In the wiki (“Sagefault’s Endgame Guide”) there is a claim that when you’re doing timeskip / resource-retrieval / levi trade to get time crystals, the optimum balance of Chrono Furnaces to Resource Retrieval levels is 5n RR to 8(n+1) CF.

This seems pretty non-obvious. The purpose of this post is to poke at it a bit and see whether it’s true. (Spoiler: it doesn’t seem to be.)

So, suppose you have f furnaces and r levels of RR. The resources produced per unit of elapsed time are simply proportional to f times r. (They’re proportional to f because in the long run your timeskip rate is proportional to your heat-absorbtion rate, which is proportional to f. They’re proportional to r because that’s how RR works.)

The cost of the next CF is 25 x 1.25^f time crystals. (Plus some relics, but if you’re fairly far along they’re worth much less than the time crystals.) The cost of the next RR is 1000 x 1.3^r time crystals.

The gain from buying a CF is proportional to r. The gain from buying a level of RR is proportional to f.

So you should be indifferent between buying CF and buying RR when the ratio (25 x 1.25^f) : (1000 x 1.3^r) equals the ratio r : f. That is, when 25 f 1.25^f = 1000 r 1.3^r.

There isn’t a nice closed-form solution to this (there’ll be something involving the Lambert W-function, I assume) so here’s a table. The columns are: RR level, nearest integer to optimal CF level, 8(r/5)+8 figure from Sagefault.

RR CF SF
 5 17 16
10 24 24
15 31 32
20 37 40
25 43 48
30 50 56
35 56 64
40 62 72
45 68 80
50 74 88

The optimal CF count doesn’t grow as fast as Sagefault suggests.

Here’s an equally simple but much better approximation: Start at RR4/CF16, and then add 5 CF per 4 RR. From RR4 up to RR60, I think this never recommends a CF count that’s off by more than 1 from optimal.

RR CF GM
 4 15 16
 8 22 21
12 27 26
16 32 31
20 37 36
24 42 41
28 47 46
32 52 51
36 57 56
40 62 61
44 66 66
48 71 71
52 76 76
56 81 81
60 85 86