View source for Hidden Power (move)/Calculation

{{main|Hidden Power (move)}}
In the Pokémon games, {{m|Hidden Power}} is a {{type|Normal}} move when the type is displayed, such as in battle and on status screens. However, the actual type of Hidden Power is determined by the Pokémon's [[individual values]], and through calculation, can be set as one of other natural types. In Generations II to V, Hidden Power's base power is also determined by the Pokémon's individual values; in Generation II, it ranges from 31 to 70, while in Generation III to V, it ranges from 30 to 70. From Generation VI onward, Hidden Power's base power is always 60.

The type of Hidden Power can be checked in certain locations in the games by talking to certain NPCs. These NPCs will refuse to tell the player the Hidden Power type of a Pokémon unless it can learn the move via [[level]] up or [[TM]] (even if it knows Hidden Power at the time).<!--they will not tell the Hidden Power type of a Smeargle that knows Hidden Power-->
*In {{game|Platinum}}, in [[Veilstone Game Corner]]'s prize house
*In {{game|HeartGold and SoulSilver|s}}, in the [[Celadon Game Corner]]
*In {{game|Black and White|s}}, in [[Mistralton City]]'s [[Pokémon Center]]
*In {{game|Black and White|s 2|Pokémon Black 2 and White 2}}, the [[Pokémon World Tournament|PWT]]
*In [[Pokémon X and Y]], in the house to the right of the Boutique in [[Anistar City]]
*In [[Pokémon Omega Ruby and Alpha Sapphire]], in the second northern house from the west in [[Fortree City]]
*In [[Pokémon Sun and Moon|Pokémon Sun, Moon]], {{pkmn|Ultra Sun and Ultra Moon}} in the [[Pokémon Nursery]] on [[Paniola Ranch]]

There is no in-game way to determine Hidden Power's [[power]] directly in games where it is not fixed.

==Generation II==
===Type===
Consider an example Pokémon, like Shellder with this set of IVs:

{{Hidden Power calculation/II|12|10|10|13|8}}

The type is determined by taking the two {{wp|least significant bit}}s of the Attack and Defense IVs, then {{wp|Concatenation|concatenating}} these two values in that order.

Mathematically, this is the equivalent of <math>HP_{type} = 4 \times a \bmod 4 + b \bmod 4</math>, where ''a'' represents the Attack IV and ''b'' represents the Defense IV.

The resulting number will correspond to a type as marked below.

{| class="roundy" style="text-align: center; color: #FFF; background: #{{platinum color}}; border: 3px solid #{{platinum color light}}; {{roundy|10px}}"
! style="color: #000; background: #{{platinum color light}}; {{roundytl|5px}}" | Number
! style="color: #000; background: #{{platinum color light}}; {{roundytr|5px}}" | Type
|- style="background: #{{Fighting color}}"
| 0
| {{typecolor|Fighting}}
|- style="background: #{{Flying color}}"
| 1
| {{typecolor|Flying}}
|- style="background: #{{poison color}}"
| 2
| {{typecolor|Poison}}
|- style="background: #{{Ground color}}"
| 3
| {{typecolor|Ground}}
|- style="background: #{{Rock color}}"
| 4
| {{typecolor|Rock}}
|- style="background: #{{Bug color}}"
| 5
| {{typecolor|Bug}}
|- style="background: #{{Ghost color}}"
| 6
| {{typecolor|Ghost}}
|- style="background: #{{Steel color}}"
| 7
| {{typecolor|Steel}}
|- style="background: #{{Fire color}}"
| 8
| {{typecolor|Fire}}
|- style="background: #{{Water color}}"
| 9
| {{typecolor|Water}}
|- style="background: #{{Grass color}}"
| 10
| {{typecolor|Grass}}
|- style="background: #{{Electric color}}"
| 11
| {{typecolor|Electric}}
|- style="background: #{{Psychic color}}"
| 12
| {{typecolor|Psychic}}
|- style="background: #{{Ice color}}"
| 13
| {{typecolor|Ice}}
|- style="background: #{{Dragon color}}"
| 14
| {{typecolor|Dragon}}
|-
| style="background: #{{Dark color}}; {{roundybl|5px}}" | 15
| style="background: #{{Dark color}}; {{roundybr|5px}}" | {{typecolor|Dark}}
|}

In our example, we get:

<math>
\begin{align}
HP_{type} & = 4 \times 10 \bmod 4 + 10 \bmod 4 \\
& = 4 \times 2 + 2 \\
& = 8 + 2 \\
& = 10
\end{align}
</math>

Since this value is equal to 10, per the table above, this means that our Shellder has a {{type|Grass}} {{m|Hidden Power}}.

