The well-known 4th order Runge-Kutta formula for the ODE $y'=f(x,y(x)$ is given by $$y_{n+1} = y_{n} + \dfrac{1}{6}h(K_1 + 2 K_2 + 2K_3 +K_4),$$ where $K_1,K_2,K_3,K_4$ are essentially approximations for the slopes of $y=y(x)$ at four points.
Why are the weights 1-2-2-1? What would happen if we change the formula to $$y_{n+1} = y_{n} + \dfrac{1}{4}h(K_1 + K_2 + K_3 +K_4)?$$
Kutta found this example when looking for 4th order methods that have symmetric final weights, $b_1=b_4$, $b_2=b_3$, together with the condition that as quadrature method, that is for trivial ODE $y'(x)=f(x)$, the method should devolve to the Simpson rule. This already fixes the $b_i$ and $c_i$.
If I remember correctly, this also fixes the matrix coefficients $a_{ij}$.
On the other hand, there are not degrees of freedom left when the strictly lower triangular matrix $a_{ij}$ has only non-zero entries on the sub-diagonal.
See How to derive 4th Runge-Kutta? for a bit more on the sub-diagonal pattern, Explanation and proof of the 4th order Runge-Kutta method for a heuristic approach to the structure of RK4.
