No Refills Low% analysis

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NoRefills-Low% analysis (for PAL and NTSC, inbounds, without SpaceTime-beam):

Definition:

It is not allowed to collect any kinds of drops and no refill-stations nor the ship are allowed to be used, such that the only ammunition is given directly by collecting items. The only exception is at the very end by obtaining health via the big Metroid during the fight against MotherBrain. Crystal Flashes are allowed (but I doubt they will be helpful in the end).

A rough estimate gives an upper bound of 23% for PAL (for TAS) and 14% is a general lower bound, but further optimization and reasoning still needs to be done.

Constraints to generate a better lower bound for the necessary number of items (for TAS):

5 Missiles are needed at least to open Bomb Torizo´s door

5 Missiles are needed at least to open the door for either early SuperMissiles or Sporespawn´s SuperMissiles

1 projectile is needed for entering the pre-golden-statues-room

1 projectile is needed to enter Dachora´s room

1 projectile is needed for the Charge-room door at the bottom right side

1 projectile is needed for the door on the right side in Noob-Bridge-room

2 projectiles are needed for the SuperMissile block at the entrance to Kraid´s Lair

1 projectile is needed for Kraid´s Eye-door

1 projectile is needed for Phantoon´s Eye-door

1 projectile is needed for either the left entrance-door to Wrecked Ship or Ice is needed, so it would be needed for either the entrance-door to Ice-beam´s area or for the entrance-door to Speedbooster´s area (or for Crystal Flash...)

1 projectile is needed for the door leading to pre-Draygon-room

1 projectile is needed for Draygon´s Eye-door

1 projectile is needed for the door leading to the ninja-pirate´s room

18 projectiles are needed for killing MotherBrain 1

1 projectile is needed for the door leading to MotherBrain´s Eye-door room

1 projectile is needed for MotherBrain´s Eye-door

Summing all of this up, we get already 10 Missiles and 31 SuperMissiles neglecting potential ammunition for zebetites and potential ammunition to get the SuperMissile packs. Hence at least 2 Missile packs and ammunition that corresponds to 7 SuperMissile packs is needed. Comparing this to what can at most be gotten in any 14%category, the difference of packs is at least 5, hence a lower bound for NoRefills-Low% is (14+5)%=19% (for both versions).

If any Crystal Flashes were used, then at least 5 more items would be needed (for 10 Missiles, 10 SuperMissiles and at least 5 more powerbombs, considering one has at most 10 powerbombs in a low%category and has to use 1 powerbomb for entering the lavapit, 1 for entering Ridley-PB-room aswell as 2 for the statue in that room) which would lead to 24%.

Now, in order to prove that no CF can be included in an optimal NoRefills-Low% option, one shall find explicit routes for upper bounds of number of items needed.

[TODO: Find complete set of optimal solutions for TAS and for humans.]

Sidenote: NoRefills-Max% is actually 100% and easily humanly doable.

