Sombrero vs. Volleyball: Difference between revisions

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imported>Bagatelle
Does anyone even use these tables? Bleh.
imported>Erathaol
Noting that the formula is incorrect.
 
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{{rewrite|The section on Wild Hares needs to be updated.  The Bandersnatch examples need to be re-written. The ML Stat formula is also incorrect (ML/3 not ML/4) meaning the math needs to be redone }}
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For a more basic analysis, see the [[Speedrun appendix]].
The two most popular stat-gain familiars can be pigeonholed into two categories: volleyballs and sombreros. For an analysis of the best familiar equipment for stat gains while using such familiars, see the [[Blood-Faced Volleyball]] and [[Hovering Sombrero]] articles. The present page is concerned with comparing volleyballs against sombreros. The main takeaway is that deciding which familiar to use is dependent on [[Monster Level]].


The two most popular stat-gain familiars can be pigeonholed into two categories: volleyballs and sombreros. For an analysis of the best familiar equipment for stat gains while using such familiars, see the [[Blood-Faced Volleyball]] and [[Hovering Sombrero]] articles. The present page is concerned with comparing volleyballs against sombreros.
==Sombrero vs. Volleyball Formula==
In the formulas below, VW is volleyball weight, SW is sombrero weight, and ML is monster level.


==Why does a Sombrero always win at ''some'' monster level?==
The formula for extra stats given by the Hovering Sombrero of weight SW is:
Computer scientists define the growth of a function using [[Wikipedia:Big O notation|Big O notation]], which describes the behavior of a function as its argument gets arbitrarily large. In the case of the stat gains for volleyballs and sombreros, volleyball stat gains are constant at a given volleyball weight. They are ''O''(''v''<sup>1/2</sup>) = ''O''(''C'').  However, the sombrero formula contains a non-constant factor in its numerator, the square root of the monster level, so its stat gains are ''O''(''m''<sup>1/2</sup>), where ''m'' is the monster level. Since sombrero stat gains always grow as ''m'' increases and volleyball stat gains are independent of ''m'', with a large enough ''m'', the sombrero will always outperform the volleyball.
 
<center>stats = (ML/4)*(0.1 + 0.005*SW)</center>
<center>(The Sombrero grants 10% of the stats you receive normally, plus an additional 0.5% per pound of Sombrero)</center>
 
And for the Blood-faced Volleyball:
 
<center>stats = 2 + VW/5</center>
 
To find the point where the sombrero matches the volleyball, we must solve for monster level in terms of the two weights. Thus, we start with:


Under extreme conditions, the [[Wild Hare|Wild Hare]]'s ability to provide extra rollover adventures will provide more (net) substats per adventure than either the volleyball or the sombrero because the benefit from the Wild Hare is ''O''(''m''). Since this requires an infinite supply of level 4000+ monsters, it's really only feasible for high-stat characters fighting against the scaling [[Fernswarthy's Basement]] and [[Special Monsters|Crimbo monsters]].
<center>2 + VW/5 = (ML/4) * (0.1 + 0.005*SW)</center>


==Sombrero vs Volleyball==
However, given that we have obtained both (or neither) familiar equipment, as well as assuming that any equipment/buffs we want can be switched between them (e.g., [[wax lips]], [[Leash of Linguini (skill)|Leash of Linguini]]), we can say that:
Given an equipment/effect configuration that includes familiar equipment, let ''v'', ''s'', and ''m'' be the resulting volleyball and sombrero weights and monster levels respectively.


According to [[Hovering Sombrero]], the sombrero's stat bonus will be (''sm'')<sup>1/2</sup>/10, whereas the volleyball's (see [[Blood-Faced Volleyball]]) will be ''v''<sup>1/2</sup>.
<center>VW = SW</center>


Next, let ''w<sub>v</sub>'', ''p<sub>v</sub>'', ''w<sub>s</sub>'', ''p<sub>s</sub>'' be the non-familiar stat bonus due to [[wax lips]] (+3 stats) or [[annoying pitchfork]] (+1.25), where a maximum of one of ''w'', ''p'' for a familiar can be non-zero (a familiar cannot equip more than one article). Given any pairing of volleyball and sombrero weights, the monster level at which the sombrero breaks even with the volleyball satisfies the equation:
Setting that value equal to W, we substitute:
:''v''<sup>1/2</sup> + ''w<sub>v</sub>'' + ''p<sub>v</sub>'' = (''sm'')<sup>1/2</sup>/10 + ''w<sub>s</sub>'' + ''p<sub>s</sub>''


Letting ''θ'' = ''w<sub>s</sub>'' + ''p<sub>s</sub>'' - ''w<sub>v</sub>'' - ''p<sub>v</sub>'', the break-even ''m'' can be shown to follow the rule:
<center>2 + W/5 = (ML/4) * (0.1 + 0.005 * W)</center>
:''m'' = 100(''v''<sup>1/2</sup> - ''θ'')<sup>2</sup>/''s''


The above is subject to the constraint ''v''<sup>1/2</sup> ≥ ''θ''; otherwise the sombrero is always better than the volleyball.
divide through by (0.1 + 0.005*W) to get:


The case of ''θ'' ≠ 0 is beyond the scope of this article, as the data tables are already bulky enough for ''θ'' = 0. The tables below calculate the monster level at which given effective weights ''v'', ''s'' provide the same bonus stats, for wax lips/pitchforks being allocated identically (on both familiars or off both familiars). In particular, we see that for ''θ'' = 0 and ''v'' = ''s'', the break-even monster level is 100.
<center>(2 + W/5) / (0.1 + 0.005 * W) = ML/4</center>


==Tables==
Re-arranging and multiplying through by 4, we get:
Having trouble reading these tables? Here's how: The vertical row of numbers are Blood-Faced Volleyball weights, in lbs. The horizontal row of numbers are Hovering Sombrero weights. Find both weights (scroll down to the next table for more sombrero weights) and match them up in the middle to find the break-even monster level. This is the monster level (rounded up) where both familiars would provide the same experience. Higher ML ends up increasing the sombrero's superiority.


