Möbius ring: Overview
2025-08-10 // IOTM Overview
Step back, adventurer. this IotM is about to Möb! It comes with one side: 11 theoretical free fights per day (ACAB -- all cops are bius!), and perhaps the most confusing turnsave ever. The Möbius ring is the August 2025 IotM, and it's here to provide some one-sided turn saving.
General Summary
The Möbius ring is an accessory with 5 basic bluetext enchantments out of the box -- +25 to HP/MP, and +10 to all stats. This seems extremely underwhelming, which is where the ring's final enchantment comes in:
- Allows interaction with your past and future
Every once in a while (numbers later), with the ring equipped, you will receive Time is a Möbius Strip, a free non-combat that lets you choose between 23 options to mess with the space-time continuum (increasing Paradoxicity), or un-mess with it (decreasing Paradoxicity). This non-combat requires one completed fight to prepare, before occurring throughout the day sporadically and at increasing intervals.
So, about that Paradoxicity thing -- the Möbius ring features a Paradoxicity counter. This does two things: first, it increases the chance (again, numbers later) of encountering a time cop. Time cops are scaling monsters that can appear in any eligible zone, at a rate related to Paradoxicity. Second, it modifies the enchantments on the ring, by changing in a first-in-first-out cycle. The following is the total list of enchantments that the ring can cycle from, of which only 5 consecutive enchantments can be active at once:
- Maximum HP +25
- Maximum MP +25
- Muscle +10
- Mysticality +10
- Moxie +10
- +15 Weapon Damage
- +2 Hot Resistance
- +100% Gear Drop
- +Paradoxicity Damage Reduction
- +50% Combat Initiative
- +5 Familiar Weight
- +2 Cold Resistance
- +100% Food Drop
- +25 Spell Damage
- +2 Stench Resistance
- +50% Booze Drop
- +25% Meat Drop
- +100% Item Drop
- +15 Ranged Damage
- +2 Spooky Resistance
- +100% Candy Drop
- +2 Familiar Experience Per Combat
- +4 Stats Per Fight
- +2 Sleaze Resistance
- +25% Pickpocket Chance
- +50% Max HP
- +50% Max MP
- +25% Muscle
For example, at 13 Paradoxicity, your ring will have:
- +25 Spell Damage
- +2 Stench Resistance
- +50% Booze Drop
- +25% Meat Drop
- +100% Item Drop
Time cops also have a 1% chance to drop their hat, which can be consumed to increase its +initiative enchantment up to +50%. This isn't extremely useful now, but maybe people will want this in a year or two.
Spadester Interlude, Part I
Time for the speedsters to step aside: the spadesters are here to take over this article for a minute. (Well, OK, it's just Scotch. And I'm not a great spade, so this is a speedster-on-speedster takeover.) I want to make sure that collective understanding of the Möbius ring is shared out as part of this article. I also want to be clear -- initial spading over the first few weeks of August hasn't revealed -exact- formulae for both the rate of NC acquisition, or exactly how Paradoxicity interacts with your time cop rate. However, there's now quite enough data for us to make a few educated guesses that are probably pretty close to correct. We will update this post when/if an exact rate is fully spaded.
First, let's cover the rate at which you get the Time is a Möbius Strip NC. For older players who've played in prior metas, cueing up this NC is done with a process not dissimilar to things like the Source Terminal's digitize.edu, the LOV Enamorang, or obtuse arrows. Once you've primed the ring by spending a single turn with it on at the start of your run, the NCs will become available in a predictable cycle, with a specific number of actual turns played before your next NC is available. To write it (very) explicitly, spading has demonstrated that:
- Your 1st NC will be available 4 turns after your 1st turn in a given day with the Möbius Ring equipped
- Your 2nd NC will be available 7 turns after your 1st NC
- Your 3rd NC will be available 13 turns after your 2nd NC
- Your 4th NC will be available 19 turns after your 3rd NC
- Your 5th NC will be available 25 turns after your 4th NC
- Your 6th NC will be available 31 turns after your 5th NC
- Your 7th NC will be available 41 turns after your 6th NC
- Your 8th NC will be available 41 turns after your 7th NC
- Your 9th NC will be available 41 turns after your 8th NC
- Your 10th NC will be available 41 turns after your 9th NC
- Your 11th NC will be available 41 turns after your 10th NC
And after your 11th, NCs 12-16 will be available every 51 turns after the previous NC, and NCs from 17 onward will be available 76 turns after the previous NC. Note that wording: the NC will become available after that amount of turns spent. Unlike digitize/enamorang, you don't have a 100% chance of hitting the NC on that turn; the percentage appears to be pretty high (likely 50% or greater), but it clearly isn't 100%. So you can expect to have about 10-15 turns of extra fishing alongside these numbers while you wait for your NC to appear.