No Refills conditions and conclusions

  • 5*M for BT red shell
  • 5*M for red shell to any of the 2 first Super Missiles
  • 5*M XOR 1*SM for pre-G4 door
  • 2*SM for Kraid lair entrance
  • 3*M XOR 1*SM for Kraid eye door; unless Ice & X-Ray & Spark allow to clip through it somehow
  • 3*M XOR 1*SM for Ridley eye door; unless Ice & X-Ray & Spark allow to clip through it somehow; using bugs with repeated knockback could possibly allow to get through, but the refill bugs couldn't be killed since they drop real drops so one would need to wait
  • 1*SM for WS Shaft green shell; unless some Moonfall or Freeze-Sink can work
  • 3*M XOR 1*SM for Phantoon eye door
  • 1*SM for Colosseum green shell
  • 3*M XOR 1*SM for Draygon eye door
  • 1*SM for Wasteland green shell
  • 1*PB for yellow shell in pre-Wasteland
  • 1*PB for PB-statue in Wasteland on way down
  • 1*PB for PB-Statue in Wasteland on way up unless having X-Ray
  • 1*PB for Gravitron yellow shell XOR Red Tower yellow shell for X-Ray, for LNF access
  • 1*SM for Noob-Bridge green shell XOR 1*SM Landing Site green shell, if one would get Etecoon-PBs
  • 1*SM XOR 3*PB XOR 5*M XOR 1*PB+3*M XOR 2*PB+1*M for Metroid Room 1 partly with Ice, unless damaging Freeze-Walk
  • 1*SM XOR 3*PB XOR 5*M XOR 1*PB+3*M XOR 2*PB+1*M for Metroid Room 2 partly with Ice, unless damaging Freeze-Walk
  • 1*SM XOR 3*PB XOR 5*M XOR 1*PB+3*M XOR 2*PB+1*M for Metroid Room 3 partly with Ice, unless damaging Freeze-Walk
  • 1*SM XOR 3*PB XOR 5*M XOR 1*PB+3*M XOR 2*PB+1*M for Metroid Room 4 partly with Ice, unless damaging Freeze-Walk
  • 5*M XOR 1*SM for Seaweed Room red shell
  • 3*M XOR 1*SM for MB eye door
  • 6 projectiles + 3000 damage, with at least 18 projectiles total, for MB1


^^^^^^^^^^^^^^^^

-> At least 31 projectiles in total after obtaining Supers (neglecting Metroid rooms, neglecting TAS Freeze-Walks). This implies 2 Missile packs initially, and at least 7 further ammo packs afterwards.

-> Without Ice, one needs at least 12 additional PBs for Metroids, so in this case at least 3 PB-Packs are needed (but actually 1 more, as follows).

[?] To be figured out: Can 5 PBs in total suffice with Ice (since without it at least 3 PB packs are needed)? In this case no CF is possible, and Reserve Tanks can only be filled with a CF and otherwise stay empty until MB3, which means no Reverse LNF is possible in this case. In NTSC there's so far no known way to get past the Green Gate next to GlassTube to Maridia for players, and Norfair together with X-Ray location implies 4 PB uses. And Red Brinstar G-Mode GGG isn't an option without Reserve energy in NSTC. And the Crateria elevator access to Maridia requires another PB if one takes this path, and storing a Rocketjump in the room under Spazer with a spark and X-Mode requires Gravity for players in order to skip past the GlassTube. And Gravity cannot be reached without at least 1 further PB usage either. So at least 1 more PB is needed which makes up to 5 powerbomb uses. And maybe another powerbomb use can be concluded...?


-> Without Ice for Mickey Mouse room clip (and also no CF there), one needs at least 1 more PB use either for opening up the acid lowering statue or opening up the pre-pillars floor. So without Ice already 12+4=16 powerbombs are at least needed (taking Metroids into account, pre-Wasteland, Wasteland downwards, X-Ray or Wasteland upwards, and the PB without Mickey-Clip), which means 4 PB-Packs are needed. So maybe this allows to already exclude options without Ice with some estimate.

-> Ball, Bombs, Varia, 3 Tanks, 2°M, 7°(M/SM mix), PBs, Ice, Charge (unless Underflow),... is needed in NTSC XOR Ball, Bombs, Varia, 3 Tanks, 2°M, 7°(M/SM mix), 4°PBs, Charge (unless Underflow),... is needed without Ice in NTSC. And in the latter case one would need to add at least 2 more items in NTSC unless Gravity is added in order to be able to beat the game as player, because Ice (for Botwoon skip) is excluded for this case distinction (and a CF to fill up Reserves for G-Mode into Maridia would add immense costs of 10*M+10*SM+11*PB which would already require too many further ammo packs to be added), and (coming necessarily from GlassTube) another Missile XOR Super XOR PB XOR E-Tank XOR R-Tank, XOR Spazer XOR Wave XOR Plasma XOR Screwattack XOR Spacejump XOR X-Ray XOR Grapple wouldn't even allow to get up onto Everest even if one had an Elevator-Flashsuit available from the ammo, and Highjump, Speedbooster, Springball remain, but none of them alone suffice either (due to CF being too costly and pre-Botwoon room only being reachable and then passable in the Gravity case given an Elevator-Flashsuit for which low HP and 1*PB is needed). And hence this option would result in 22% items already for players on NTSC unless Gravity is combined with an ELevator-Flashsuit, and there exists a known route for 22% already, so if there is a lower item combination, it must include the former case or Gravity with an Elevator-Flashsuit for this case.{And maybe this reasoning can be translated to still hold in PAL.}