Jump to:
<center>ML = 4 * (2 + W/5) / (0.1 + 0.005*W)</center>
*[[#1-10 Pound Sombrero]]s
*[[#11-20 Pound Sombrero]]s
*[[#21-30 Pound Sombrero]]s
*[[#31-40 Pound Sombrero]]s
*[[#41-50 Pound Sombrero]]s
*[[#51-60 Pound Sombrero]]s
*[[#61-70 Pound Sombrero]]s
*[[#71-80 Pound Sombrero]]s
*[[#81-90 Pound Sombrero]]s
*[[#91-100 Pound Sombrero]]s


===1-10 Pound Sombrero===
===Interpretation===
{|border="1" class="wikitable" style="text-align:center"
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!8
||800||400||267||200||160||134||115||100||89||80
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!9
||900||450||300||225||180||150||129||113||100||90
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!10
||1000||500||334||250||200||167||143||125||112||100
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!11
||1100||550||367||275||220||184||158||138||123||110
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!12
||1200||600||400||300||240||200||172||150||134||120
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!13
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!14
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||5100||2550||1700||1275||1020||850||729||638||567||510
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||5200||2600||1734||1300||1040||867||743||650||578||520
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!75
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!100
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===11-20 Pound Sombrero===
This cannot be simplified any more until either a weight (W) or Monster Level (ML) are provided.  The below chart showing break-even points plots this formula graphically.
{|border="1" class="wikitable" style="text-align:center"
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!width="72"|BFV\HS!!width="75"|11!!width="75"|12!!width="75"|13!!width="75"|14!!width="75"|15!!width="75"|16!!width="75"|17!!width="75"|18!!width="75"|19!!width="75"|20
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!8
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!9
||82||75||70||65||60||57||53||50||48||45
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!10
||91||84||77||72||67||63||59||56||53||50
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|-
!91
||828||759||700||650||607||569||536||506||479||455
|-
!92
||837||767||708||658||614||575||542||512||485||460
|-
!93
||846||775||716||665||620||582||548||517||490||465
|-
!94
||855||784||724||672||627||588||553||523||495||470
|-
!95
||864||792||731||679||634||594||559||528||500||475
|-
!96
||873||800||739||686||640||600||565||534||506||480
|-
!97
||882||809||747||693||647||607||571||539||511||485
|-
!98
||891||817||754||700||654||613||577||545||516||490
|-
!99
||900||825||762||708||660||619||583||550||522||495
|-
!100
||910||834||770||715||667||625||589||556||527||500
|-
|}


