Edit: Optimal spins can now be found using
this tool, which removes the need for the tables.
For colorful tables and charts, see
here. (WARNING: very large and slow-to-load spreadsheets!)
IntroductionWhile calculating the
special spin drop rate, the dictator, ColorlessGreen, Pineapple, and I were considering what spin strategies would maximize the drop rate or the value of each special spin. The strategies that we came up with, called "weak SS", "priority SS/SD", "focus SD/DD", and "max DD", take into account only the rares showing on the slots and the number of spins left. However, to get the maximum value out of a special spin with the information available to the player, the rares that have been seen in the spin, and how many times they have been seen, must also be accounted for.
The special spin works as follows: first four rares (out of 27) are chosen, with replacement. This means that there could be four, three, or even only two rares to be spun for (ignoring the possibility of only one rare being chosen, because then it is automatically won). Then each time a slot is spun, including when the slots are first shown, one of the four rares is picked at random. Given the information that the player can see, it starts out as rather likely that the four rares are all different, but the exact probabilities change as more spins are shown. For example, if there is only one spin left on each slot and only two rares have been seen, it becomes more probable that there were only two rares being picked from.
I wrote a program to use a recursive strategy to determine the value of each possible situation (combination of the current state of the slots and what has been seen previously in the same spin), and from there to generate the best spin for that situation. In doing this, I made the assumption that
the player only has two categories of rares, with each category having the same value. This seemed to be most similar to the cases of farming to get a specific rare (the categories being the rare being farmed for, and all other rares) or to complete full sets of rares (the categories being the rares that are not complete, and the rares that have full sets already). Assigning individual values to all rares would have resulted in a prohibitively large number of situations.
NotationThe class of rares with the higher value is called the "target(s)".
I gave each situation a 12-character code representing that situation, for convenience of space. This code consists of 4 parts:
First three characters: What is currently showing on the slots. The letters tuvw represent targets, while abcd represent other rares. The first three characters can be "abc" (three different non-target rares), "tbc" (one target rare and two different non-target rares), "tuc" (two different target rares and one non-target rare), "tuv" (three different target rares), "abb" (a pair of non-target rares and a different non-target rare), "tbb" (a pair of non-target rares and a target rare), "auu" (a pair of target rares and a non-target rare), or "tuu" (a pair of target rares and a different target rare).
Second three characters: What other rares have been seen. The first character in this section is the total number of rares seen, and the other two characters are any seen rares that are not showing, with the letters tuvw representing targets and abcd representing other rares. For example, "4vd" means that four rares in total have been seen, and of the two that are not showing, one is a target and one is not. Underscores mean that there are fewer than two hidden rares, e.g. "3__" means that all three rares that have been seen are also currently shown.
Third three characters: How many spins are remaining for each slot. These three positions correspond to the first three positions, so if the first three positions are "tbc" and the third three are "123", the slot with one spin left is the one showing the target rare.
Last three characters: How many times each rare has been seen, if two or three rares have been seen (if four different rares have shown up, it does not matter how many times they have shown up). These are in the same order as the rares are listed in the first six characters. Every time a slot comes up as a rare, including the initial states of the slots, the count of that rare is increased by 1. If four different rares have shown up, this section is three underscores ("___"). If only two different rares have shown up, the last character will be an underscore.
Example: "tbb3c_233121" means that the slots show one target rare with two spins remaining and a pair of a non-target rare, with three spins remaining on both slots. Another non-target rare has been seen once, but is not currently showing. This situation could arise if I started with a pair of rare "b" and a single rare "c", then spun "c" and it came up as a target rare "t".
Pairs are always listed as the second and third slots, making the single rare the first slot. If all three slots are different, targets are listed first ("tuc" instead of "auv" or "tbv"). If there is a choice, rares and slots are ordered such that the number of spins left is in increasing order, and if there is still a tie, the number of times seen is in increasing order. For example, "tuv3__121341" is instead listed as "tuv3__112134".
