[Guide] Self-sustaining condition for unbirthday party

[RazorbackPirate]

Would you be willing to apply this to your calculations?

I'm not sure if it's applicable. The reason for wanting to create a perpetual motion machine is as an unlimited supply of FP packs. Yes, they will get spent eventually, but at the time of generation, you'd most often want to store them.

To me, it breaks the model when you use the output you want to store to sustain the generation engine. IMO, the output of coins and supplies as the only inputs is the correct way to go. If anything, it makes the math conservative, meaning a few extra levels than what's truly needed. This means enough coins and supplies to carry through the times that FP packs are few and far between.

Either way, the formulas above and @ozyman-tremble-weaklings' spreadsheet are a great base to apply your own variations.

 

[DeletedUser35867]

I'm not sure if it's applicable. The reason for wanting to create a perpetual motion machine is as an unlimited supply of FP packs. Yes, they will get spent eventually, but at the time of generation, you'd most often want to store them.

To me, it breaks the model when you use the output you want to store to sustain the generation engine. IMO, the output of coins and supplies as the only inputs is the correct way to go. If anything, it makes the math conservative, meaning a few extra levels than what's truly needed. This means enough coins and supplies to carry through the times that FP packs are few and far between.

Either way, the formulas above and @ozyman-tremble-weaklings' spreadsheet are a great base to apply your own variations.

That's the whole point of using the second and third quest slots, though. Maximizing the efficiency of the engine. Therefore, it's beyond applicable! It's an important aspect of the whole plan! Plus, you don't have to use the FP on CF. You can spend them on research, investments on other's GBs,other GBs of your own, or any other reason. You just have to have the quest queued up while you spend them is all, so it applies to literally any player with a Chateau. If I'm trying to make unlimited FP, I want to start that ASAP. If it means I need to focus my FP spending on my own CF for a while, then save up when CF is a few levels over the minimum, that's fine with me. That's kinda the whole point of this concept anyway. In fact, there's no time where you're spending a lot of FP and you SHOULDN'T have that quest up.

I'm just interested in figuring out a more exact lower limit to when I can start utilizing CF as a perpetual engine. So I'd like to ask this person, who's already done so much math for me (I was about to buckle down and do this kind of thing for myself when a friend of mine pointed me to this thread), if he'd be willing to account for this "extra quest slot" in his calculations.

I appreciate the conservative level estimate (and how difficult it is to calculate an exact minimum), but I don't want to know when I'm a few levels over, I want to know when I'm a few levels under!

Edit: Storing too many FP can really slow you down. Obviously keep enough of a stockpile to pay out on good investments (so higher lv Arc= larger FP stock), but you really do have to invest in yourself with this game. Gotta spend FP to make FP

 

[DeletedUser27525]

Would you be willing to apply this to your calculations?

Apologies for my late reply. I agree with @RazorbackPirate that coins and supplies are the main variables here. Balancing cost and reward makes the math a bit tidier. I do see your point regarding the "spend x FPs" RQ, and this can be incorporated into the analysis. Keep in mind that the L2 and L3 numbers I posted here are estimates since players will almost always overshoot in "collect x coins/supplies" and hence perform worse than the fractional rewards model. The FPs earned from UB and used to fulfill "spend x FPs" will offset that somewhat, although the exact amount is unknown to me at this point.

In practice, however, if you're in the vicinity of L2 and L3, city production and other quests will allow you to do UB almost as much as you want on a daily basis. So even if you couldn't do a literally infinite amount from a theoretical point of view, you probably wouldn't notice it.

When I have time I can take a look at this but I'm afraid I can't attach too many figures in one post.

 

[DeletedUser31440]

@NINJAxPENGUIN92 I have no plans of adding fp's spent to my calculator.

Below is with no CF

1/14 chance of getting 5 fp's from quests, 1/14 for large coins/supplies & 2/14 for small coins/supplies
IA - Avg 42 quests before getting a free one from spending fp's.
IA - Avg 2 free UBQ's for every 14 completed.

EMA - Avg 70 quests before getting a free one from spending fp's
EMA - Avg 1.8 free UBQ's for every 14 completed.

HMA - Avg 112 quests before getting a free one from spending fp's
HMA - Avg 1.5 free UBQ's for every 14 completed.

 

[DeletedUser32824]

Just a quick tip/strat for anyone looking. If you increase your daily supply and coin production, you can boost your UBQs per day. This is most efficiently done in IA and LMA. I'm in LMA so here is what that looks like. The way you can accomplish this without leveling an LOA and RAH and filling your city with premium supply buildings is by leveling a voyager and Dynamic tower which can be leveled to 70 for about 16.5k FP using 1.9 help. This is opposed to dumping around the same amount or more into getting a CF to a ridiculous level (lvl 120 is like 7k FP self contribution).

lvl 70 CF (5.5x the rewards) with 200k coins/supplies per day you can complete 205 quests for about 63 FP
with 800k coins/supplies you can complete 417 quests for about 139 FP (lvl 70 voyager or DT)
with 1.3m coins/supplies you can complete 629 quests for about 215 FP (lvl 70 voyager AND DT)

I'd wager that's more quests than a person would want to do in one day. And these calculations are without the double dipping of a 2nd quest slot.