===21-30 Pound Sombrero===
It should be noted that with equally weighted familiars, a Sombrero will always be better than a Volleyball ''at 160 or more Monster Level.''  This is because the limit as W approaches infinity of the ML formula above is 160.
{|border="1" class="wikitable" style="text-align:center"
* According to Wolfram Alpha, <pre>limit (4 * (2 + W/5) / (0.1 + 0.005*W)) as W->infinity</pre> = [http://www.wolframalpha.com/input/?i=limit+%284+*+%282+%2B+W%2F5%29+%2F+%280.1+%2B+0.005*W%29%29+as+W-%3Einfinity 160 Monster Level].
|+
|-
!width="72"|BFV\HS!!width="75"|21!!width="75"|22!!width="75"|23!!width="75"|24!!width="75"|25!!width="75"|26!!width="75"|27!!width="75"|28!!width="75"|29!!width="75"|30
|-
!1
||5||5||5||5||4||4||4||4||4||4
|-
!2
||10||10||9||9||8||8||8||8||7||7
|-
!3
||15||14||14||13||12||12||12||11||11||10
|-
!4
||20||19||18||17||16||16||15||15||14||14
|-
!5
||24||23||22||21||20||20||19||18||18||17
|-
!6
||29||28||27||25||24||24||23||22||21||20
|-
!7
||34||32||31||30||28||27||26||25||25||24
|-
!8
||39||37||35||34||32||31||30||29||28||27
|-
!9
||43||41||40||38||36||35||34||33||32||30
|-
!10
||48||46||44||42||40||39||38||36||35||34
|-
!11
||53||50||48||46||44||43||41||40||38||37
|-
!12
||58||55||53||50||48||47||45||43||42||40
|-
!13
||62||60||57||55||52||50||49||47||45||44
|-
!14
||67||64||61||59||56||54||52||50||49||47
|-
!15
||72||69||66||63||60||58||56||54||52||50
|-
!16
||77||73||70||67||64||62||60||58||56||54
|-
!17
||81||78||74||71||68||66||63||61||59||57
|-
!18
||86||82||79||75||72||70||67||65||63||60
|-
!19
||91||87||83||80||76||74||71||68||66||64
|-
!20
||96||91||87||84||80||77||75||72||69||67
|-
!21
||100||96||92||88||84||81||78||75||73||70
|-
!22
||105||100||96||92||88||85||82||79||76||74
|-
!23
||110||105||100||96||92||89||86||83||80||77
|-
!24
||115||110||105||100||96||93||89||86||83||80
|-
!25
||120||114||109||105||100||97||93||90||87||84
|-
!26
||124||119||114||109||104||100||97||93||90||87
|-
!27
||129||123||118||113||108||104||100||97||94||90
|-
!28
||134||128||122||117||112||108||104||100||97||94
|-
!29
||139||132||127||121||116||112||108||104||100||97
|-
!30
||143||137||131||125||120||116||112||108||104||100
|-
!31
||148||141||135||130||124||120||115||111||107||104
|-
!32
||153||146||140||134||128||124||119||115||111||107
|-
!33
||158||150||144||138||132||127||123||118||114||110
|-
!34
||162||155||148||142||136||131||126||122||118||114
|-
!35
||167||160||153||146||140||135||130||125||121||117
|-
!36
||172||164||157||150||144||139||134||129||125||120
|-
!37
||177||169||161||155||148||143||138||133||128||124
|-
!38
||181||173||166||159||152||147||141||136||132||127
|-
!39
||186||178||170||163||156||150||145||140||135||130
|-
!40
||191||182||174||167||160||154||149||143||138||134
|-
!41
||196||187||179||171||164||158||152||147||142||137
|-
!42
||200||191||183||175||168||162||156||150||145||140
|-
!43
||205||196||187||180||172||166||160||154||149||144
|-
!44
||210||200||192||184||176||170||163||158||152||147
|-
!45
||215||205||196||188||180||174||167||161||156||150
|-
!46
||220||210||200||192||184||177||171||165||159||154
|-
!47
||224||214||205||196||188||181||175||168||163||157
|-
!48
||229||219||209||200||192||185||178||172||166||160
|-
!49
||234||223||214||205||196||189||182||175||169||164
|-
!50
||239||228||218||209||200||193||186||179||173||167
|-
!51
||243||232||222||213||204||197||189||183||176||170
|-
!52
||248||237||227||217||208||200||193||186||180||174
|-
!53
||253||241||231||221||212||204||197||190||183||177
|-
!54
||258||246||235||225||216||208||200||193||187||180
|-
!55
||262||250||240||230||220||212||204||197||190||184
|-
!56
||267||255||244||234||224||216||208||200||194||187
|-
!57
||272||260||248||238||228||220||212||204||197||190
|-
!58
||277||264||253||242||232||224||215||208||200||194
|-
!59
||281||269||257||246||236||227||219||211||204||197
|-
!60
||286||273||261||250||240||231||223||215||207||200
|-
!61
||291||278||266||255||244||235||226||218||211||204
|-
!62
||296||282||270||259||248||239||230||222||214||207
|-
!63
||300||287||274||263||252||243||234||225||218||210
|-
!64
||305||291||279||267||256||247||238||229||221||214
|-
!65
||310||296||283||271||260||250||241||233||225||217
|-
!66
||315||300||287||275||264||254||245||236||228||220
|-
!67
||320||305||292||280||268||258||249||240||232||224
|-
!68
||324||310||296||284||272||262||252||243||235||227
|-
!69
||329||314||300||288||276||266||256||247||238||230
|-
!70
||334||319||305||292||280||270||260||250||242||234
|-
!71
||339||323||309||296||284||274||263||254||245||237
|-
!72
||343||328||314||300||288||277||267||258||249||240
|-
!73
||348||332||318||305||292||281||271||261||252||244
|-
!74
||353||337||322||309||296||285||275||265||256||247
|-
!75
||358||341||327||313||300||289||278||268||259||250
|-
!76
||362||346||331||317||304||293||282||272||263||254
|-
!77
||367||350||335||321||308||297||286||275||266||257
|-
!78
||372||355||340||325||312||300||289||279||269||260
|-
!79
||377||360||344||330||316||304||293||283||273||264
|-
!80
||381||364||348||334||320||308||297||286||276||267
|-
!81
||386||369||353||338||324||312||300||290||280||270
|-
!82
||391||373||357||342||328||316||304||293||283||274
|-
!83
||396||378||361||346||332||320||308||297||287||277
|-
!84
||400||382||366||350||336||324||312||300||290||280
|-
!85
||405||387||370||355||340||327||315||304||294||284
|-
!86
||410||391||374||359||344||331||319||308||297||287
|-
!87
||415||396||379||363||348||335||323||311||300||290
|-
!88
||420||400||383||367||352||339||326||315||304||294
|-
!89
||424||405||387||371||356||343||330||318||307||297
|-
!90
||429||410||392||375||360||347||334||322||311||300
|-
!91
||434||414||396||380||364||350||338||325||314||304
|-
!92
||439||419||400||384||368||354||341||329||318||307
|-
!93
||443||423||405||388||372||358||345||333||321||310
|-
!94
||448||428||409||392||376||362||349||336||325||314
|-
!95
||453||432||414||396||380||366||352||340||328||317
|-
!96
||458||437||418||400||384||370||356||343||332||320
|-
!97
||462||441||422||405||388||374||360||347||335||324
|-
!98
||467||446||427||409||392||377||363||350||338||327
|-
!99
||472||450||431||413||396||381||367||354||342||330
|-
!100
||477||455||435||417||400||385||371||358||345||334
|-
|}