Having discussed the NCs you use to pump up your Paradoxicity, let's examine the rate at which your ring generates time cops. There is a clear relationship between time cops and Paradoxicity, and one that makes a certain narrative sense with the IOTM's whole milieu: the more you've messed with the timeline, the higher chance a cop will accost you to try and stop you. However, spading thus far hasn't demonstrated a pure linear increasing relationship between your Paradoxicity and your time cop rate. Instead, it's shown a bit of a step-function relationship, where you switch between five "tiers" of time cop rates over the course of your run:
- At 0 to 4 Paradoxicity, time cops appear approximately 2% of the time
- At 5 to 9 Paradoxicity, time cops appear approximately 4% of the time
- At 0 to 14 Paradoxicity, time cops appear approximately 8% of the time
- At 0 to 19 Paradoxicity, time cops appear approximately 16% of the time
- At 20+ Paradoxicity, time cops appear approximately 32% of the time
I will again note: this is absolutely not finalized spading. The data is pretty close to this, but still about 1-2% fuzzy around the edges, which is enough that gives me some pause. But this is clearly close enough that it can be used as a (very) general rule, and it's good enough for us to do some basic math on encounter rate later on in this article.
Speedrun Applicability
The first 11 time cops fought per day do not cost an adventure. Dumping one of these in a delay zone is obviously good, as they won't cost a turn and also provide scaling statgain. In a universe where you can get all 11 into delay zones every day, this would be an incredibly strong IOTM.
Additionally, some of the enchantments are quite powerful. At 3 Paradoxicity, the ring offers +100% Gear Drop, useful for Knob Goblin Harem Girl items, the amulet of plot significance, and the mohawk wig. At 8 Paradoxicity it gets +100% Food drop for blackberries and blasting soda. At 13 Paradoxicity it gets a whopping +100% item drops of all kinds, useful for... well, anything.
To make matters more complicated, the Möbius Strip NC offers lots of beneficial effects when increasing or decreasing Paradoxicity, such as +30 ML, +5 fxp for 100 adventures, and more. But any NC choice that repairs the timeline will lower your Paradoxicity, making it less likely you encounter time cops! What a hassle. It'll be a bit difficult to extract full value from the ring's various (singular) dimension.
Spadester Interlude, Part II
Scotch here! I'm back! Just like the dinosaurs in "We're Back, A Dinosaur Story." Having gone over the general formula behind the NC rate and the time cop rate, I think it's worth synergizing our understanding of those two formulae into something that will make the whole thing make a bit more sense.
Since you roll for a time cop every entry into an adventure.php zone with the ring equipped (and multiple times when you have things like free-exit NCs), you can figure out a rough amount of predicted cops at each section of your run. For simplicity's sake, I will do this in a 1-day context and leave 2+ day as an exercise for the reader. Assuming that you are never taking any of the timeline repair options (and are singularly focused on juicing your Paradoxicity), you will need 4 + 7 + 13 + 19 + 25 + a few slush turns rolling for NCs = ** ~ 74 turns** to go from the 2% to 4% time cop rates. In those 74 turns, if you kept the ring on, you would expect ~ 1.4 time cops (74 * 0.02), assuming no freebies in delay or freeruns. This leads to the following setup, in a 400 turn 1-day run, if you never ever take your Möbius ring off:
- You expect 1.4 time cops from turns 0 to 74 (at a 2% rate)
- You expect 8.1 time cops from turns 75 to 320 (at a 4% rate)
- You expect 5.3 time cops in turns 321 to the sorceress' tower at turn 388 (at a 8% rate)
It's worth noting: these numbers are significantly higher than what you're actually going to hit in-run, unless you don't care about putting them in delay at all and keep the ring equipped for your entire run. You spend about 60% of your run outside of zones where you want wanderers to hit. Time cops outside of delay zones (or zones where free wanderers can advance timers, like Spookyraven Kitchen or 8-Bit Realm) are of pretty limited value to a speedrunner. There are a few extra factors here too, though: getting a few of them out of delay zones isn't a particularly big deal, but there are certain zones where there's simply no reason to wear the ring. For one key example: your 42 war turns. A time cop encountered in the battlefield does literally nothing for you.