In specifying what to spin, some table entries contain "+" or "-" at the end. "+" means to spin the slot with more spins left, while "-" means to spin the slot with fewer spins. For example, "tbb3c_123141" is listed as "b-", meaning that the correct spin is the b slot with fewer spins left. "b+" would mean the b slot with more spins left.
Using the tablesAs it turns out, some situations depend on both the number of targets and the value of targets relative to other rares. Some other situations depend on the value of targets but not the number, and the rest of the situations don't depend on either. For each of the eight possible configurations shown on the slots, a "default" spin is provided, as well as up to three tables.
Table 1 lists some situations and what to spin in those situations. These situations do not depend on the value or number of targets.
Table 2 lists situations that depend on the value of the targets but not how many targets. To use this table, find your current situation and then compare your value for rares to the one listed in the table. For example, the situation "tuu4cd123___" has the entry "t (to 7.24x) u+ (above)". This means that if you value a target rare more than 7.24 non-target rares, you should spin u+ (the u slot with more spins left), otherwise you should spin t.
Table 3 lists situations that depend on the value of the targets and how many targets there are, from 1 to 26 (0 targets can be treated as 1 target with a value of 1). This table is used the same as the second table, except that you should find the cell corresponding to the current situation and your number of targets.
If a situation is not shown in any of the three tables, the default spin should be used.
Spoiler for Example:
As an example, let's say that my targets are the shards (12 targets) and that I value a shard at two times the value of a non-shard rare.
I get a special spin and it looks like this:
I have a pair of a non-target rare and a single target rare, so I go to the "tbb" section. Two rares have been seen, there are three spins left on each slot, "t" has come up once, and "b" has come up twice, making this situation "tbb2__33312_". "tbb2__33312_" is in the third table, where the entry for 12 targets is "t (to 1.390x) b (above)". Since I value a shard more than 1.390x any other rare, I spin "b", which is either of the slots showing an Owl's Eye.
The slot comes up as a Shard of Void. Now I have two different target rares and a non-target rare, so I go to the "tuc" section. Because the Shard of Void has fewer spins remaining than the Shard of Freedom, Shard of Void is "t" and Shard of Freedom is "u", instead of the other way around. Three rares have been seen, one target has two spins left and the other two slots have three spins left, "t" has come up once, "u" has come up once, and "c" has come up twice, so this is "tuc3__233112". "tuc3__233112" isn't in any of the tables, so I use the default spin, which is "c". This indicates that the correct spin is the slot with the Owl's Eye.
Now I have a pair of targets and a single target, which is the "tuu" section. Three rares have been seen, and the one that is not on the slots is not a target. The single target has two spins remaining and has been seen once, while the pair has 2 and 3 spins and has seen twice. The hidden rare has also been seen twice, so this is "tuu3c_223122". This also isn't in the tables, so the default spin is used. The default spin is "t", so I spin the Shard of Void.
This situation is now "tuu3c_123222", which is in the third table. The entry for twelve targets reads "t (to 30.8x) u- (above)". I value my targets at less than 30.8 times other rares, so I spin "t"—the Shard of Void—again.
With no spins left on one slot, the strategy becomes to try to match that slot. It does not matter in what order the other two slots are spun.
Notice that the letter corresponding to each rare is not the same throughout. The Owl's Eye is sometimes "a", sometimes "b", and sometimes "c". The Shard of Freedom is sometimes "t" and sometimes "u". Which rare gets which letter depends on the situation:
Pairs are always listed as the second and third slots, making the single rare the first slot. If all three slots are different, targets are listed first ("tuc" instead of "auv" or "tbv"). If there is a choice, rares and slots are ordered such that the number of spins left is in increasing order, and if there is still a tie, the number of times seen is in increasing order. For example, "tuv3__121341" is instead listed as "tuv3__112134".
Out of 1926 possible situations with at least one spin left for each slot, 802 of them can possibly be reached with optimal strategy for some number and value of targets. Of these 802, 21 are in Table 1, 41 are in Table 2, 63 are in Table 3, and the remaining 677 use the default spin.