===31-40 Pound Sombrero===
Solving the formula for 1 pound familiars, it can also be noted that with equally weighted familiars, ''at 83 or less Monster Level,'' a Volleyball is always better than a Sombrero.
{|border="1" class="wikitable" style="text-align:center"
|+
|-
!width="72"|BFV\HS!!width="75"|31!!width="75"|32!!width="75"|33!!width="75"|34!!width="75"|35!!width="75"|36!!width="75"|37!!width="75"|38!!width="75"|39!!width="75"|40
|-
!1
||4||4||4||3||3||3||3||3||3||3
|-
!2
||7||7||7||6||6||6||6||6||6||5
|-
!3
||10||10||10||9||9||9||9||8||8||8
|-
!4
||13||13||13||12||12||12||11||11||11||10
|-
!5
||17||16||16||15||15||14||14||14||13||13
|-
!6
||20||19||19||18||18||17||17||16||16||15
|-
!7
||23||22||22||21||20||20||19||19||18||18
|-
!8
||26||25||25||24||23||23||22||22||21||20
|-
!9
||30||29||28||27||26||25||25||24||24||23
|-
!10
||33||32||31||30||29||28||28||27||26||25
|-
!11
||36||35||34||33||32||31||30||29||29||28
|-
!12
||39||38||37||36||35||34||33||32||31||30
|-
!13
||42||41||40||39||38||37||36||35||34||33
|-
!14
||46||44||43||42||40||39||38||37||36||35
|-
!15
||49||47||46||45||43||42||41||40||39||38
|-
!16
||52||50||49||48||46||45||44||43||42||40
|-
!17
||55||54||52||50||49||48||46||45||44||43
|-
!18
||59||57||55||53||52||50||49||48||47||45
|-
!19
||62||60||58||56||55||53||52||50||49||48
|-
!20
||65||63||61||59||58||56||55||53||52||50
|-
!21
||68||66||64||62||60||59||57||56||54||53
|-
!22
||71||69||67||65||63||62||60||58||57||55
|-
!23
||75||72||70||68||66||64||63||61||59||58
|-
!24
||78||75||73||71||69||67||65||64||62||60
|-
!25
||81||79||76||74||72||70||68||66||65||63
|-
!26
||84||82||79||77||75||73||71||69||67||65
|-
!27
||88||85||82||80||78||75||73||72||70||68
|-
!28
||91||88||85||83||80||78||76||74||72||70
|-
!29
||94||91||88||86||83||81||79||77||75||73
|-
!30
||97||94||91||89||86||84||82||79||77||75
|-
!31
||100||97||94||92||89||87||84||82||80||78
|-
!32
||104||100||97||95||92||89||87||85||83||80
|-
!33
||107||104||100||98||95||92||90||87||85||83
|-
!34
||110||107||104||100||98||95||92||90||88||85
|-
!35
||113||110||107||103||100||98||95||93||90||88
|-
!36
||117||113||110||106||103||100||98||95||93||90
|-
!37
||120||116||113||109||106||103||100||98||95||93
|-
!38
||123||119||116||112||109||106||103||100||98||95
|-
!39
||126||122||119||115||112||109||106||103||100||98
|-
!40
||130||125||122||118||115||112||109||106||103||100
|-
!41
||133||129||125||121||118||114||111||108||106||103
|-
!42
||136||132||128||124||120||117||114||111||108||105
|-
!43
||139||135||131||127||123||120||117||114||111||108
|-
!44
||142||138||134||130||126||123||119||116||113||110
|-
!45
||146||141||137||133||129||125||122||119||116||113
|-
!46
||149||144||140||136||132||128||125||122||118||115
|-
!47
||152||147||143||139||135||131||128||124||121||118
|-
!48
||155||150||146||142||138||134||130||127||124||120
|-
!49
||159||154||149||145||140||137||133||129||126||123
|-
!50
||162||157||152||148||143||139||136||132||129||125
|-
!51
||165||160||155||150||146||142||138||135||131||128
|-
!52
||168||163||158||153||149||145||141||137||134||130
|-
!53
||171||166||161||156||152||148||144||140||136||133
|-
!54
||175||169||164||159||155||150||146||143||139||135
|-
!55
||178||172||167||162||158||153||149||145||142||138
|-
!56
||181||175||170||165||160||156||152||148||144||140
|-
!57
||184||179||173||168||163||159||155||150||147||143
|-
!58
||188||182||176||171||166||162||157||153||149||145
|-
!59
||191||185||179||174||169||164||160||156||152||148
|-
!60
||194||188||182||177||172||167||163||158||154||150
|-
!61
||197||191||185||180||175||170||165||161||157||153
|-
!62
||200||194||188||183||178||173||168||164||159||155
|-
!63
||204||197||191||186||180||175||171||166||162||158
|-
!64
||207||200||194||189||183||178||173||169||165||160
|-
!65
||210||204||197||192||186||181||176||172||167||163
|-
!66
||213||207||200||195||189||184||179||174||170||165
|-
!67
||217||210||204||198||192||187||182||177||172||168
|-
!68
||220||213||207||200||195||189||184||179||175||170
|-
!69
||223||216||210||203||198||192||187||182||177||173
|-
!70
||226||219||213||206||200||195||190||185||180||175
|-
!71
||230||222||216||209||203||198||192||187||183||178
|-
!72
||233||225||219||212||206||200||195||190||185||180
|-
!73
||236||229||222||215||209||203||198||193||188||183
|-
!74
||239||232||225||218||212||206||200||195||190||185
|-
!75
||242||235||228||221||215||209||203||198||193||188
|-
!76
||246||238||231||224||218||212||206||200||195||190
|-
!77
||249||241||234||227||220||214||209||203||198||193
|-
!78
||252||244||237||230||223||217||211||206||200||195
|-
!79
||255||247||240||233||226||220||214||208||203||198
|-
!80
||259||250||243||236||229||223||217||211||206||200
|-
!81
||262||254||246||239||232||225||219||214||208||203
|-
!82
||265||257||249||242||235||228||222||216||211||205
|-
!83
||268||260||252||245||238||231||225||219||213||208
|-
!84
||271||263||255||248||240||234||228||222||216||210
|-
!85
||275||266||258||250||243||237||230||224||218||213
|-
!86
||278||269||261||253||246||239||233||227||221||215
|-
!87
||281||272||264||256||249||242||236||229||224||218
|-
!88
||284||275||267||259||252||245||238||232||226||220
|-
!89
||288||279||270||262||255||248||241||235||229||223
|-
!90
||291||282||273||265||258||250||244||237||231||225
|-
!91
||294||285||276||268||260||253||246||240||234||228
|-
!92
||297||288||279||271||263||256||249||243||236||230
|-
!93
||300||291||282||274||266||259||252||245||239||233
|-
!94
||304||294||285||277||269||262||255||248||242||235
|-
!95
||307||297||288||280||272||264||257||250||244||238
|-
!96
||310||300||291||283||275||267||260||253||247||240
|-
!97
||313||304||294||286||278||270||263||256||249||243
|-
!98
||317||307||297||289||280||273||265||258||252||245
|-
!99
||320||310||300||292||283||275||268||261||254||248
|-
!100
||323||313||304||295||286||278||271||264||257||250
|-
|}