Remaining options: (1)(*) Ball, Bombs, Varia, 3 Tanks, 4°M, 5°(M/SM mix), 4°PBs, Charge (unless Underflow), Gravity (with Elev-Flash). (2) Ball, Bombs, Varia, 3 Tanks, 2°M, 7°(M/SM mix), PBs, Ice, Charge (unless Underflow),...

(*) Since an Elevator-Flashsuit is required in that case, and previous Missile ammo is emptied, at least 2 more Missile packs will need to be collected in order to be able to CF, which implies the 4°M, 5°(M/SM mix) part. In this case however, one would at most have 25 Supers to use altogether. And in this case unless one gets another item, one cannot travel through Maridia to get to Gravity, so additionally at least 1*PB for the moat-door is needed aswell as 1*SM for the WS Entrance-Hall.

Option (1) needs at least 1 further item as proven as follows. 4 PB-Packs provide 20 powerbombs of which 12 need to be used for Metroids, 1 for at least one required Elevator-Flashsuit (or CF in general for Flashsuit), and 1 for Gravitron, 1 for pre-Wasteland, 2 due to no X-Ray for Wasteland in both directions, and because no Ice is involved, 1 powerbomb for pre-Pillars floor and 1 for GlassTube and 1 for Snail-Room. Furthermore 1 for moat since the other path cannot be taken to get Gravity. But this adds up to already 21 powerbombs needed which is impossible with 21% in NTSC for players in the case (1). So at most the Ice-less Gravity Elevator-Flashsuit option can be extended to 22% to constitute another item-combination for Low%NoRefills, provided 22% turns out to be the minimum.

-> This means only extensions of (2) can anymore possibly lead to lower item counts!

For this purpose, question [?] will be analysed regarding the minimal number of powerbombs used (to possibly conclude that at least 2 PB-Packs are needed). Gravitron forces 1 powerbomb, another 1 is needed for pre-Wasteland, and Wasteland needs 2 more (of which 1 can and likely will be for X-Ray instead), which so far makes up for 4 powerbombs. Now one cannot go via Wrecked Ship to Maridia due to moat and Maridia elevator yellow shells which would make up 6. So in NTSC (otherwise the Freeze-Walk Gate is an option to Maridia), one enters Maridia via GlassTube either by using 1 powerbomb there or by Rocketjumping past it which requires X-Ray, Gravity and Speedbooster (in the 1 PB-Pack limit case) together with the moat shell for the 5th powerbomb.

-> So either the 3 additional items are exactly these, or the GlassTube requires the final 5th powerbomb.

Branches of the remaining option (2):

(2.1) Ball, Bombs, Varia, 3 Tanks, 2°M, 7°(M/SM mix), PBs, Ice, Charge (unless Underflow), X-Ray, Gravity, Speed. (2.2) Ball, Bombs, Varia, 3 Tanks, 2°M, 1°SM+6°(M/SM mix), PBs, Ice, Charge (unless Underflow),...

