Let’s start by de­fin­ing which vari­ables we’re dis­cuss­ing. We will dis­cuss the main-game vari­ables, (

$x-\xi$

We can split the pro­gres­sion through the game into three sec­tions:

• From 0 to ee4310, this sec­tion in­cludes the un­lock­ing of all of the vari­ables, fin­ish­ing with the un­lock­ing of

  $\xi$

  at ee791. It also in­cludes the pur­chase of

  $y$

  power up­grades, of which we will dis­cuss more later.

• From ee9160 to ee47362, this is the range when we buy su­prem­acy up­grades known as “psi3” up­grades.

• Fi­nally, the fi­nal sec­tion is after ee47362, where we see the title of strongest vari­able change hands a few more times with a fi­nal change at ~ee70000.

▶︎ Pro­gram Permalink: pro­gram#

$f(t)$

$f(t)$

$f(t)$

$x$

$x_8$

$f(t)$

$x_8$

$x_8$

The cost of the first levels of each vari­able range from Free to an ex­pens­ive ee791. Here’s a table with the cost of the first level of each vari­able:

Here’s a graph of vari­able power up to ee50

$f(t)$

$y$

$\beta$

Com­puted every ee1 from ee1 to ee50.

And here’s a graph with the same

$f(t)$

$f(t)$

Com­puted every ee1 from ee1 to ee50.

And here’s a graph of vari­able power up to the pur­chase of the fi­nal vari­able at ee791:

Com­puted every ee1 from ee1 to ee850.

And the per­cent­age graph:

Com­puted every ee1 from ee1 to ee850. A bit more spiky ;)

You may have no­ticed the ab­rupt jumps in the above graphs. We’ll now dis­cuss why that oc­curs.

After we su­prem­acy for the first time at ee50, we are given a cur­rency

$\psi$

$y$

$0.2$

$y$

$y^{1.2}$

$y^{9.0}$

$40$

Be­cause the power each vari­able has is propag­ated down all of its lower vari­ables (ex.

$z \rightarrow y \rightarrow x$

The cost of each up­grade from

$y^{1.2}$

$y^{1.8}$

$\psi = 1.5 \times 2^{2(x-1)}$

The cost model changes after this, and from

$y^{2.0}$

$y^{4.0}$

$\psi = 0.488281 \times 2^{3(x-1)}$

The cost model changes one fi­nal time, and from

$y^{4.2}$

$y^{9.0}$

$\psi = 0.0000000794093 \times 2^{5(x-1)}$

With all of these cal­cu­la­tions in place, here is the new vari­able power graph, now ran­ging up to ee5000:

Com­puted every ee1 from ee1 to ee5000.

Each small jump present in the graph marks the pur­chase of an ad­ded

$0.2$

$y$

And fi­nally, here is the per­cent­age of total power for each vari­able:

Com­puted every ee1 from ee1 to ee5000.

Very spiky!

After we buy

$y^{9.0}$

$\psi$

$f(t)$

$\psi$

$\psi$

$f(t)$

These up­grades help delay the de­cay play­ers would oth­er­wise ex­per­i­ence from ee20k-ee50k as their the­or­ies slow down and they gain less

$\tau$

The first psi3 up­grade in­creases

$z$

$z^{1.04}$

$z^{1.08}$

$s$

$1.08$

$\eta$

$1.08$

To il­lus­trate the ef­fect these pur­chases have, let’s use an ex­ample.

Let’s say these are the cur­rent equa­tions for your first five vari­ables:

$x = 4.7893e7y^9 + 2^3 \times 165.2$

$y = 19891z^{1.08} + 2^{419} \times 87522$

$z = 18082s^{1.04} + 2^{376} \times 70017$

$s = 16433u + 2^{341} \times 56007$

$u = 14933v + 2^{323} \times 48973$

And let’s say that shortly af­ter­ward you pur­chased a new psi3 level:

$x = 4.7893e7y^9 + 2^3 \times 165.2$

$y = 19891z^{1.08} + 2^{419} \times 87522$

$z = 18082s^{1.08} + 2^{376} \times 70017$

$s = 16433u + 2^{341} \times 56007$

$u = 14933v + 2^{323} \times 48973$

With this new level, the power of

$s$

$u$

$x$

$0.04$

$s$

$x$

$y$

$z$

But don’t just take my word for it. Let’s look at the data from the pro­gram around this up­grade.

