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User:Tsukuyomai

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Server: Tarlach
On Hiatus


Maximum Possible Protection (Renovation)

Updated as of 5/13/2016

Note: Homunculus enchant has been taken out of consideration as it is too hard to keep below level 15.
Note 2: Animal Taming temporary stat boosts were not taken into consideration as there is not enough information, and the boost is highly situational.
Note 3: It is also possible to get +6.3 prot with the Hard Shelled score, but the score can no longer be obtained.
Note 4: It is possible for giants to have even more protection with the Wind Guard Protection reforge on armor, however there is not enough data on the wiki to add to this build.

Additional Prot Bonuses:


Max Prot set (Human):

Head: (+9.5 prot)

  • Any heavy headgear (or special headgear with 1 prot) (+1 prot)
  • Enchants
  • Lv 5 Protection reforge (+7.5 prot)

Body: (+30.5 prot w/ Light Armour) (+38.5 prot w/ Heavy Armour)

  • Bhafel Slayer Guard (M/F) (+10 prot)
  • Upgrades: (+16 prot)
    • Surface Enhancement 2 (+1 prot)
    • Surface Enhancement 2 (+1 prot)
    • Surface Enhancement 2/3/4 (+1 prot)
    • Edern's Heavy Armor Modification (+3 prot)
    • Magic Armor Artisan (+10 prot)

OR

  • Languhiris Chaser Armor (M/F) or Caswyn's Armor/Pihne's Armor (+5 prot)
  • Upgrades: (+15 prot)
    • Surface Enhancement 2 (+1 prot)
    • Surface Enhancement 2/3 (+1 prot)
    • Surface Enhancement 2/3 (+1 prot)
    • Meles' Light Armor Upgrade (+2 prot)
    • Magic Armor Artisan (+10 prot)


  • Enchants (+3 prot Light Armour) (+5 prot Heavy Armour)
  • Lv 5 Protection reforge (+7.5 prot)

Gloves: (+13.5 prot) (+17.5 prot with barrier spikes up)

  • Languhiris Chaser Gloves (+5 prot)
  • Enchants (+1 prot) (+4 prot with barrier spikes up)
    • Prefix: Deep (+1 prot) (+4 prot with barrier spikes up)
  • Lv 5 Protection reforge (+7.5 prot)

Boots: (+12.5 prot)

  • Bhafel Slayer Greves (M/F) or Languhiris Chaser Boots (M/F) (+4 prot)
  • Enchants
  • Lv 5 Protection reforge (+7.5 prot)

Shield: (20.5 prot)

  • Divine Shield (+6 prot)
  • Upgrades: (+5 prot)
    • Artisan Upgrade with max prot roll (+2 prot)
    • Gem update (+3 prot)
  • Enchants (+2 prot)
  • Lv 5 Protection reforge (+7.5 prot)

Accessories: (+13.5 prot) (+27 prot for two)

  • Drosera (+3 prot) (+6 prot)
  • Enchants (+3 prot) (+6 prot)
    • Suffix: Taurus (+3 prot) (+6 prot)
  • Lv 5 Protection reforge (+7.5 prot) (+15 prot)

Weapon: (+1 prot)

  • Any enchantable metal 1h weapon.
  • Enchants (+1 prot)

OR:

  • Any enchantable cylinder
  • Enchants (+1 prot)

Title: (+22 prot)

  • Barrier spikes master (+12 prot)
  • SAO Kirito (Secondary Title) (+10 prot)

Skills: (+15 prot Light Armor) (+20 prot Heavy Armor)

  • Shield Mastery Rank 3+ (+10 prot (Medium shield bonus))
  • Heavy Armour Mastery Rank 1 (+10 prot)
  • Light Armour Master Rank 1 (+5 prot)

Totem: (+0)


