Previous: Theory 9 to Endgame
Disclaimer: This is a simplified version of the guide. The guide will skip over things, and is not completely optimal. Click here for a more polished, in-depth, and optimal guide.
When discussion strategy, this section may use more jargon then the rest of the guide. If you are confused, please check out the discord server and ask.
SL is based of a variation of stirlings approximation for e. Stirlings approximation is an expression that approaches e and x approches infinity:
\(lim_{x \rightarrow \infty} \frac{x}{\sqrt[x]{x!}} = e\)
In the theory \(\frac{x}{\sqrt[x]{x!}}\) is represented the the symbol \(\gamma\)
Essentially, the higher x is, the closer \(\gamma\) is to e
Stirlings approximation is linear, that is, doubling x, makes \(\gamma\) twice as close to e
Approximations like this are often used to calculate the value of e by plugging in larger and larger values for x
\(\dot{\rho_1} = \frac{\sqrt{\rho_2}}{e - \gamma}\)
\(\gamma = \frac{\rho_3}{\sqrt[\rho_3]{\rho_3!}}\)
\(\dot{\rho_2} = a_1a_2a_3^{-ln(\rho_3)}\)
\(\dot{\rho_3} = b_1b_2\)
\(a_3 = 2\)
The first equation shows that \(\dot{\rho_1}\) is affected by \(\rho_2\) (half effectiveness) and how close \(\gamma\) is to e
Half effectiveness means that quadrupling \(\rho_2\) only doubles \(\rho_1\).
The 2nd equation is stirlings approximation, and shows us that the closeness of gamma to e is proportional to \(\rho_3\)
The next equations are simple and show that \(a_{12}\) increases \(\rho_2\) and \(b_{12}\) increases \(\rho_3\)
Interestingly, \(\dot{\rho_2}\) is DECREASED by \(a_3\) and \(\rho_3\), meaning to increase \(\rho_2\) you should decrease those
The final equation shows the value of \(a_3\) which can be lowered through milestones
Overall the effectiveness of each variable is
| \(a_1\) | Increases \(\dot{\rho_2}\) by approx 52%, though due to the square root this is only a 26% increase in \(\rho_1\) |
| \(a_2\) | Increases \(\dot{\rho_2}\) by 2x |
| \(b_1\) | Increases \(\dot{\rho_3}\) by approx 60% |
| \(b_2\) | Increases \(\dot{\rho_3}\) by 2x |
Every variable does not affect \(\rho_1\) directly, affecting either \(\rho_2\) or \(\rho_3\), which usually takes a while to build up. Therefore, near the end of a publication, you should stop buying them, because by the time the upgrades have built up, you've past the optimal pub multi. Thus we have the strat SLStopAd. I would like to remove the Mod part for simplicity, but I am afraid I may make the start unviable. Therefore, a simplified form, is not included, and the form with mod10 is listed only for notice, as the speed gain from SLStopAd is rarely enough to justify using over the idle strat SLStopA. We recommend only using SLStopA and SLMS
✔️ - Always buy, ❌ - Never buy
| var | Pub multi < 4.5 | Pub multi 4.5 - 6 | Pub multi > 6 |
|---|---|---|---|
| \(a_1\) | ✔️ | ❌ | ❌ |
| \(a_2\) | ✔️ | ❌ | ❌ |
| \(b_1\) | ✔️ | ✔️ | ❌ |
| \(b_2\) | ✔️ | ✔️ | ❌ |
✔️ - Always buy, ❌ - Never buy
We do not recommend this strategy
| var | Pub multi < 4.5 | Pub multi 4.5 - 6 | Pub multi > 6 |
|---|---|---|---|
| \(a_1\) | if lvl % 3 = 0 ✔️, else at \(\frac{1}{lvl \% 3}\) of a2 | ❌ | ❌ |
| \(a_2\) | ✔️ | ❌ | ❌ |
| \(b_1\) | if lvl % 4 = 0 ✔️, else at \(\frac{1}{lvl \% 4}\) of b2 | ✔️ | ❌ |
| \(b_2\) | ✔️ | ✔️ | ❌ |
SL has a MS strat as well. This MS strat is quite difficult, however, without manual buying it might be not too bad. It's also good at get some
MS practice in, and if I have to suffer SLMS you all do too. Therefore I have decided to include SLMS here as well (again, SLMSd is not
included)
SLMS is a DIFFICULT strat to perform. If you find it overly difficult, please feel free to ignore this section
| State 1 | \(4 \rightarrow 3 \rightarrow 1 \rightarrow 2\) (prioritieses \(\rho_3\)) |
| State 2 | \(2 \rightarrow 1 \rightarrow 4 \rightarrow 3\) (prioritieses \(\rho_2\)) |
| State 3 | \(1 \rightarrow 2 \rightarrow 4 \rightarrow 3\) (prioritieses \(\rho_1\)) |
This notation tells you the priority order to buy milestones. For example \(4 \rightarrow 3 \rightarrow 1 \rightarrow 2\) means to max out the 4th milestone (from the top) then the 3rd then the 1st then the 2nd
| \(\rho\) | State 1 | State 2 | State 3 |
|---|---|---|---|
| 1e25 - 1e50 | 5 | 4 | - |
| 1e50 - 1e75 | 7 | 6 | - |
| 1e75 - 1e100 | 12 | 10 | - |
| 1e100 - 1e150 | 20 | 15 | - |
| 1e150 - 1e175 | 8 | 6 | - |
| 1e175 - 1e200 | 1.5 | 1 | SKIP |
| 1e200 - 1e275 | 3 | SKIP | - |
| 1e275 - 1e300 | 2 | SKIP | - |
This table tells you when to swap to each state. Start in state 1 and read the number in the box (I'm calling it "n"). When your \(\rho_1\) reaches "n" times less then the next \(b_{12}\) upgrade, swap to State 2. Now you have a new "n". When your \(\rho_1\) again reaches "n" times less then the next \(b_{12}\) upgrade with the new "n", swap to State 3 (note that for one of the stages there is no swapping to state 3.). Finally, when you can afford and buy the upgrade, or your \(\rho\) drops back down due to another upgrade, swap back to State 1
For variable purchase, follow the SLStopA strat. Therefore you don't have to worry about variables and just focus on the MS. If you want to take a break, just leave it in state 1 and come back. Any amount of active play is beneficial. (SLMSd is just the same thing with the SLStopAd strat anyway)
This strategy is over 2x faster then SLStopA, so I suggest doing it if you can. Plus, it's more fun then just waiting