The increase in FP production from the first lvl 70 DT/Voy is 76 FP. The second is also 76 FP. That's more than cape! They would pay for themselves in about 215 days.

 

[mamboking053]

Just a quick tip/strat for anyone looking. If you increase your daily supply and coin production, you can boost your UBQs per day. This is most efficiently done in IA and LMA. I'm in LMA so here is what that looks like. The way you can accomplish this without leveling an LOA and RAH and filling your city with premium supply buildings is by leveling a voyager and Dynamic tower which can be leveled to 70 for about 16.5k FP using 1.9 help. This is opposed to dumping around the same amount or more into getting a CF to a ridiculous level (lvl 120 is like 7k FP self contribution).

lvl 70 CF (5.5x the rewards) with 200k coins/supplies per day you can complete 205 quests for about 63 FP
with 800k coins/supplies you can complete 417 quests for about 139 FP (lvl 70 voyager or DT)
with 1.3m coins/supplies you can complete 629 quests for about 215 FP (lvl 70 voyager AND DT)

I'd wager that's more quests than a person would want to do in one day. And these calculations are without the double dipping of a 2nd quest slot.

The increase in FP production from the first lvl 70 DT/Voy is 76 FP. The second is also 76 FP. That's more than cape! They would pay for themselves in about 215 days.

Yeah, at some point it seems like it will get ridiculous. Even making it halfway to achieving a perpetual quest machine, whose going to sit there and click for it and not end up pulling their hair, teeth, and eyes out?

 

[DreadfulCadillac]

Introduction
The unbirthday party quest (UB) is a recurring quest that is available in all eras. The player is asked to pay a certain amount of coins and supplies for a random reward. In this guide we ask whether it is possible to achieve self-sustaining UB. We define self-sustaining as having no net loss in coins and supplies.

Notation and Definition
The condition for UB to be self-sustaining is simply

P(large coins) x large coins + P(small coin) x small coin >= C(coins)
P(large supplies) x large supplies + P(small supplies) x small supplies >= C(supplies)​

where P() denotes the probability and C() denotes the cost of UB.

To simplify the notation, we define:

R1 = small coins/supplies reward
R2 = large coins/supplies reward
P1 = P(small coins) = P(small supplies) = 1/7
P2 = P(large coins) = P(large supplies) = 1/14​

so that the self-sustaining condition is rewritten as

P1R1 + P2R2 >= C​

for both coins and supplies.

UB with CF boost
Using the data for "random rewards" provided on wikia, it is easy to see that UB cannot be self-sustaining with its base rewards. However, Chateau Frontenac provides a boost factor of

View attachment 11012​

for level L > 10. Rearranging terms in the self-sustaining condition, we obtain the critical level of CF for self-sustaining UB:

View attachment 11013​

for both coins and supplies. We denote the corresponding critical level as Lc and Ls, respectively, and effectively L = max (Lc, Ls).

Results
1. UB without other recurring quests
In this section we first consider the case of UB alone without other recurring quests. This shall give us an upper bound for self-sustaining UB.

The results for all eras are plotted below and tabulated in the Appendix (values for BA are theoretical since one cannot build CF in BA). We see that the critical level for self-sustaining UB is the lowest in IA, where L = 73. Generally speaking, the critical level increases (but not monotonically) through the eras from IA to CA and remains fairly stable till AF, beyond which the critical level becomes dauntingly high (385 for OF and 583 for VF).

View attachment 11002

2. UB with 2nd recurring quest
It is possible to use the coins/supplies reward from UB to fulfill the "collect x coins/supplies" recurring quest, which may in turn give you coins and supplies. A complete solution to this problem is too difficult and therefore we make a simple approximation to obtain an analytical solution.

Let us recall the self-sustaining condition without a 2nd quest:

B(P1R1 + P2R2) >= C​

The rewarded coins/supplies can be used to fulfill a 2nd quest that requires you to collect Q coins/supplies. (The values for Q are obtained from "recurring quests" on wikia.) We make an approximation that the 2nd reward can be immediately retrieved as a fraction of the 2nd quest. The self-sustaining condition then becomes

View attachment 11019​

We make the variable substitution X = P1R1 + P2R2 to simplify the algebra:

BX ( 1 + B X / Q ) >= C
X^2 B^2 + Q X B - Q C >= 0.​

Solving for the positive root of the quadratic equation, we obtain:

View attachment 11021​

Results are plotted in the following figure (as crosses) together with the upper bound obtained in the previous section (as dots). They are also tabulated as L2c, L2s, L2 = max(L2c, L2s) in the Appendix. Note that:

• Starred numbers are theoretical since the 2nd recurring quest is not possible in those ages.
• Only two eras require a CF level below 100: IA (73) and LMA (96).
• Overall, we see a reduction of about 20% in the critical CF level to reach self-sustaining UB.