Now in neither of these cases, one can use/lay a single further powerbomb anymore if only 1 PB-Pack is involved, but powerbombs initially need to be obtained in some way at least. And the only viable options to do this without at least 1 powerbomb cost are: (a) Etecoon-PBs (strictly after Ice, since X-Ray needs PBs, but only 1 PB pack is taken) (b) Pink-Brinstar-PBs (strictly after Ice, and needs Spazer XOR Wave) (c) Crocomire-PBs --- Furthermore, with just 1 PB-Pack, one cannot go down past the Ice-Gates powerbomb floor nor the Beetom-Hall (initially to get to Speed), so without further PB-Packs, the red shell and green shell to Bubble Mountain need to be opened which make up 2 bonus projectiles.

-> Can (2.1) even work? In this case, the red shell to X-Ray needs to be opened, and red shell towards Speed (but possibly not the green one) needs to be opened. Also (since one could not enter Wrecked Ship from behind with these items to get Gravity), the WS-Entrance green shell would need to be opened. This adds up (together with the 2 bonus shells towards Bubble Mountain) to at least 5 more projectiles needed, so this makes for 31+5=36 projectiles are needed (neglecting Metroid rooms, neglecting TAS Freeze-Walks, while making sure that the 5 additional projectiles don't come from already previously covered conditions). And this would mean another Ammo pack would be needed, so such an option would already be at the 22% level. This means only option (2.2) which uses a powerbomb for the GlassTube remains.

If one in NTSC does not take X-Ray in option (2.2), then 4 powerbombs are used up in Norfair, and one cannot get into Maridia via WS due to 2 more yellow doors if only 1 PB-Pack is taken, so GlassTube needs to be blown up. And this means one would need to backdoor into Wrecked Ship without going through Snail room which would need another powerbomb, which means the green gate and bottom green shell of Toilet needs to be opened. And traversing the Sand Hall rooms would require at least 1 additional mobility item, which could only be either Highjump (which requires a red shell) [or Speed (which requires also at least 1 red shell) but would be really hard], which together with the above red and green shell towards Bubble Mountain make up 5 additional needed projectiles which make up at least 1 more ammo item. This means at most 1 further item (other than X-Ray) could be taken in addition to either of the following 2 cases for players in NTSC: (2.2.1) Ball, Bombs, Varia, 3 Tanks, 2°M, 1°SM+7°(M/SM mix), PBs, Ice, Charge (unless Underflow), Highjump. (2.2.2) Ball, Bombs, Varia, 3 Tanks, 2°M, 1°SM+7°(M/SM mix), PBs, Ice, Charge (unless Underflow), Speed. Now, this item needs to be Gravity in both cases, or one could not get back out of pre-Draygon room.

Green shells in Wasteland, Colosseum, WS Shaft, and Noob-Bridge XOR Landing Site, together with 2 Super Missiles at the entrance to Kraid's Lair already requires 6 Super Missiles, so at least 2 SM-Packs are required in any case.

So if 21% is possible for players in NTSC with only 1 PB-Pack taken, then the item combination needs to have one of the following 3 forms: (2.2.1+) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+6°(M/SM mix), PBs, Ice, Charge (unless Underflow), Highjump, Gravity. (2.2.2+) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+6°(M/SM mix), PBs, Ice, Charge (unless Underflow), Speed, Gravity. (2.2.3) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+5°(M/SM mix), PBs, Ice, Charge (unless Underflow), X-Ray,...

But now in the cases (2.2.1+) and (2.2.2+), one would need a powerbomb in Snail-room eitherway since one also needs to get to Draygon and there's no free item slot anymore so one cannot do Botwoon skip (Ice-Shield would require a further powerbomb use), unless [*] is possible.