===41-50 Pound Sombrero===
== Charts ==
{|border="1" class="wikitable" style="text-align:center"
* Comparing different weight Sombreros and Volleyballs is much more useful than comparing equal weight ones, but figuring out where to stop using one familiar and start using another can be tricky.  The following graph attempts to make that process easier by giving a visualization of the familiars at various weights and Monster Levels, showing how many stats they give.
|+
* [[Image:bugbear_vs_sombrero_may_2014_update_thumb.png|link=http://ben.bloomroad.com/kol/bugbear_vs_sombrero_may_2014_update.png|Baby Bugged Bugbear (wearing a balaclava) vs Volleyball vs Sombrero]]
|-
* The Sombrero's stat cap will always be higher as compared to an equivalent weight volleyball (or baby bugged bugbear wearing a balaclava).  The following graph shows those break-even Monster Levels, and how many stats you'd get from the Sombrero (or Volleyball/Bugbear) at the break-even point of ML for each weight.
!width="72"|BFV\HS!!width="75"|41!!width="75"|42!!width="75"|43!!width="75"|44!!width="75"|45!!width="75"|46!!width="75"|47!!width="75"|48!!width="75"|49!!width="75"|50
*[[Image:bugbear_vs_sombrero_breakeven_thumb.png|link=http://ben.bloomroad.com/kol/bugbear_vs_sombrero_breakeven.png|Breakeven ML for Volleyball (and Bugbear) vs. Sombrero]]
|-
!1
||3||3||3||3||3||3||3||3||3||2
|-
!2
||5||5||5||5||5||5||5||5||5||4
|-
!3
||8||8||7||7||7||7||7||7||7||6
|-
!4
||10||10||10||10||9||9||9||9||9||8
|-
!5
||13||12||12||12||12||11||11||11||11||10
|-
!6
||15||15||14||14||14||14||13||13||13||12
|-
!7
||18||17||17||16||16||16||15||15||15||14
|-
!8
||20||20||19||19||18||18||18||17||17||16
|-
!9
||22||22||21||21||20||20||20||19||19||18
|-
!10
||25||24||24||23||23||22||22||21||21||20
|-
!11
||27||27||26||25||25||24||24||23||23||22
|-
!12
||30||29||28||28||27||27||26||25||25||24
|-
!13
||32||31||31||30||29||29||28||28||27||26
|-
!14
||35||34||33||32||32||31||30||30||29||28
|-
!15
||37||36||35||35||34||33||32||32||31||30
|-
!16
||40||39||38||37||36||35||35||34||33||32
|-
!17
||42||41||40||39||38||37||37||36||35||34
|-
!18
||44||43||42||41||40||40||39||38||37||36
|-
!19
||47||46||45||44||43||42||41||40||39||38
|-
!20
||49||48||47||46||45||44||43||42||41||40
|-
!21
||52||50||49||48||47||46||45||44||43||42
|-
!22
||54||53||52||50||49||48||47||46||45||44
|-
!23
||57||55||54||53||52||50||49||48||47||46
|-
!24
||59||58||56||55||54||53||52||50||49||48
|-
!25
||61||60||59||57||56||55||54||53||52||50
|-
!26
||64||62||61||60||58||57||56||55||54||52
|-
!27
||66||65||63||62||60||59||58||57||56||54
|-
!28
||69||67||66||64||63||61||60||59||58||56
|-
!29
||71||70||68||66||65||64||62||61||60||58
|-
!30
||74||72||70||69||67||66||64||63||62||60
|-
!31
||76||74||73||71||69||68||66||65||64||62
|-
!32
||79||77||75||73||72||70||69||67||66||64
|-
!33
||81||79||77||75||74||72||71||69||68||66
|-
!34
||83||81||80||78||76||74||73||71||70||68
|-
!35
||86||84||82||80||78||77||75||73||72||70
|-
!36
||88||86||84||82||80||79||77||75||74||72
|-
!37
||91||89||87||85||83||81||79||78||76||74
|-
!38
||93||91||89||87||85||83||81||80||78||76
|-
!39
||96||93||91||89||87||85||83||82||80||78
|-
!40
||98||96||94||91||89||87||86||84||82||80
|-
!41
||100||98||96||94||92||90||88||86||84||82
|-
!42
||103||100||98||96||94||92||90||88||86||84
|-
!43
||105||103||100||98||96||94||92||90||88||86
|-
!44
||108||105||103||100||98||96||94||92||90||88
|-
!45
||110||108||105||103||100||98||96||94||92||90
|-
!46
||113||110||107||105||103||100||98||96||94||92
|-
!47
||115||112||110||107||105||103||100||98||96||94
|-
!48
||118||115||112||110||107||105||103||100||98||96
|-
!49
||120||117||114||112||109||107||105||103||100||98
|-
!50
||122||120||117||114||112||109||107||105||103||100
|-
!51
||125||122||119||116||114||111||109||107||105||102
|-
!52
||127||124||121||119||116||114||111||109||107||104
|-
!53
||130||127||124||121||118||116||113||111||109||106
|-
!54
||132||129||126||123||120||118||115||113||111||108
|-
!55
||135||131||128||125||123||120||118||115||113||110
|-
!56
||137||134||131||128||125||122||120||117||115||112
|-
!57
||140||136||133||130||127||124||122||119||117||114
|-
!58
||142||139||135||132||129||127||124||121||119||116
|-
!59
||144||141||138||135||132||129||126||123||121||118
|-
!60
||147||143||140||137||134||131||128||125||123||120
|-
!61
||149||146||142||139||136||133||130||128||125||122
|-
!62
||152||148||145||141||138||135||132||130||127||124
|-
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||154||150||147||144||140||137||135||132||129||126
|-
!64
||157||153||149||146||143||140||137||134||131||128
|-
!65
||159||155||152||148||145||142||139||136||133||130
|-
!66
||161||158||154||150||147||144||141||138||135||132
|-
!67
||164||160||156||153||149||146||143||140||137||134
|-
!68
||166||162||159||155||152||148||145||142||139||136
|-
!69
||169||165||161||157||154||150||147||144||141||138
|-
!70
||171||167||163||160||156||153||149||146||143||140
|-
!71
||174||170||166||162||158||155||152||148||145||142
|-
!72
||176||172||168||164||160||157||154||150||147||144
|-
!73
||179||174||170||166||163||159||156||153||149||146
|-
!74
||181||177||173||169||165||161||158||155||152||148
|-
!75
||183||179||175||171||167||164||160||157||154||150
|-
!76
||186||181||177||173||169||166||162||159||156||152
|-
!77
||188||184||180||175||172||168||164||161||158||154
|-
!78
||191||186||182||178||174||170||166||163||160||156
|-
!79
||193||189||184||180||176||172||169||165||162||158
|-
!80
||196||191||187||182||178||174||171||167||164||160
|-
!81
||198||193||189||185||180||177||173||169||166||162
|-
!82
||200||196||191||187||183||179||175||171||168||164
|-
!83
||203||198||194||189||185||181||177||173||170||166
|-
!84
||205||200||196||191||187||183||179||175||172||168
|-
!85
||208||203||198||194||189||185||181||178||174||170
|-
!86
||210||205||200||196||192||187||183||180||176||172
|-
!87
||213||208||203||198||194||190||186||182||178||174
|-
!88
||215||210||205||200||196||192||188||184||180||176
|-
!89
||218||212||207||203||198||194||190||186||182||178
|-
!90
||220||215||210||205||200||196||192||188||184||180
|-
!91
||222||217||212||207||203||198||194||190||186||182
|-
!92
||225||220||214||210||205||200||196||192||188||184
|-
!93
||227||222||217||212||207||203||198||194||190||186
|-
!94
||230||224||219||214||209||205||200||196||192||188
|-
!95
||232||227||221||216||212||207||203||198||194||190
|-
!96
||235||229||224||219||214||209||205||200||196||192
|-
!97
||237||231||226||221||216||211||207||203||198||194
|-
!98
||240||234||228||223||218||214||209||205||200||196
|-
!99
||242||236||231||225||220||216||211||207||203||198
|-
!100
||244||239||233||228||223||218||213||209||205||200
|-
|}