A rough accounting of all the delay in a normal KOL run leads you to roughly 120 turns of delay, and about 50 turns where wanderers like time cops are as good as a native fight. If you were to (somehow) hold every single one of those turns for after you hit the 8% time cop rate, you'd be looking at well over 11 expected timecops, and could plop them all in delay. As is, I think you can hold off a good portion of those, but not -that- many. And some get eaten by the natural use of resources (for instance, losing anywhere from 8 to 18 with NC forcers + accountant fights in the Hidden City) or fights you actually need to do for alternate reasons (like quiet healers and burly sidekicks). My educated guess would be that most runs will need to run through 20-25 turns of delay in the 2% time cop zone, 80% of the remaining delay in the 4% time cop zone, and the remainder over the final 10-70 timecop zone (with turns differentiated by how fast you expect you end up going).
If you only wear the ring in delay, the expected numbers from above turn into:
- You expect 0.4 time cops from turns 0 to 74 (at a 2% rate, with approximately 20 runnable turns)
- You expect 5.2 time cops from turns 75 to 320 (at a 4% rate, with approximately 130 runnable turns)
- You expect 1.6 time cops in your remaining turns to the sorceress' tower at turn 388 (at a 8% rate, with approximately 20 runnable turns)
... for a total of 7.2 expected time cops per run. This roughly matches what speedsters are seeing; as an example, I got 9 in a recent 1-day run where I wore it outside of delay a decent bit, which led to 6 time cops in delay and 3 time cops out of delay. Which is all to say: this thing is incredibly complicated, almost to an annoying extent, but actual usage isn't super complicated. Leave it on for delay turns, take it off if a time cop doesn't help you (outside of the turns where you're rolling for your NCs). With mild management of your route, 6-8 cops in delay over your whole run is about what you can expect, with that amount being highly variable from run to run, and able to be tweaked ever-so-slightly upwards with careful management of delay to ensure a maximal amount is burned at the very end of your run. It also will be more powerful for slower runners; cops are markedly easier to encounter after around turn 300, which can be far less than 100 turns at bleeding edge but almost 200 turns in the mid silver range.
It's also worth noting that you get the benefit of faster NC acquisition in 2-day runs; for example, in a bleeding edge 2-day 300 turn run, you can expect 7 NCs D1 and 7 NCs day 2, with the critical 11th NC coming with around 70 turns left to go. That's right: this is yet another IOTM that wants a (relatively) short D1. Nasty work.
2025 In-Standard Synergies
The Möbius ring interacts with quite a few other IOTMs:
- The Prismatic Beret (2025) can give you a slightly better shot at getting time cop time hats, as it has an associated boost to hat drop.
- Anything that benefits from extra free fights (such as pushing for familiar XP with the Chest Mimic (2024) and Cooler Yeti (2025)), will appreciate free time cop fights. And the 100-duration +5 familiar XP buff helps even more!
- Because time cops have a higher priority than most non-superlikely adventures, you can pair it with multiple things as a bonus chance at +1 turnsave. Things like banish-runaways, Everything Looks Red, Everything Looks Green, or even the escapable free noncom in every zone from the Peridot of Peril (2025).
Overall Rating
We'd rate the Möbius ring a tier 2 IOTM, because of its dimensionally multi-one-sided skill ceiling. Although it can save 10+ turns in a day, even after everything gets figured it, it's going to be very difficult to squeeze full value out of the ring. Your expectation should be essentially 6-7 cops in delay over your whole run, and a handful of cute little enchantments that may shave a turn or so. Time may be a flat circle, but this ring is a difficult metaphor.