Stand alone total Prot (Heavy Armor Build): 164.5 prot +3 w/ barrier spikes up

Stand alone total Prot (Light Armor Build): 151.5 prot +3 w/ barrier spikes up



Buffs: +22.064 prot

  • Song of Endurance Rank 1 (+5 prot)
    • With Rank 1 playing, rank 1 song, and perfect playing (5 prot) * 1.30 * 1.3
  • Equipment (5 prot) * 1.46 * 1.3
    • Instrument:
      • Demonic Gloomy Sunday (+16 music buff)
  • With Bard (buffer) enchants: (16 prot) * 1.62 * 1.3
    • Instrument:
    • Headgear
      • Suffix: Chorus (+3 music buff)
    • Body:
      • Prefix: Encore (+3 music buff)
    • Glove:
      • Prefix: Solo (+2 music buff))
    • Shoes:
      • Prefix: Solo (+2 music buff)
  • Bard (buffer) Reforges: (8 prot) * 1.82 * 1.4
    • Instrument:
      • Perfect play bonus Lv 10 (+10% perfect play bonus)
      • Buff Effect Lv 20 (+20 music buff)
    • Accesories:
      • Song of Endurance Protection (not enough data)
    • Hand gear:
      • Song of Endurance Protection Lv 3 (+3 prot)
  • Bard (buffer) Title: (8 prot) * 1.92 * 1.4
  • Bard (buffer) Talent: Grandmaster Bard: (8 prot) * 1.97 * 1.4

Total prot with buffs (Heavy): 186.564 prot +3 w/ barrier spikes up

Stand alone total Prot (Light): 173.564 prot +3 w/ barrier spikes up

Max prot set (giant):
Same as human except:

  • May use windbreaker (+100 prot)

Stand alone total prot (Heavy): 164.5 prot +3 w/ barrier spikes up +100 w/ windbreaker
Total prot with buffs: 186.564 prot +3 w/ barrier spikes up +100 w/ windbreaker

Stand alone total prot (Light): 151.5 prot +3 w/ barrier spikes up +100 w/ windbreaker
Total prot with buffs: 173.564 prot +3 w/ barrier spikes up +100 w/ windbreaker

Max prot set (elf):
Same as human

Mostly confirmed things:

Dan Test Revamp

  • Dan ranks now give a boost to skills.
  • In addition to the skills listed below, the following life skills also have dan ranks:
    • Tailoring
    • Blacksmithing
    • Potion Making
    • Hillwen Engineering
    • Magic Crafting

Combat Mastery

Rank Dan 1 Dan 2 Dan 3
Max HP (Human) 160 170 180
Max Damage (Human) 20 22 24
Min Damage (Human) 10 12 14
Max HP (Giant) 160 170 180
Max Damage (Giant) 22 24 26
Min Damage (Giant) 11 12 13
Max HP (Elf) 160 170 180
Max Damage (Elf) 11 12 13
Min Damage (Elf) 7 7 8

Counter

Rank Dan 1 Dan 2 Dan 3
Damage From Opponent (%) (Human) 210 220 230
Damage From Self (%) (Human) 165 170 175
Extra Crit Chance (%) (Human) 10 10 13
Damage From Opponent (%) (Giant) 210 220 230
Damage From Self (%) (Giant) 210 220 230
Extra Crit Chance (%) (Giant) 10 10 13
Damage From Opponent (%) (Elf) 160 170 180
Damage From Self (%) (Elf) 165 170 175
Extra Crit Chance (%) (Elf) 10 10 13

Defense

Rank Dan 1 Dan 2 Dan 3
Base Def (Human) 22 24 26
Additional Def on Use (Human) 70 75 80
Base Def (Giant) 34 38 42
Additional Def on Use (Giant) 90 100 110
Base Def (Elf) 16 17 18
Additional Def on Use (Elf) 70 75 80


Smash

Rank Dan 1 Dan 2 Dan 3
Damage (%) (Human/Elf) 540 580 620
Damage (%) (Giant) 640 680 720

Windmill

Rank Dan 1 Dan 2 Dan 3
Damage (%) (Human/Giant) 270 290 310
Damage (%) (Elf) 210 220 230