===51-60 Pound Sombrero===
==Examples==
{|border="1" class="wikitable" style="text-align:center"
Example: Picking a weight such as 20 pounds gives: ML = 4*(2+20/5) / (0.1 + 0.005*20) = 4*(6) / (0.2) = 120.  So when both the Volleyball and Sombrero are 20 pounds: ''if you are fighting monsters more than 120 ML'', use the '''sombrero'''; and ''less than 120 ML'', use the '''volleyball'''.
|+
|-
!width="72"|BFV\HS!!width="75"|51!!width="75"|52!!width="75"|53!!width="75"|54!!width="75"|55!!width="75"|56!!width="75"|57!!width="75"|58!!width="75"|59!!width="75"|60
|-
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||59||58||57||56||55||54||53||52||51||50
|-
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||61||60||59||58||57||56||55||54||53||52
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||65||64||63||62||60||59||58||57||56||55
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||67||66||65||63||62||61||60||59||58||57
|-
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||69||68||67||65||64||63||62||61||60||59
|-
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||71||70||68||67||66||65||64||63||62||60
|-
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|-
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||75||74||72||71||70||68||67||66||65||64
|-
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||77||75||74||73||71||70||69||68||67||65
|-
!40
||79||77||76||75||73||72||71||69||68||67
|-
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||81||79||78||76||75||74||72||71||70||69
|-
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||83||81||80||78||77||75||74||73||72||70
|-
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||85||83||82||80||79||77||76||75||73||72
|-
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||87||85||84||82||80||79||78||76||75||74
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||89||87||85||84||82||81||79||78||77||75
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||91||89||87||86||84||83||81||80||78||77
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|-
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||95||93||91||89||88||86||85||83||82||80
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||100||99||97||95||93||92||90||88||87||85
|-
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||102||100||99||97||95||93||92||90||89||87
|-
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||104||102||100||99||97||95||93||92||90||89
|-
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||106||104||102||100||99||97||95||94||92||90
|-
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||108||106||104||102||100||99||97||95||94||92
|-
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||110||108||106||104||102||100||99||97||95||94
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|-
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||118||116||114||112||110||108||106||104||102||100
|-
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||120||118||116||113||111||109||108||106||104||102
|-
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||122||120||117||115||113||111||109||107||106||104
|-
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||124||122||119||117||115||113||111||109||107||105
|-
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||126||124||121||119||117||115||113||111||109||107
|-
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||128||125||123||121||119||117||115||113||111||109
|-
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||130||127||125||123||120||118||116||114||112||110
|-
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||132||129||127||125||122||120||118||116||114||112
|-
!68
||134||131||129||126||124||122||120||118||116||114
|-
!69
||136||133||131||128||126||124||122||119||117||115
|-
!70
||138||135||133||130||128||125||123||121||119||117
|-
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||140||137||134||132||130||127||125||123||121||119
|-
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||142||139||136||134||131||129||127||125||123||120
|-
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||144||141||138||136||133||131||129||126||124||122
|-
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||146||143||140||138||135||133||130||128||126||124
|-
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||148||145||142||139||137||134||132||130||128||125
|-
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||150||147||144||141||139||136||134||132||129||127
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||151||149||146||143||140||138||136||133||131||129
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||153||150||148||145||142||140||137||135||133||130
|-
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||155||152||150||147||144||142||139||137||134||132
|-
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||157||154||151||149||146||143||141||138||136||134
|-
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||159||156||153||150||148||145||143||140||138||135
|-
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||161||158||155||152||150||147||144||142||139||137
|-
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||163||160||157||154||151||149||146||144||141||139
|-
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||165||162||159||156||153||150||148||145||143||140
|-
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||167||164||161||158||155||152||150||147||145||142
|-
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||169||166||163||160||157||154||151||149||146||144
|-
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||171||168||165||162||159||156||153||150||148||145
|-
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||173||170||167||163||160||158||155||152||150||147
|-
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||175||172||168||165||162||159||157||154||151||149
|-
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||177||174||170||167||164||161||158||156||153||150
|-
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||179||175||172||169||166||163||160||157||155||152
|-
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||181||177||174||171||168||165||162||159||156||154
|-
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||183||179||176||173||170||167||164||161||158||155
|-
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||185||181||178||175||171||168||165||163||160||157
|-
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||187||183||180||176||173||170||167||164||162||159
|-
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||189||185||182||178||175||172||169||166||163||160
|-
!97
||191||187||184||180||177||174||171||168||165||162
|-
!98
||193||189||185||182||179||175||172||169||167||164
|-
!99
||195||191||187||184||180||177||174||171||168||165
|-
!100
||197||193||189||186||182||179||176||173||170||167
|-
|}