The fractional-reward approximation in the above analysis is a lower bound estimate for self-sustaining UB with a 2nd quest, since in practice the player can only do worse. Hence, the actual L2 should be somewhere between L and L2, but probably closer to L2.

View attachment 11022

3. UB with 3rd recurring quest
Similar to the previous section, we make some simplifying assumptions:

• 1 quest for coins and 1 quest for supplies. (The cases of 2 coins quests or 2 supplies quests are ignored.)
• Fractional rewards (as explained above).

This model allows us to store both coins and supplies from UB. The self-sustaining condition now contains an additional cross term that involves both coins and supplies, and we use the respective subscripts c and s to distinguish them. For example, the condition for coins is:

View attachment 11032​

and the condition for supplies can be similarly derived. The two conditions are rewritten as

View attachment 11029​

where X_{c,s} are P1R1 + P2R2 for coins and supplies, respectively. The solution is

View attachment 11030​

Results are plotted below (as open squares) and tabulated in the Appendix. The critical CF level is reduced by about 30-40% compared to UB alone. This reduction is most significant in VF, where the CF level is brought down from 583 to 323.
View attachment 11031

Final Remarks
To assess the self-sustaining condition for UB, I have provided an exact upper bound on the CF level (without 2nd recurring quest) and an approximate lower bound with 2nd and 3rd recurring quest. For most players with 2 recurring quests, L2 should serve as a good estimate.

"Self-sustaining" ignores city production by definition. In practice, city production will enable players to achieve a pseudo-self-sustaining state more easily.

Happy Unbirthday!

Appendix

Age	Lc	Ls	L	L2c	L2s	L2	% change between L2 and L	L3c	L3s	L3	% change between L3 and L
BA*	296	240	296	226	182	226	-23.6	186	157	186	-37.2
IA	72	73	73	49*	49*	49*	-32.9*	37*	37*	37*	-49.3*
EMA	116	116	116	84*	84*	84*	-27.6*	67*	67*	67*	-42.2*
HMA	147	147	147	109*	109*	109*	-25.9*	90*	90*	90*	-38.8*
LMA	100	128	128	74	96	96	-25.0	61	77	77	-39.8
CA	210	215	215	168	169	169	-21.4	142	145	145	-32.6
IndA	255	194	255	205	156	205	-19.6	172	137	172	-32.5
PE	240	179	240	198	147	198	-17.5	168	131	168	-30.0
ME	257	189	257	211	157	211	-17.9	182	139	182	-29.2
PME	240	167	240	195	137	195	-18.8	165	121	165	-31.2
CE	209	174	209	165	141	165	-21.1	141	121	141	-32.5
TE	177	167	177	140	135	140	-20.9	121	115	121	-31.6
FE	206	240	240	157	185	185	-22.9	134	152	152	-36.7
AF	198	222	222	154	170	170	-23.4	130	143	143	-35.6
OF	349	385	385	245	296	296	-23.1	215	232	232	-39.7
VF	535	583	583	400	369	400	-31.4	303	323	323	-44.6

(* = theoretical value)

Way to confusing, but happy birthday

 

[DeletedUser32348]

i liked the graphical representation very much

 

[DeletedUser28324]

I assume that after LMA the numbers are in a sense not obtainable or at least impractical. The highest CF in the game is level 167. Most people do not push it this high

 

[DeletedUser27525]

I assume that after LMA the numbers are in a sense not obtainable or at least impractical. The highest CF in the game is level 167. Most people do not push it this high

For 3 RQ slots in TE the CF level is 121, which is feasible. The next batch is CA, CE, AF in the 140's. The other eras are not very practical, but then in reality, who would be doing an infinite amount of unbirthdays anyway? This was more of a math exercise.

 

[RazorbackPirate]

I've never viewed this as more than a 'proof of concept.' Although the math shows it can be done, I can't imagine ever spinning enough RQs to make the investment worthwhile. A fun math problem, but nothing I'd ever actually do. You never know, though. Goals can change.

 

[DeletedUser29726]

Well I mean when FP used to be hard to find, the ability to drive your chateau to the point where you could make ~85 FP an hour was significant (4 quests a minute or 240 quests an hour was what I clocked myself at in iron age - would be a little more than that in the new quest interface and in ages with less quests to skip per completion). You didn't need to get to infinite though, just cheap enough for as much as you want to do which is a significantly easier bar to clear in low ages - chateau already adding a decent lump of flat coins you just needed a supply strategy (which now would be as simple as himeji castle).