Com­puted every ee1 from ee13000 to ee15000. For the pur­pose of this visu­al­iz­a­tion, only the four vari­ables dis­cussed are plot­ted.

As can be seen in the im­age, both

$y$

$z$

$s$

$u$

$\xi$

This leads to an in­ter­est­ing ef­fect where every two psi3 up­grades one of the vari­able’s power stops get­ting boos­ted from the up­grade, so we see a line sep­ar­ate from the rest every two jumps:

Com­puted every ee1 from ee1 to ee50000. This took a while to run…

And here’s a rather in­ter­est­ing plot with the per­cent­age of total power each vari­able has.

Com­puted every ee1 from ee1 to ee50000.

We’ve made it. The end of the su­prem­acy up­grades is upon us. What’s next?

Well, now is when

$f(t)$

Com­puted every ee5 from ee35000 to ee80000.

Im­me­di­ately after the fi­nal su­prem­acy up­grade, we gain a lot less

$x$

$f(t)$

$17 f(t)$

$e1$

$x$

Shortly af­ter­ward, dur­ing the period up to ~ee52000,

$\xi$

$77 f(t)$

$e1$

$x$

For­tu­nately,

$\xi$

$\eta$

$f(t)$

$\eta$

$45 f(t)$

$e1$

$x$

Fi­nally, at ee70000,

$\eta$

$y$

$y$

$y$

$22 f(t)$

$e1$

$x$

$17 f(t)$

Class: strat;
Cap­tion:

$F(t)$

A plot with only

$y$

,

$\eta$

, and

$\xi$

to il­lus­trate the dif­fer­ence in their scal­ing and where each vari­able be­comes dom­in­ant.

Even be­fore

$y$

$y$

At ee48000, the su­prem­acy equa­tion we re­com­mend us­ing skips most of the psi3 su­prem­acy up­grades. In fact, it only su­prem­a­cies at

$e52, e411, e511, e531, e551,$

$e571d\psi$

$d\psi$

Com­puted every ee10 from ee1 to ee50000

Very in­ter­est­ing! So it looks like we should be buy­ing

$y$

$\psi$

$y$

$y$

$y$

After we reach ee52000, we are re­com­men­ded to use a new su­prem­acy equa­tion. This equa­tion only su­prem­a­cies at e52, e511, and e571

$d\psi$

Com­puted every ee10 from ee1 to ee50000

Once again, it looks like we should be buy­ing

$y$

$y$

$\psi$

Fi­nally, some­where between ee58000 and ee60000 (we aren’t en­tirely sure where within this range) we switch to our fi­nal su­prem­acy equa­tion. This su­prem­acy equa­tion only su­prem­a­cies at e52

$d\psi$

$d\psi$

Com­puted every ee10 from ee1 to ee50000

There we go! Now we should buy

$y$

$\psi$

Where I put all of the in­ter­est­ing plots that did­n’t fit well in any sec­tion.

Full Graph
The plots of vari­able power from ee1 to ee75000.

Com­puted every ee5 from ee1 to ee75000.

And the plot with per­cent­age of total power for each vari­able:

Com­puted every ee5 from ee1 to ee75000.

▶︎ Log Per­cent­age Total Power Plots Permalink: log-per­cent­age-total-power-plots#

More than just the reg­u­lar per­cent­age total power plots, I also tried log­ging the per­cent­age so we can see more than just 0% power for al­most all of the vari­ables, cre­at­ing fas­cin­at­ing res­ults. Ig­nore the ini­tial xi/

$\xi$

Com­puted every ee1 from ee1 to ee120.

Com­puted every ee1 from ee1 to ee1000.

Com­puted every ee1 from ee1 to ee3000.

Com­puted every ee1 from ee1 to ee20000.

Com­puted every ee10 from ee1 to ee75000.

So many stor­ies told in one plot!