[These slides](https://www.cc.gatech.edu/classes/AY2015/cs4496_spring/slides/DiffEqu.pdf) (slide 42) give a table (same as Table 1.6 given in Butcher's [General Linear Methdos](https://www.cambridge.org/core/journals/acta-numerica/article/div-classtitlegeneral-linear-methodsdiv/68B6D07A0CBC9AC5DE06ED4048A22A3F) of the minimum number of stages $s$ for a Runge-Kutta type numerical method of order $p$ (the slides use $q$, but all the other references I'm working with use $q$ to refer to stage order and $s$ to refer to the number of stages, so I'm sticking with that convention). *Neither table specifies that this only applies to explicit methods.*


The table is reproduced below:



```
p   s
-----
1   1
2   2
3   3
4   4
5   6
6   7
7   9
8   11
9   12-17 (lecture notes only)
10  13-17 (lecture notes only)

```

For a numerical method (implicit or explicit) of order $p=2$, the minimum number of stages is $s=2$, but the Implicit Midpoint/Gauss-Legendre 2 method is has $p=2$, $s=1$. The same applies to Gauss-Legendre 4 ($p=4$, $s=2$), and so on (Gauss-Legendre methods have $p=2s$).


The slides state "The minimum number of stages necessary for an explicit method to attain order $p$ is still an open problem." This suggests that for implicit methods, it is not an open problem.


In Butcher's General Linear Methods, we also have the following.



> 
> According to Table 1.6, it is suggested that, for p ≥ 5, it is necessary that s > p. We will now prove this result.
> 
> 
> **Theorem 1.7.** There does not exist an explicit Runge–Kutta method with order $p = s \ge 5$.
> 
> 
> 


This along with the existence of [Gauss-Legendre and other implicit Runge-Kutta methods](https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods#Implicit_methods) with fewer stages than what's listed in this table, suggests that higher order implicit methods require fewer stages than explicit methods.


**So what's up with this table?**



What is the minimum number of stages $s$ required for a Runge-Kutta type numerical method of given order $p$?
For implicit methods, you can achieve order $2s$ with $s$ stages. Note that this result is the same if one considers the simpler problem of numerical integration (quadrature).


For explicit methods, the table you have given contains everything that is currently known.


