HiHandEV shows the EV of the hands that have the lowest EV that are still above my cutoff. This time there is no difference depending upon which cutoffs are used. Each pair of cutoffs reside in the same gap. (Reminder: InBetween shows the number of dealt hands out of 2,598,960 that would be counted differently depending upon which set of cutoffs is used. And this time that number is always zero.)

*Obviously, the Wizard did NOT calculate these. I was able to duplicate the calculations that he made for DDB, and I believe that I did them in the same manner here for 9/6 Jacks or Better.

Something troublesome, which could indicate that I missed something unless this can be verified by the Wizard or YH, is that I calculated an EV of only 97.6130%. I did use the 11% frequency for 2x bonus appearing. Maybe this is wrong or I have an error somewhere in my spreadsheet.

I say this is troubling because we saw that the 98.98% base game for DDB became 99.01% under Fortune X. And I understand a lot of games with novel features like multipliers have an EV that is designed to be equal to or just a touch above that of the base game. Clearly 97.61% is pretty underwhelming for a 99.54% base game.
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Quote: drrock

I can also report that the trigger percentages for these games have been updated today at videopoker.com.

There are 11 different trigger points listed for the 13 game types. In general, it looks like the games with the higher variances have lower trigger points and vice versa. This would be expected due to the advantage of being able to choose which hands get a multiplier when there is a higher probability of more extreme winning dealt hands.link to original post

800.00000000      4  800.00000000
 50.00000000     40  125.00000000
 25.00000000    664   31.02409639
 19.68085098    760   29.59126539
 19.65957451    796   29.14209344
 19.59574509    820   28.86268812
 19.55319214    848   28.55529910
 18.55319214    944   27.53813568
 18.53191566    992   27.10235084
 18.48936081   1316   24.98182746
 18.46808434   1436   24.43750352
 18.42553139   1468   24.30645236
 18.40425491   1504   24.16517636
 18.36170197   1600   23.81696789
  9.00000000   5344   13.43621793