Assault Slash

Rank Dan 1 Dan 2 Dan 3
Damage (%) 220 240 260

Ranged Attack

Rank Dan 1 Dan 2 Dan 3
Max Damage (Human) 28 31 34
Min Damage (Human) 12 14 16
Max Damage (Elf) 17 19 21
Min Damage (Elf) 12 14 16

Magnum Shot

Rank Dan 1 Dan 2 Dan 3
Damage (%) 520 540 560

Crash Shot

Rank Dan 1 Dan 2 Dan 3
Fragment Damage (%) 260 270 280

Firebolt

Rank Dan 1 Dan 2 Dan 3
Max Damage 122 124 126
Min Damage 62 64 66

Icebolt

Rank Dan 1 Dan 2 Dan 3
Max Damage 88 96 104
Min Damage 77 84 91

Lightning Bolt

Rank Dan 1 Dan 2 Dan 3
Max Damage 160 170 180
Min Damage 48 56 64

Fireball

Rank Dan 1 Dan 2 Dan 3
Max Damage 2520 2640 2760

Ice Spear

Rank Dan 1 Dan 2 Dan 3
Max Damage 272 304 336
Min Damage 172 184 196

Thunder

Rank Dan 1 Dan 2 Dan 3
Max Damage 427 454 481
Min Damage 111 116 121

Lightning Rod

Rank Dan 1 Dan 2 Dan 3
Max Damage 475 485 500
Min Damage 141 148 155
Full Charge Bonus (%) 210 220 230

Healing

Rank Dan 1 Dan 2 Dan 3
Max Per Charge 47 49 51
Min Per Charge 39 43 47

Water Cannon

Rank Dan 1 Dan 2 Dan 3
Max Damage 149 162 175
Min Damage 130 141 152

Sand Burst

  • Dan 3 increases blind duration.
Rank Dan 1 Dan 2 Dan 3
Max Damage 64 68 72
Min Damage 32 34 36

Flame Burst

Rank Dan 1 Dan 2 Dan 3
Max Damage 44 45 46
Min Damage 22 23 24

Lance Charge

Rank Dan 1 Dan 2 Dan 3
Damage (%) 265 280 295

Charging Strike

Rank Dan 1 Dan 2 Dan 3
Damage (%) 420 440 460

Spinning Uppercut

Rank Dan 1 Dan 2 Dan 3
Damage (%) 630 660 690
Def Reduction 31 32 33

Pummel

Rank Dan 1 Dan 2 Dan 3
Damage (%) 930 960 1000

Act 4

Rank Dan 1 Dan 2 Dan 3
Damage (%) 320 340 360

Act 6

  • Dan 3 increases range to 6.5m. Puppet prot and def also increase as you rank it.
Rank Dan 1 Dan 2 Dan 3
Damage (%) 205 210 215

Flash Launcher

Rank Dan 1 Dan 2 Dan 3
Damage (%) 850 900 950

Shooting Rush

Rank Dan 1 Dan 2 Dan 3
Damage (%) 440 480 520

Shuriken Charge

Rank Dan 1 Dan 2 Dan 3
Damage (%) 440 450 460

Kunai Storm

Rank Dan 1 Dan 2 Dan 3
Damage (%) 420 440 460

Mission Success Rate Formula

min((min(str, mission max str) + min(div, mission max div) + ...) / (mission max str + mission max div + ...), 95%) - (stress above 50%)

i.e. If a mission's max requirements are str 5, div 7, and will 8, and the selected squires stats are all 6, the success rate will be: (5 (str) + 6 (div) + 6 (will)) / (5 + 7 + 8) = 17/20 = 85%

Magic Craft Crafting Ranges

  • Rank 9
    • Lucky Rabbit Feet: 1-8 luck
  • Rank 6
    • Barbaric Wands: 0-3% Magic Damage
    • Drosera: 1-6 def, 1-3 prot
  • Rank 5
    • Hermit Staff: 0-5%
  • Rank 4
    • Hermit Staff: 0-6%
  • Rank 2
    • Hermit Staff: 0-7%
  • Rank 1
    • Hermit Staff: 1-8%
    • Savage Wands: 1-5%