===Power===
The [[power]] of the Hidden Power is calculated using the following formula:

<math display="block">HP_{power} = \bigg\lfloor \frac{5 \times (v + 2w + 4x + 8y) + Z}{2} + 31\bigg\rfloor</math>

* The variables ''v'' through ''y'' (the "damage bits") represent the {{wp|most significant bit}} of each IV. If a variable is less than eight, this bit is 0; otherwise, it is 1.
:* ''v'' depends on the Special IV.
:* ''w'' depends on the Speed IV.
:* ''x'' depends on the Defense IV.
:* ''y'' depends on the Attack IV.
* ''Z'' is equal to the Special IV mod 4 (its remainder when divided by 4).

Hidden Power's base power is therefore a number ranging from 31 to 70, inclusively. 

In our example, we get:

{{Hidden Power calculation/II|12<br>''N/A''|10<br>''1''|10<br>''1''|13<br>''1''|8<br>''1''}}

<math>
\begin{align}
HP_{power} & = \bigg\lfloor \frac{5 \times (1 + 2 \cdot 1 + 4 \cdot 1 + 8 \cdot 1) + 0}{2}\bigg\rfloor + 31 \\
& = \bigg\lfloor \frac{5 \times (1 + 2 + 4 + 8)}{2}\bigg\rfloor + 31 \\
& = \bigg\lfloor \frac{5 \times 15}{2}\bigg\rfloor + 31 \\
& = \bigg\lfloor \frac{75}{2}\bigg\rfloor + 31 \\
& = \lfloor 37.5 \rfloor + 31 \\
& = 37 + 31 \\
& = 68
\end{align}
</math>

Which means that our Shellder's Hidden Power's power is '''68'''.

{{Hidden Power calculation/II|12|10|10|13|8}}
{| class="roundy" style="width: 650px; border: 2px solid #{{Grass color light}}" 
|- style="text-align: center; color: #FFF"
| style="width: 33%; background: #{{Grass color}}; {{roundyleft|10px}}" | {{mcolor|Hidden Power|FFF}}:
| style="background: #{{Grass color}}" | Type:<br>'''{{tcolor|Grass|FFF}}'''
| style="background: #{{Grass color}}; {{roundyright|10px}}" | Power:<br>'''68'''
|}

=== Remarks ===
* Because {{Shiny|Shininess}} is also determined by IVs, Shiny Pokémon can only have a Grass-type or Dragon-type Hidden Power with a power of 49 or 69.
* Because gender is also determined by the Attack IV, female Pokémon of {{cat|Pokémon with a gender ratio of seven males to one female|species with a seven male to one female ratio}} can only have a physical Hidden Power.

==Generation III onward==
===Type===
Consider an Unown with this set of IVs:

{{Hidden Power calculation/IV|25|2|12|17|5|8|201Unown.png{{!}}100px}}

Hidden Power's type of a Pokémon with given IVs is represented by a number, calculated with this formula:
<math display="block">HP_{type} = \left\lfloor \frac{(a + 2b + 4c + 8d + 16e + 32f) \times 15}{63}\right\rfloor</math>

where ''a'', ''b'', ''c'', ''d'', ''e'', ''f'' (the "type bits") are the {{wp|least significant bit}} of their respective IVs. If a number is odd, its least significant bit is 1; otherwise (if the number is even), it is 0.

* ''a'' depends on the HP IV.
* ''b'' and ''c'' depend on the Attack and Defense IVs respectively.
* ''d'' depends on the Speed IV.
* ''e'' and ''f'' depend on the Special Attack and Special Defense IVs respectively.

The sum inside the formula may range from 0 (when all IVs are even) to 63 (when all IVs are odd), inclusive. This sum is equivalent to constructing the {{wp|binary number}} (''fedcba'')<sub>2</sub>.

