The Math Behind Pokemon Levels

Calculus Level 4

When Pokemon was being designed, its creator Satoshi Tajiri had to make decisions about the gameplay. He decided that the amount of experience points which a defeated Pokemon yields should be directly proportional to its level. He also wanted the gameplay to last longer at higher levels, and decided that in order to level up, the number of Pokemon you needed to defeat should be directly proportional to your level.

Based on these decisions, he created EXP(L), which denoted the total number of experience points you needed to reach level L. What is the shape of EXP(L)?

Image credit: Indie Games Ichiban

Cubic Linear Exponential Quadratic

2 solutions

Christian Nader
Jan 29, 2014

If you try to build the governing equation from the bottom up you will get to something similar to: $EXP(L+1)=EXP(L)+L^{2}\times EXP(1)$

This is of the form : $f(x)=f(x-1)+kx^{2}$

By recursive analysis :

$f(x)=f(x-1)$ is obviously a constant function

$f(x)=f(x-1)+k$ is a linear function because its derivative is constant

$f(x)=f(x-1)+kx$ is a quadratic function because its derivative is linear

Therefore $f(x)=f(x-1)+kx^{2}$ is cubic .

Okay so here is what I thought: Let $EXP(1) = k$ . Then $EXP(2) = k + 2k = 3k$ . Obviously it follows a triangular pattern, namely $EXP(n) = k\frac{n(n+1)}{2}$ , which is quadratic.

Does the first condition imply that you can only fight pokemon of a certain level when your own pokemon is that level? Is that how it works in pokemon (Honestly I have no idea, never played the game). I thought that was somewhat of a red herring since I didn't know that you could only play a certain level of pokemon.

Michael Tong - 7 years, 4 months ago

hi michael

keep in mind that $EXP(L)$ is the experience gained thru the whole game up to this stage, therefore, $EXP(2)=EXP(1)+1\times xp\times 1\times n$ where $xp$ is the experience point yeilded by a defeated pokemon of your level, and $n$ is the number of pokemons you need when you're at a specific level.

Then, by the rules of the problem, if you're going from level $L$ to level $L+1$ you will need to defeat $L\times n$ pokemons, each of them yielding $L\times xp$ experience points (let $k=xp\times n$ ) and dont forget all the experience you've collected up to this point which is equal to $EXP(L)$ , that makes the total in-game experience up to level $L+1$ equal to: $EXP(L+1)=EXP(L)+L^{2}k$

christian nader - 7 years, 4 months ago

total exp point earned in level L = k L^2, so to reach level 2 point earned in level 1=k 1^2, ......to reach level L points earned at L-1 level is k (L-1)^2, so total points to reach level L= k (1^2+2^2+....(L-1)^2)=k*[(L-1)^3/2+(L-1)^2/3+(L-1)/6] which is cubic.

Shahriar Rizwan - 7 years, 4 months ago

Samuraiwarm Tsunayoshi
Aug 5, 2014

For someone who don't know the game system, the game recounts the EXP for every level. Example, we need 0/50 exp to lvl from 1 to 2. If we reach 50/50, we need 0/100 for 2 to 3. And so on...

Let $EXP(1\rightarrow2) = n$ (we need k Pokemons to lvl up from 1 to 2)

Since it's directly proportional to the number of Pokemons to lvl.

Therefore, $EXP(2\rightarrow3) = kn$ for constant k

$EXP(3\rightarrow4) = 2kn$

$EXP(4\rightarrow5) = 3kn$

...

$EXP(n-1\rightarrow n) = (n-2)kn$

Sum all these together we get

$EXP(1\rightarrow n) = \displaystyle \frac{k(n-1)(n-2)n}{2}$

Which gives you a cubic equation.

PS: This may or may not be the actual formula the Pokemon game.

The Math Behind Pokemon Levels

Image credit: Indie Games Ichiban

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