===61-70 Pound Sombrero===
==Why does a Sombrero always win at ''some'' monster level?==
{|border="1" class="wikitable" style="text-align:center"
Because the Sombrero's stat bonuses grow with monster level, and the volleyball's does not.
|+
|-
!width="72"|BFV\HS!!width="75"|61!!width="75"|62!!width="75"|63!!width="75"|64!!width="75"|65!!width="75"|66!!width="75"|67!!width="75"|68!!width="75"|69!!width="75"|70
|-
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||12||12||12||11||11||11||11||11||11||10
|-
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||14||13||13||13||13||13||12||12||12||12
|-
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||15||15||15||15||14||14||14||14||14||13
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||17||17||16||16||16||16||15||15||15||15
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||19||18||18||18||17||17||17||17||16||16
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|-
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|-
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||23||23||23||22||22||22||21||21||21||20
|-
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|-
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||27||26||26||25||25||25||24||24||24||23
|-
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||30||30||29||29||28||28||27||27||27||26
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||32||31||31||30||30||29||29||28||28||28
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!20
||33||33||32||32||31||31||30||30||29||29
|-
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||35||34||34||33||33||32||32||31||31||30
|-
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||37||36||35||35||34||34||33||33||32||32
|-
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||38||38||37||36||36||35||35||34||34||33
|-
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||40||39||39||38||37||37||36||36||35||35
|-
!25
||41||41||40||40||39||38||38||37||37||36
|-
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||43||42||42||41||40||40||39||39||38||38
|-
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|-
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||46||46||45||44||44||43||42||42||41||40
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||48||47||47||46||45||44||44||43||43||42
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||53||52||51||50||50||49||48||48||47||46
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||55||54||53||52||51||50||50||49||48||48
|-
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||56||55||54||54||53||52||51||50||50||49
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||58||57||56||55||54||54||53||52||51||50
|-
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||60||59||58||57||56||55||54||53||53||52
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||63||62||61||60||59||58||57||56||56||55
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||64||63||62||61||60||60||59||58||57||56
|-
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||66||65||64||63||62||61||60||59||58||58
|-
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||68||67||66||65||64||63||62||61||60||59
|-
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||69||68||67||66||65||64||63||62||61||60
|-
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||71||70||69||68||67||66||65||64||63||62
|-
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||73||71||70||69||68||67||66||65||64||63
|-
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||74||73||72||71||70||69||68||67||66||65
|-
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||76||75||74||72||71||70||69||68||67||66
|-
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||78||76||75||74||73||72||71||70||69||68
|-
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||79||78||77||75||74||73||72||71||70||69
|-
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||81||80||78||77||76||75||74||73||72||70
|-
!50
||82||81||80||79||77||76||75||74||73||72
|-
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||84||83||81||80||79||78||77||75||74||73
|-
!52
||86||84||83||82||80||79||78||77||76||75
|-
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||87||86||85||83||82||81||80||78||77||76
|-
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||89||88||86||85||84||82||81||80||79||78
|-
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||91||89||88||86||85||84||83||81||80||79
|-
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||92||91||89||88||87||85||84||83||82||80
|-
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||94||92||91||90||88||87||86||84||83||82
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||96||94||93||91||90||88||87||86||85||83
|-
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||97||96||94||93||91||90||89||87||86||85
|-
!60
||99||97||96||94||93||91||90||89||87||86
|-
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||100||99||97||96||94||93||92||90||89||88
|-
!62
||102||100||99||97||96||94||93||92||90||89
|-
!63
||104||102||100||99||97||96||95||93||92||90
|-
!64
||105||104||102||100||99||97||96||95||93||92
|-
!65
||107||105||104||102||100||99||98||96||95||93
|-
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||109||107||105||104||102||100||99||98||96||95
|-
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||110||109||107||105||104||102||100||99||98||96
|-
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||112||110||108||107||105||104||102||100||99||98
|-
!69
||114||112||110||108||107||105||103||102||100||99
|-
!70
||115||113||112||110||108||107||105||103||102||100
|-
!71
||117||115||113||111||110||108||106||105||103||102
|-
!72
||119||117||115||113||111||110||108||106||105||103
|-
!73
||120||118||116||115||113||111||109||108||106||105
|-
!74
||122||120||118||116||114||113||111||109||108||106
|-
!75
||123||121||120||118||116||114||112||111||109||108
|-
!76
||125||123||121||119||117||116||114||112||111||109
|-
!77
||127||125||123||121||119||117||115||114||112||110
|-
!78
||128||126||124||122||120||119||117||115||114||112
|-
!79
||130||128||126||124||122||120||118||117||115||113
|-
!80
||132||130||127||125||124||122||120||118||116||115
|-
!81
||133||131||129||127||125||123||121||120||118||116
|-
!82
||135||133||131||129||127||125||123||121||119||118
|-
!83
||137||134||132||130||128||126||124||123||121||119
|-
!84
||138||136||134||132||130||128||126||124||122||120
|-
!85
||140||138||135||133||131||129||127||125||124||122
|-
!86
||141||139||137||135||133||131||129||127||125||123
|-
!87
||143||141||139||136||134||132||130||128||127||125
|-
!88
||145||142||140||138||136||134||132||130||128||126
|-
!89
||146||144||142||140||137||135||133||131||129||128
|-
!90
||148||146||143||141||139||137||135||133||131||129
|-
!91
||150||147||145||143||140||138||136||134||132||130
|-
!92
||151||149||147||144||142||140||138||136||134||132
|-
!93
||153||150||148||146||144||141||139||137||135||133
|-
!94
||155||152||150||147||145||143||141||139||137||135
|-
!95
||156||154||151||149||147||144||142||140||138||136
|-
!96
||158||155||153||150||148||146||144||142||140||138
|-
!97
||160||157||154||152||150||147||145||143||141||139
|-
!98
||161||159||156||154||151||149||147||145||143||140
|-
!99
||163||160||158||155||153||150||148||146||144||142
|-
!100
||164||162||159||157||154||152||150||148||145||143
|-
|}


===71-80 Pound Sombrero===
Computer scientists define the growth of a function using [[Wikipedia:Big O notation|Big O notation]], which describes the behavior of a function as its argument gets arbitrarily large.  In the case of the stat gains for volleyballs and sombreros, volleyball stat gains are constant at a given volleyball weight.  They are ''O''(''v'') = ''O''(''C'') = ''O''(''1'').  However, the sombrero formula contains a non-constant factor in its numerator, the monster level, so its stat gains are ''O''(''m''), where ''m'' is the monster level.  Since sombrero stat gains always grow as ''m'' increases and volleyball stat gains are independent of ''m'', with a large enough ''m'', the sombrero will always outperform the volleyball.  This ignores the fact that the sombrero has a stat cap, because in order to reach the current stat cap of 230, a volleyball would need to weigh 230 = 2 + w/5 => w = 228*5 = 1140 pounds, which is unrealistic.
{|border="1" class="wikitable" style="text-align:center"
|+
|-
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|-
|}