Bard Skill bonus calculation

Source: gameabout
Currently, the effects of bard skills can be increased with higher ranked playing as well as enchants, special upgrades, and how well you play (e.g. great success vs success).
The formula seems to be: bard skill effect * (1 + playing rank(15% for rank 1, 14% for rank 2, etc) + reforge bonus + enchant bonus) * playing level(1.3 for perfect, 1.1 for very good? 1 for normal?)

Fire Shield Effectiveness

* NOTE: This information was compiled pre renovation. As such, it may be outdated.

  • Each individual test used the same equipment and same fireballer throughout the test.
  • Test 1: Rank 4 fire shield
    • Damage without: 4700~5100
    • Damage with: ~724
    • Conclusion: ~85% fireball damage reduction
  • Test 2: Rank f fire shield
    • Damage without: ~4500
    • Damage with: 1100~1300
    • Conclusion: ~75% fireball damage reduction
  • User:Blargel: 90% fireball damage reduction at r1

Special Upgrade Analysis

Summary

Melee/Archery/Martial Arts/Guns will benefit more from R type upgrades after achieving some amount of a combination of crit/bal/dmg/critical skill rank.
Magic will benefit more from R type upgrades once you hit over roughly 500 MATK depending on the spell.
Alchemy is largely unaffected by one or both special upgrade types with the exceptions being Flame Burst, Water Cannon, Heat Buster, and Frozen burst.

For physical weapons (Melee/Archery/Martial Arts/Guns), after a certain amount of damage, a special upgrade type R weapon will have more dps than a special upgrade type S weapon. The following are the results of my calculations
Where:

  • min means minimum damage
  • max means maximum damage
  • x is the damage vector (min, max)
  • x# is the damage vector ((max - min) * balance + min, max)
  • xb is the damage vector (min, (max - min) * balance + min)
  • 50% of the time damage will be between the values of x# while the other 50% of the time the damage will be between the values of xb
  • dmg is the damage vector: 0.5 * x# + 0.5 * xb

= 0.5 * (x# + xb)
= 0.5 * (((max - min) * balance + min, max) + (min, (max - min) * balance + min))
= 0.5 * ((max - min) * balance + 2min, (max - min) * balance + min + max)

  • Calculation for S type upgrades is: (dmg with S type bonus) * (1 - crit rate) + (dmg with S type bonus) * crit rank multiplier * crit rate
  • Calculation for R type upgrades is: dmg * (1 - crit rate) + dmg * (crit rank multiplier + R type bonus) * crit rate

Formula

The formula for calculating when R type starts to have more DPS than S type is then:
R Type Damage = S Type Damage
dmg * (1 - crit rate) + dmg * (crit rank multiplier + R type bonus) * crit rate = (dmg with S type bonus) * (1 - crit rate) + (dmg with S type bonus) * crit rank multiplier * crit rate
dmg * ((1 - crit rate) + (crit rank multiplier + R type bonus) * crit rate) = (dmg with S type bonus) * ((1 - crit rate) + crit rank multiplier * crit rate)
(0.5 * ((max - min) * balance + 2min, (max - min) * balance + min + max)) * ((1 - crit rate) + (crit rank multiplier + R type bonus) * crit rate) = ((0.5 * ((max - min) * balance + 2min, (max - min) * balance + min + max)) with S type bonus) * ((1 - crit rate) + crit rank multiplier * crit rate)

Calculations

(NOTE: The calculations only give you one number; you may adjust the derived "max + min = some number" equations into the form y = mx + b and plot the cutoff line for when R type upgrades are better than S type.)
(NOTE 2: I will not be taking dual wielding into account nor will I calculate for cylinders.)
Assuming:

  • Crit is rank 1
  • Crit rate is 33.3%
  • Balance is 80%

The formula is therefore:
dmg * (2/3 + (2.5 + R type bonus)/3) = (dmg with S type bonus) * (2/3 + 2.5/3)
dmg = dmg with S type bonus * (2/3 + 2.5/3) /(2/3 + (2.5 + R type bonus)/3)
dmg = dmg with S type bonus * 1.5 /(1.5 + R type bonus / 3)

Where: dmg = 0.5 * ((max - min) * 0.8 + 2min, (max - min) * 0.8 + min + max)
= ((max - min) * 0.4 + min, (max - min) * 0.4 + (max + min) * 0.5)
= ((max - min) / 2.5 + min * 2.5 / 2.5, (max - min) / 2.5 + (max + min) / 2)
= ((max + 1.5min) / 2.5, (2max - 2min) / 5 + (2.5max + 2.5min)/5)
= (0.4max + 0.6min, (4.5max + 0.5min)/5)
= (0.4max + 0.6min, 0.9max + 0.1min)
and: dmg with S type bonus = (0.4(max + S type max bonus) + 0.6(min + S type min bonus), 0.9(max + S type max bonus) + 0.1(min + S type min bonus))

1 Handed Axes

Level 3 dmg = dmg with S type bonus * 1.5 / (1.5 + 0.23 / 3)
(0.4max + 0.6min, 0.9max + 0.1min) = (0.4(max + 16) + 0.6(min + 8), 0.9(max + 16) + 0.1(min + 8)) * 0.95137420718816067653276955602537
(0.4max + 0.6min, 0.9max + 0.1min) = (0.38(max + 16) + 0.57(min + 8), 0.856(max + 16) + 0.095(min + 8))
(0.4max + 0.6min, 0.9max + 0.1min) = (0.38max + 0.57min + 10.64, 0.856max + 0.095min + 14.456))
(0.4max + 0.6min, 0.9max + 0.1min) - (0.38max + 0.57min, 0.856max + 0.095min)) = (10.64, 14.456)
(0.02max + 0.03min, 0.044max + 0.005min) = (10.64, 14.456)
Thus we end up with:
0.02max + 0.03min = 10.64 and 0.044max + 0.005min = 14.456

Solving for max:
0.02max + 0.03min = 10.64
-(0.044max + 0.005min = 14.456) * 6

0.02max + 0.03min = 10.64
-(0.264max + 0.03min = 86.736
-0.244max = -76.096

max = 311.86885245901639344262295081967

Solving for min:
0.02max + 0.03min = 10.64
6.2373770491803278688524590163934 + 0.03min = 10.64
min = (10.64 - 6.2373770491803278688524590163934)/0.03
min = 146.75409836065573770491803278689

146.75~311.86

Level 5 dmg = dmg with S type bonus * 1.5 / (1.5 + 0.43 / 3)
(0.4max + 0.6min, 0.9max + 0.1min) = (0.4(max + 30) + 0.6(min + 16), 0.9(max + 30) + 0.1(min + 16)) * 0.91277890466531440162271805273834
(0.4max + 0.6min, 0.9max + 0.1min) = (0.365(max + 30) + 0.548(min + 16), 0.822(max + 30) + 0.0913(min + 16))
(0.4max + 0.6min, 0.9max + 0.1min) = (0.365max + 0.548min + 20.294, 0.822max + 0.0913min + 26.1208)
(0.4max + 0.6min, 0.9max + 0.1min) - (0.365max + 0.548min, 0.822max + 0.0913min) = (20.294, 26.1208)
(0.035max + 0.052min, 0.078max + 0.0087min) = (20.294, 26.1208)

Therefore:

  • 0.035max + 0.052min = 20.294
  • 0.078max + 0.0087min = 26.1208

Solving for max:
0.035max + 0.052min = 20.294
-(0.078max + 0.0087min = 26.1208) * 5.9770114942528735632183908045977
-0.43120689655172413793103448275862max = -135.83032183908045977011494252874

max = 315.00034652805544448887111821938

Solving for min:
0.035max + 0.052min = 20.294
min = (20.294 - 0.035max)/0.052
min = 178.24976675996268159402905504465