The sum is then multiplied by 15 and divided by 63. This ensures that the number representing the type ranges from 0 to 15, inclusive. The [[wp:Floor function|floor]] of this value is taken, meaning that if the result is not an {{wp|integer}}, it will be rounded down to the nearest integer less than it.

The resulting number will correspond to a type as marked below.

{| class="roundy" style="text-align: center; color: #FFF; background: #{{platinum color}}; border: 3px solid #{{platinum color light}}; {{roundy|10px}}"
! style="color: #000; background: #{{platinum color light}}; {{roundytl|5px}}" | Number
! style="color: #000; background: #{{platinum color light}}; {{roundytr|5px}}" | Type
|- style="background: #{{Fighting color}}"
| 0
| {{typecolor|Fighting}}
|- style="background: #{{Flying color}}"
| 1
| {{typecolor|Flying}}
|- style="background: #{{poison color}}"
| 2
| {{typecolor|Poison}}
|- style="background: #{{Ground color}}"
| 3
| {{typecolor|Ground}}
|- style="background: #{{Rock color}}"
| 4
| {{typecolor|Rock}}
|- style="background: #{{Bug color}}"
| 5
| {{typecolor|Bug}}
|- style="background: #{{Ghost color}}"
| 6
| {{typecolor|Ghost}}
|- style="background: #{{Steel color}}"
| 7
| {{typecolor|Steel}}
|- style="background: #{{Fire color}}"
| 8
| {{typecolor|Fire}}
|- style="background: #{{Water color}}"
| 9
| {{typecolor|Water}}
|- style="background: #{{Grass color}}"
| 10
| {{typecolor|Grass}}
|- style="background: #{{Electric color}}"
| 11
| {{typecolor|Electric}}
|- style="background: #{{Psychic color}}"
| 12
| {{typecolor|Psychic}}
|- style="background: #{{Ice color}}"
| 13
| {{typecolor|Ice}}
|- style="background: #{{Dragon color}}"
| 14
| {{typecolor|Dragon}}
|-
| style="background: #{{Dark color}}; {{roundybl|5px}}" | 15
| style="background: #{{Dark color}}; {{roundybr|5px}}" | {{typecolor|Dark}}
|}

In our example, we get:

{{Hidden Power calculation/IV|25<br>''1''|2<br>''0''|12<br>''0''|17<br>''1''|5<br>''1''|8<br>''0''|201Unown.png{{!}}100px}}

<math>
\begin{align}
HP_{type} & = \left\lfloor \frac{(1 \cdot 1 + 2 \cdot 0 + 4 \cdot 0 + 8 \cdot 1 + 16 \cdot 1 + 32 \cdot 0) \times 15}{63}\right\rfloor \\
& = \left\lfloor \frac{(1 + 0 + 0 + 8 + 16 + 0) \times 15}{63}\right\rfloor \\
& = \left\lfloor \frac{25 \times 15}{63}\right\rfloor \\
& = \left\lfloor \frac{375}{63}\right\rfloor & \bigg(\frac{690}{63} \approx 5.952\bigg)\\
& = 5
\end{align}
</math>

Which means that our Unown has a {{type|Bug}} {{m|Hidden Power}}.

===Damage===
Damage of the Hidden Power is calculated in a manner very similar to that of its type, using the following formula:
<math display="block">HP_{power} = \left\lfloor \frac{(u + 2v + 4w + 8x + 16y + 32z) \times 40}{63}\right\rfloor + 30</math>

* The variables ''u'' through ''z'' (the "damage bits") represent the second least significant bit of each IV. If a variable has a remainder of 2 or 3 when divided by 4, this bit is 1; otherwise, the bit is 0.
:* ''u'' depends on the HP stat.
:* ''v'' and ''w'' depend on the Attack and Defense stats respectively.
:* ''x'' depends on the Speed stat.
:* ''y'' and ''z'' depend on the Special Attack and Special Defense stats respectively.

Like before, the sum may range from 0 to 63, inclusively. The calculated number is then multiplied by 40 and divided by 63 to make sure that the fraction will not exceed 40. Then, the number is increased by 30 and rounded down, making Hidden Power's power a number ranging from 30 to 70, inclusively. 