In particular, (b) is not an option for (2.2.1+) and (2.2.2+), since no item slot is free for 21% with 1 PB-Pack. For players, it is not realistically possible to clip through any of the 4 Metroid room doors without Wave XOR Spazer, so in a situation with just 1 PB-Pack, at least 4 more projectiles are required. Due to the green shell to Crocomire and the then induced needed additional ammo, (c) is not an option for (2.2.1+) and (2.2.2+) either. So (a) would need to be used in the cases (2.2.1+) and (2.2.2+). And this means one would need to get to Ice before having powerbombs. Hence the only possible path to do this goes through Pink Brinstar which adds shell ammo requirements and would hence result in 1 more ammo being needed. This is a contradiction to those options being viable for 21% or less in NTSC for players if only 1 PB-Pack is used, and since one cannot add any further items to these options, they are impossible for 21% or less.

-> Hence, if there exists an item combination that just uses 1 PB-Pack with 21% or less, it must be of the form (2.2.3). If Etecoon-PBs are taken before Ice, then a 6th powerbomb would be needed in this case and then 2 PB-Packs would be required. So if only 1 PB-Pack can be used, then one needs to go through Pink Brinstar and open the bottom right green shell with 1 more projectile (which is a condition not already included/used above). Now, either Wave XOR Spazer is involved in this option, or 4 bonus projectiles from the Metroid rooms are needed here aswell. Hence if there's an option that only uses 1 PB-Pack, then it is of one of the following forms: (2.2.3.1) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+5°(M/SM mix), PBs, Ice, Charge (unless Underflow), X-Ray, Wave,... (2.2.3.2) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+5°(M/SM mix), PBs, Ice, Charge (unless Underflow), X-Ray, Spazer,... (2.2.3.3) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+6°(M/SM mix), PBs, Ice, Charge (unless Underflow), X-Ray,... In each of these 3 cases, at most 1 item can be added in order to stay at 21% or lower.

With just 1 PB-Pack, in NTSC for players, the 5 powerbombs need to be exactly used for Gravitron, pre-Wasteland, Wasteland towards Ridley, GlassTube (because Maridia could not be entered via WS and Forgotten Highway due to 2 yellow shells), and then either to get X-Ray or for going up through Wasteland. In particular, this means that any NTSC route for player with just 1 PB-Pack needs to backdoor enter Wrecked Ship through Maridia (obviously without Gravity), either entirely through the Sand Halls or through Snail room and then with X-Ray.



And the only other type of 21% candidate (with at least 2 PB-Packs) would be of this form: (2.2.3.0) Ball, Bombs, Varia, 3 Tanks, 2°M, 2°SM+5°(M/SM mix), 2*PBs, Ice, Charge (unless Underflow),... In this separate case, up to 2 items can be added in order to stay at 21% or lower.


[*] With Ice (and Gravity) it might be humanly possible to freeze 2 crabs in pre Snail-room in order to moonfall between them to skip the PB floor in Snail room.


^^^^^^^^^^^^^^ the above section is to be properly reformatted and organized



Concrete route approaches for playthroughs

NTSC/PAL 22% approach (without Crystal Flash) using Moonwalk:

Ball (0,0,0) -> 1.Missile (5,0,0) -> Bombs (0,0,0) -> last100%Missile (5,0,0) -> early SM (0,5,0) -> Etecoon-SM (0,8,0) -> Etecoon-PB (0,8,5) -> Charge (0,8,4) -> RedTower (past Noob-Bridge) (0,6,4) -> Phantoon (0,3,2) -> left WS-SM (0,8,2) -> right WS-SM (0,13,2) -> Varia (0,11,2) -> Kraid-ET (0,11,2) -> Gravity (0,10,2) -> Ice (0,9,2) -> LNF (0,7,1) -> Ridley-PB (via Mickey-Clip) (0,7,5) -> Ridley (0,5,4) -> Ridley-ET (0,5,4) -> Mainstreet-SM (0,10,2) -> Beach-SM (0,15,2) -> Snail-SM (0,19,1) -> Draygon (0,17,1) -> G4 (0,16,1) -> Gauntlet-ET (0,16,1) -> leftGauntlet-M. (5,16,1) -> rightGauntlet-M. (10,16,1) -> Big Metroid room (10,12,1) -> MB-Room (10,9,1) -> Ship.