178.25~315

Level 6 dmg = dmg with S type bonus * 1.5 / (1.5 + 0.53 / 3)
(0.4max + 0.6min, 0.9max + 0.1min) = (0.4(max + 37) + 0.6(min + 20), 0.9(max + 37) + 0.1(min + 20)) * 0.89463220675944333996023856858847
(0.4max + 0.6min, 0.9max + 0.1min) = (0.358(max + 37) + 0.537(min + 20), 0.805(max + 37) + 0.0895(min + 20))
(0.4max + 0.6min, 0.9max + 0.1min) = (0.358max + 0.537min + 23.986, 0.805max + 0.0895min + 31.575)
(0.042max + 0.063min, 0.095max + 0.0105min) = (23.986, 31.575)

Therefore:

  • 0.042max + 0.063min = 23.986
  • 0.095max + 0.0105min = 31.575

Solving for max:
0.042max + 0.063min = 23.986
-(0.095max + 0.0105min = 31.575) * 6
-0.528max = -165.464

max = 313.37878787878787878787878787879

Solving for min:
0.042max + 0.063min = 23.986
min = (23.986 - 0.042max)/0.063
min = 171.81096681096681096681096681097

171.81~313.38

1 Handed Weapons

Level 6 dmg = dmg with S type bonus * 1.5 /(1.5 + 0.42 / 3)
(0.4max + 0.6min, 0.9max + 0.1min) = (0.4(max + 31) + 0.6(min + 15), 0.9(max + 31) + 0.1(min + 15)) * 0.91463414634146341463414634146341
(0.4max + 0.6min, 0.9max + 0.1min) = (0.366(max + 31) + 0.549(min + 15), 0.823(max + 31) + 0.0915(min + 15))
(0.4max + 0.6min, 0.9max + 0.1min) = (0.366max + 0.549min + 19.581, 0.823max + 0.0915min + 26.8855)
(0.034max + 0.051min, 0.077max + 0.0085min) = (19.581, 26.8855)

Therefore:

  • 0.034max + 0.051min = 19.581
  • 0.077max + 0.0085min = 26.8855

Solving for max:
0.034max + 0.051min = 19.581
-(0.077max + 0.0085min = 26.8855) * 6
-0.428max = -141.732

max = 331.14953271028037383177570093458

Solving for min:
0.034max + 0.051min = 19.581
min = (19.581 - 0.034max) / 0.051
min = 163.1748213304013194062671797691

163.17~331.15


2 Handed Weapons

Level 6 dmg = dmg with S type bonus * 1.5 /(1.5 + 0.62 / 3)
(0.4max + 0.6min, 0.9max + 0.1min) = (0.4(max + 48) + 0.6(min + 25), 0.9(max + 48) + 0.1(min + 25)) * 0.87890625
(0.4max + 0.6min, 0.9max + 0.1min) = (0.352(max + 48) + 0.527(min + 25), 0.791(max + 48) + 0.0879(min + 25))
(0.4max + 0.6min, 0.9max + 0.1min) = (0.352max + 0.527min + 30.071, 0.791max + 0.0879min + 40.1655)
(0.048max + 0.073min, 0.109max + 0.0121min) = (30.071, 40.1655)

Therefore:

  • 0.048max + 0.073min = 30.071
  • 0.109max + 0.0121min = 40.1655

Solving for max:
0.048max + 0.073min = 30.071
-(0.109max + 0.0121min = 40.1655) * 6.0330578512396694214876033057851
-0.60960330578512396694214876033058max = -212.24978512396694214876033057851

max = 348.17689325126759035817900816138

Solving for min:
0.048max + 0.073min = 30.071
min = (30.071 - 0.048max) / 0.073
min = 182.99327567039939264119736449663