In our example, we get:

{{Hidden Power calculation/IV|25<br>''0''|2<br>''1''|12<br>''0''|17<br>''0''|5<br>''0''|8<br>''0''|201Unown.png{{!}}100px}}

<math>
\begin{align}
HP_{power} &= \left\lfloor \frac{(1 \cdot 0 + 2 \cdot 1 + 4 \cdot 0 + 8 \cdot 0 + 16 \cdot 0 + 32 \cdot 0) \times 40}{63}\right\rfloor + 30 \\
&= \left\lfloor \frac{(0 + 2 + 0 + 0 + 0 + 0) \times 40}{63}\right\rfloor + 30 \\
&= \left\lfloor \frac{2 \times 40}{63}\right\rfloor + 30 \\
&= \left\lfloor \frac{80}{63}\right\rfloor + 30 \\
&= \lfloor 1 \rfloor + 30 \\
&= 1 + 30 \\
&= 31 \\
\end{align}
</math>

Which means that our Unown's Hidden Power's power is '''31'''.

{{Hidden Power calculation/IV|25|2|12|17|5|8|201Unown.png{{!}}100px}}{{-}}
{| class="roundy" style="width: 650px; border: 2px solid #{{Bug color light}}" 
|- style="text-align: center"
| style="width: 33%; background: #{{Bug color}}; {{roundyleft|10px}}" | {{mcolor|Hidden Power|000}}:
| style="background: #{{Bug color}}" | Type:<br>'''{{tcolor|Bug|000}}'''
| style="background: #{{Bug color}}; {{roundyright|10px}}" | Power:<br>'''31'''
|}

===Number of possible Hidden Powers===
As there are 6 IVs, ranging from 0 to 31 (32 in total), the number of different possible Hidden Powers should be 32<sup>6</sup>=2<sup>30</sup>, which is more than one billion possibilities. But let us consider two Pokémon with one different IV:

{{Hidden Power calculation/IV|'''30'''|31|31|31|30|31|025Pikachu.png{{!}}100px}}

and

{{Hidden Power calculation/IV|'''26'''|31|31|31|30|31|025Pikachu.png{{!}}100px}}

As we see, both 26 and 30 are divisible by 2 and give the remainder of 2 when divided by 4. So, in both cases the algorithms will interpret the IVs of those Pokémon in the same way, returning Grass-type Hidden Power with 70 power. It means that for the mentioned algorithms an IV of 30 is treated in the same way like IVs of 2,6,10,14,18,22 and 26 (8 in total). 

In fact, there are only four essentially different types of IV when calculating Hidden Power:

{|border="0" style="border-collapse: collapse;" cellspacing="1" cellpadding="3" 
|-
|1.
|IV that gives a remainder of 0 when divided by 4—has damage bit 0 and type bit 0:
|0, 4, 8, 12, 16, 20, 24, 28
|-
|2.
|IV that gives a remainder of 1 when divided by 4—has damage bit 0 and type bit 1:
|1, 5, 9, 13, 17, 21, 25, 29
|-
|3.
|IV that gives a remainder of 2 when divided by 4—has damage bit 1 and type bit 0:
|2, 6, 10, 14, 18, 22, 26, 30
|-
|4.
|IV that gives a remainder of 3 when divided by 4—has damage bit 1 and type bit 1:
|3, 7, 11, 15, 19, 23, 27, 31
|}

In other words: only four IVs that give different remainders when divided by four would cover all possible Hidden Power types and powers.

If so, the number of possible Hidden Powers should be 4<sup>6</sup> = 4096. This number, however is again far too large, as the real number of possible variations of Hidden power is simply 16*41=656, as Hidden Power exists in 16 types and has 41 different powers. The significant difference between 656 and 4096 is explained by the fact that the floor function results in distinct values (e.g. 10.952 and 10.476) being mapped to the same integer result (in this case 10).

===Percentage distribution of different variations of Hidden Power===
Due to the fact that both Hidden Power's type and its power are the products of rounding down, chances of getting a Pokémon with certain Hidden Power are not equal, as one might have thought, and are different for every Hidden Power's type and damage value. 

As it was stated before, the number of theoretically different Hidden Powers is 4096, which is the result of multiplying theoretical values of possible types (64) and powers (also 64). It means that every of 64 "types" comes into 64 "powers". After rounding down, however, the number of 64 "types" is reduced to 16 and the number of 64 "powers" to 41.  

