How to get previous ELO score from current and match result I want to get a previous ELO value knowing the opponent's value and the result of the match. There are two equations, one for a loss and one for a win:
Win
$$ELO_{Current} = ELO_{Previous} + K \cdot \left(1 - \frac{1}{1 + 10^{{(ELO_{Opponent} - ELO_{Previous})}/{400}}}\right)$$
Loss
$$ELO_{Current} = ELO_{Previous} + K \cdot \left(0 - \frac{1}{1 + 10^{{(ELO_{Opponent} - ELO_{Previous})}/{400}}}\right)$$
$K, ELO_{Opponent}, ELO_{Current}$ and whether it was a win or not are known, which is why I figured that it should be possible to get $ELO_{Previous}$ by rearranging the equation, but I have no clue how to do it.
 A: To make the formulas a little easier to read and write, let
\begin{align}
E_C &= ELO_{Current}, \\
E_P &= ELO_{Previous}, \\
E_O &= ELO_{Opponent}. \\
\end{align} 
If we let $w = 1$ for a win, $w = 0$ for a loss, then we can summarize the two equations as
$$
E_C = E_P + K \left(w - \frac1{1 + 10^{(E_O - E_P)/400}}\right).
$$
The unknown value is $E_P.$
Now let's start rearranging this equation in an attempt to solve it.
Distributing the $K$ over the parentheses and collecting all terms on the left side,
we get
$$
 \frac K{1 + 10^{(E_O - E_P)/400}} + E_C - wK - E_P = 0.
$$
Clear the fraction and separate the exponents:
$$
 K + (E_C - wK - E_P) \left(1 + 10^{E_O/400}\left(10^{-1/400}\right)^{E_P} \right) = 0
$$
There are still other ways to try to arrange the terms, but the fact that $E_P$
occurs as a linear term in on of the factors and an exponent in another factor seems to be a clear indication that this equation will not have a solution using ordinary operations and functions.
The best you can do is to use numerical methods, which is a kind of sophisticated guesswork.