Slightly easier routing of the above approach:

Ball (0,0,0) -> 1.Missile (5,0,0) -> Bombs (0,0,0) -> last100%Missile (5,0,0) -> early SM (0,5,0) -> Charge (0,4,0) -> Etecoon-SM (0,8,0) -> Etecoon-PB (0,8,5) -> RedTower (past Noob-Bridge) (0,6,4) -> Varia (0,4,4) -> Kraid-ET (0,4,4) -> Phantoon (0,0,2) -> left WS-SM (0,5,2) -> right WS-SM (0,10,2) -> Gravity (0,10,2) -> Ice (0,9,2) -> LNF (0,7,1) -> Ridley-PB (via Mickey-Clip) (0,7,5) -> Ridley (0,5,4) -> Ridley-ET (0,5,4) -> Mainstreet-SM (0,10,2) -> Beach-SM (0,15,2) -> Snail-SM (0,19,1) -> Draygon (0,17,1) -> G4 (0,16,1) -> Gauntlet-ET (0,16,1) -> leftGauntlet-M. (5,16,1) -> rightGauntlet-M. (10,16,1) -> Big Metroid room (10,12,1) -> MB-Room (10,9,1) -> Ship.


Another PAL 22% approach (without Crystal Flash) using MoonWalk:

Ball (0,0,0) -> 1.Missile (5,0,0) -> Bombs (0,0,0) -> last100%Missile (5,0,0) -> early SM (0,5,0) -> Etecoon-SM (direct Moonfall) (0,9,0) -> Etecoon-PB (0,9,5) -> BlueBrinstar-PB (0,9,8) -> Charge (0,9,8) -> Phantoon (via Landing Site) (0,5,7) -> left WS-SM (0,10,7) -> right WS-SM (0,15,7) -> Gravity (0,15,7) -> Red Tower (via Blue Brinstar) (0,14,6) -> Varia (0,12,6) -> Kraid-ET (0,12,6) -> Ice (0,10,6) -> LNF (0,8,5) -> Ridley (via Mickey-Clip) (0,6,3) -> Ridley-ET (0,6,3) -> Mainstreet-SM (0,11,1) -> Beach-SM (0,16,1) -> Snail-SM (0,20,0) -> Draygon (0,18,0) -> G4 (via Forgotten Highway) (0,17,0) -> Gauntlet-ET (0,17,0) -> leftGauntlet-M. (5,17,0) -> rightGauntlet-M. (10,17,0) -> Big Metroid room (10,13,0) -> MB-Room (10,10,0) -> Ship.

Playthrough planning:

Moonwalk needs to be turned on, but should probably be turned off once it isn't needed anymore.

Etecoon Beetoms are preferably to be dodged and especially Phantoon with 42 charged shots without taking flame drops after Moat-HIBJ will be rough (preferably if possible one should survive with 61+ HP for Bowling but CIBJ could be done either way), so this would mean 14 rounds of 3 hits against Phantoon starting from a far left Phantoon 2nd phase spawn each time, with 199 HP where getting hit means 20 damage every time, so one could get hit 9 times at most which should be doable; then Mini-Kraid.

One might need to check how much damage is taken at Mickey-Mouse room Ice-Clip, since Ridley follows soon afterwards! 160 damage from 1st Desgeega is to be expected, and a Multiviola does 12 damage. Ridley afterwards requires 200 charged Ice shots (except 1 free powerbomb that could be used for 200 damage; since at Wasteland exit, with diagonal bombjump out of the tunnel, the firesteam does only 5 damage, and the first Plowerhouse Holtzes can be dodged by just running off the doorway). And Draygon needs 67 charged Ice shots without getting hit much, and while dodging 3 active Turrets.

Big Metroid skip is needed aswell, and some finetuned Metroid kills using only 1 Super per room.