{| style="margin: auto"
|- style="vertical-align: top"
|
{| class="roundy" style="text-align: center; color: #FFF; background: #{{platinum color}}; border: 3px solid #{{platinum color light}}; {{roundy|10px}}"
|+ style="color: #000" | '''Percentage distribution of Hidden Power's type'''
! style="color: #000; background: #{{platinum color light}}; {{roundytl|5px}}" | Number
! style="color: #000; background: #{{platinum color light}}" | Type
! style="color: #000; background: #{{platinum color light}}" | Qty
! style="color: #000; background: #{{platinum color light}}; {{roundytr|5px}}" | %
|- style="background: #{{Fighting color}}"
| 0
| Fighting
| 320
| 7.8125%
|- style="background: #{{Flying color}}"
| 1
| Flying
| 256
| 6.25%
|- style="background: #{{poison color}}"
| 2
| Poison
| 256
| 6.25%
|- style="background: #{{Ground color}}"
| 3
| Ground
| 256
| 6.25%
|- style="background: #{{Rock color}}"
| 4
| Rock
| 256
| 6.25%
|- style="background: #{{Bug color}}"
| 5
| Bug
| 320
| 7.8125%
|- style="background: #{{Ghost color}}"
| 6
| Ghost
| 256
| 6.25%
|- style="background: #{{Steel color}}"
| 7
| Steel
| 256
| 6.25%
|- style="background: #{{Fire color}}"
| 8
| Fire
| 256
| 6.25%
|- style="background: #{{Water color}}"
| 9
| Water
| 256
| 6.25%
|- style="background: #{{Grass color}}"
| 10
| Grass
| 320
| 7.8125%
|- style="background: #{{Electric color}}"
| 11
| Electric
| 256
| 6.25%
|- style="background: #{{Psychic color}}"
| 12
| Psychic
| 256
| 6.25%
|- style="background: #{{Ice color}}"
| 13
| Ice
| 256
| 6.25%
|- style="background: #{{Dragon color}}"
| 14
| Dragon
| 256
| 6.25%
|-
| style="background: #{{Dark color}}; {{roundybl|5px}}" | 15
| style="background: #{{Dark color}}" | Dark
| style="background: #{{Dark color}}" | 64
| style="background: #{{Dark color}}; {{roundybr|5px}}" | 1.5625%
|}
| style="width: 40px" |
|
{| class="roundy" style="background: #{{platinum color}}; border: 3px solid #{{platinum color light}}; {{roundy|10px}}"
|+ '''Percentage distribution of Hidden Power's powers (prior to Generation VI)'''
! style="background: #{{platinum color light}}; {{roundytl|5px}}" | Powers
! style="background: #{{platinum color light}}" | Qty
! style="background: #{{platinum color light}}; {{roundytr|5px}}" | %
|- style="background: #FFF"
| 30, 31, 33, 35, 36, 38,<br>40, 42, 43, 45, 47, 49,<br>50, 52, 54, 56, 57, 59,<br>61, 63, 64, 66, 68
| 128 each (23×)
| 3.125% each (23×)
|-
| style="background: #FFF; {{roundybl|5px}}" | 32, 34, 37, 39, 41, 44,<br>46, 48, 51, 53, 55, 58,<br>60, 62, 65, 67, 69, 70 
| style="background: #FFF" | 64 each (18×)
| style="background: #FFF; {{roundybr|5px}}" | 1.5625% each (18×)
|}
|}

==External links==
Many Pokémon related sites prepared web-based calculators, which allows to compute Hidden Power of a Pokémon with given IVs:
* [http://www.psypokes.com/gsc/dvguide.php Psypoke's Hidden Power calculator (Generation II)]
* [http://www.psypokes.com/dex/hp.php Psypoke's Hidden Power calculator (Generation III and on)]
* [http://www.legendarypokemon.net/javacalc.html Legendary Pokémon Hidden Power calculator (Generation III and on)]  

{{Project Games notice|game mechanic}}

[[Category:Subpages]]

[[de:Kraftreserve/Berechnung]]
[[es:Cálculo del poder oculto]]
[[it:Introforza/Determinazione e calcolo]]
[[ja:めざめるパワー計算法]]
[[zh:觉醒力量（招式）/计算]]