182.99~348.18

Wands and Staves

  • Calculations are different than for melee weapons.
  • I will be calculating based on the magic damage formula above.
  • I will be assuming critical hit is rank 1.
  • I will be assuming critical hit is 33.3% and magic balance is 100%.
  • Only the first section of the magic damage formula ((Base Damage + (Spell Constant * Magic Attack))) and the crit bonus are affected by special upgrades.
  • Therefore I will be comparing (Base Damage + (Spell Constant * (Magic Attack + S type bonus))) * 2.5 with (Base Damage + (Spell Constant * Magic Attack)) * (2.5 + R type bonus).
    • Where Base Damage = 0.5 * ((max - min) * balance + 2min, (max - min) * balance + min + max)
      • = 0.5 * ((max - min) + 2min, (max - min) + min + max)
      • = 0.5 * (max + min, 2 max)
      • = (0.5 max + 0.5 min, max)
  • Therefore the equation is ((0.5 max + 0.5 min, max) + ((min const, max const) * (Magic Attack + S type bonus))) * (1 * (1 - crit rate) + 2.5 * crit rate) vs ((0.5 max + 0.5 min, max) + ((min const, max const) * Magic Attack)) * ((1 - crit rate) + (2.5 + R type bonus) * crit rate)

Calculating Expected Number of Stones Required for Special Upgrading

Summary

From
To
0 1 2 3 4 5 6 7
0 N/A 1 4 9 17.332 29.735704 68.294734288 230.9824476266
1 N/A N/A 3 8 16.332 28.735704 67.294734288 229.9824476266
2 N/A N/A N/A 5 13.332 25.735704 64.294734288 226.9824476266
3 N/A N/A N/A N/A 8.332 20.735704 59.294734288 221.9824476266
4 N/A N/A N/A N/A N/A 12.403704 50.962734288 213.6504476266
5 N/A N/A N/A N/A N/A N/A 38.559030288 201.2467436266
6 N/A N/A N/A N/A N/A N/A N/A 162.6877133386
7 N/A N/A N/A N/A N/A N/A N/A N/A

Math Behind it

To solve this problem, I basically considered the expected number of attempts required for each step and multiplied one less than that by the previous amount of attempts required.

  • To get from step 0->1, 1 attempt is expected.
  • To get from step 1->2, 3 attempts are expected. The first attempt (1 attempt) will downgrade the wand by one, which will require you to go back to step 0->1, costing you the number of attempts to get from step 0->1 (1 attempt). The second attempt is expected to work (1 attempt). This results in a total of:
    • (Expected attempts - 1) * (attempts for previous step + 1) + 1
    • = (2 - 1) * (1 + 1) + 1
    • =3

Using the same formula, we can determine the number of attempts required for each successive rank:

  • Step 2->3: (2 - 1) * (3 + 1) + 1 = 4 + 1 = 5
  • Step 3->4: (2.222 - 1) * (5 + 1) + 1 = 8.332
  • Step 4->5: (2.222 - 1) * 9.332 + 1 = 12.403

Then, to get the expected number of attempts required to get to a particular step, you simply add the expected attempts at each step together

  • Step 0->1 = 1
  • Step 0->2 = Step 0->1 + Step 1->2 = 1 + 3 = 4
  • Step 0->3 = Step 0->2 + Step 2->3 = 4 + 5 = 9
  • Step 0->4 = Step 0->3 + Step 3->4 = 9 + 8.332 = 17.332
  • Step 0->5 = Step 0->4 + Step 4->5 = 17.332 + 12.403 = 29.735

Finally, we can solve for the special case of Step 6, as this resets the step from 6->0 (technically) (by explosion):

  • Step 5->6 = (Expected attempts - 1) * (Step 0->5 + 1) + 1
    • = (2.222 - 1) * (29.735 + 1) + 1
    • = 1.222 * 30.735 + 1
    • = 38.558
  • Step 0->6 = Step 0->5 + Step 5->6 = 29.735 + 38.558 = 68.293