===81-90 Pound Sombrero===
Under extreme conditions, the [[Wild Hare|Wild Hare]]'s ability to provide extra rollover adventures will provide more (net) substats per adventure than either the volleyball or the sombrero because the benefit from the Wild Hare is ''O''(''m'').  Since this requires an infinite supply of level 9200+ monsters, it's really only feasible for high-stat characters fighting against scaling monsters such as the [[Sloppy Seconds Diner]] (cap at 11,111 ml), [[Fernswarthy's Basement]] and [[Special Monsters|Crimbo monsters]].
{|border="1" class="wikitable" style="text-align:center"
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|}


===91-100 Pound Sombrero===
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[[Category:Familiars| Sombrero vs Volleyball]]
[[Category:Familiars| Sombrero vs Volleyball]]
[[Category:Game Mechanics]]
[[Category:Game Mechanics]]

Latest revision as of 02:34, 3 November 2022

The Hermit humbly requests that this page be rewritten or expanded.

The section on Wild Hares needs to be updated. The Bandersnatch examples need to be re-written. The ML Stat formula is also incorrect (ML/3 not ML/4) meaning the math needs to be redone

The two most popular stat-gain familiars can be pigeonholed into two categories: volleyballs and sombreros. For an analysis of the best familiar equipment for stat gains while using such familiars, see the Blood-Faced Volleyball and Hovering Sombrero articles. The present page is concerned with comparing volleyballs against sombreros. The main takeaway is that deciding which familiar to use is dependent on Monster Level.

Sombrero vs. Volleyball Formula

In the formulas below, VW is volleyball weight, SW is sombrero weight, and ML is monster level.

The formula for extra stats given by the Hovering Sombrero of weight SW is:

stats = (ML/4)*(0.1 + 0.005*SW)
(The Sombrero grants 10% of the stats you receive normally, plus an additional 0.5% per pound of Sombrero)

And for the Blood-faced Volleyball:

stats = 2 + VW/5

To find the point where the sombrero matches the volleyball, we must solve for monster level in terms of the two weights. Thus, we start with:

2 + VW/5 = (ML/4) * (0.1 + 0.005*SW)

However, given that we have obtained both (or neither) familiar equipment, as well as assuming that any equipment/buffs we want can be switched between them (e.g., wax lips, Leash of Linguini), we can say that:

VW = SW

Setting that value equal to W, we substitute:

2 + W/5 = (ML/4) * (0.1 + 0.005 * W)

divide through by (0.1 + 0.005*W) to get:

(2 + W/5) / (0.1 + 0.005 * W) = ML/4

Re-arranging and multiplying through by 4, we get:

ML = 4 * (2 + W/5) / (0.1 + 0.005*W)

Interpretation

This cannot be simplified any more until either a weight (W) or Monster Level (ML) are provided. The below chart showing break-even points plots this formula graphically.

It should be noted that with equally weighted familiars, a Sombrero will always be better than a Volleyball at 160 or more Monster Level. This is because the limit as W approaches infinity of the ML formula above is 160.

  • According to Wolfram Alpha, 
    limit (4 * (2 + W/5) / (0.1 + 0.005*W)) as W->infinity
    = 160 Monster Level.

Solving the formula for 1 pound familiars, it can also be noted that with equally weighted familiars, at 83 or less Monster Level, a Volleyball is always better than a Sombrero.

Charts

  • Comparing different weight Sombreros and Volleyballs is much more useful than comparing equal weight ones, but figuring out where to stop using one familiar and start using another can be tricky. The following graph attempts to make that process easier by giving a visualization of the familiars at various weights and Monster Levels, showing how many stats they give.
  • Baby Bugged Bugbear (wearing a balaclava) vs Volleyball vs Sombrero
  • The Sombrero's stat cap will always be higher as compared to an equivalent weight volleyball (or baby bugged bugbear wearing a balaclava). The following graph shows those break-even Monster Levels, and how many stats you'd get from the Sombrero (or Volleyball/Bugbear) at the break-even point of ML for each weight.
  • Breakeven ML for Volleyball (and Bugbear) vs. Sombrero

Examples

Example: Picking a weight such as 20 pounds gives: ML = 4*(2+20/5) / (0.1 + 0.005*20) = 4*(6) / (0.2) = 120. So when both the Volleyball and Sombrero are 20 pounds: if you are fighting monsters more than 120 ML, use the sombrero; and less than 120 ML, use the volleyball.

Why does a Sombrero always win at some monster level?

Because the Sombrero's stat bonuses grow with monster level, and the volleyball's does not.

Computer scientists define the growth of a function using Big O notation, which describes the behavior of a function as its argument gets arbitrarily large. In the case of the stat gains for volleyballs and sombreros, volleyball stat gains are constant at a given volleyball weight. They are O(v) = O(C) = O(1). However, the sombrero formula contains a non-constant factor in its numerator, the monster level, so its stat gains are O(m), where m is the monster level. Since sombrero stat gains always grow as m increases and volleyball stat gains are independent of m, with a large enough m, the sombrero will always outperform the volleyball. This ignores the fact that the sombrero has a stat cap, because in order to reach the current stat cap of 230, a volleyball would need to weigh 230 = 2 + w/5 => w = 228*5 = 1140 pounds, which is unrealistic.

Under extreme conditions, the Wild Hare's ability to provide extra rollover adventures will provide more (net) substats per adventure than either the volleyball or the sombrero because the benefit from the Wild Hare is O(m). Since this requires an infinite supply of level 9200+ monsters, it's really only feasible for high-stat characters fighting against scaling monsters such as the Sloppy Seconds Diner (cap at 11,111 ml), Fernswarthy's Basement and Crimbo monsters.

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