Pen-and-paper educational games I'm teaching maths at the 8-16 year level, one-on-one tuition.
What are some good pen-and-paper maths games I can play with these kids?
Googling "math games" just gets me a load of interactive animated junk and most games in this list are not actually educational and most are not really mathematical either (in the high-school-level sense -- they are combinatorial games).
It's important that these games teach or give exercise in something other than arithmetic.
Ideally these games would have an element of competition (teacher and pupil racing to complete something, for example) but I need to be able to invent instances of the game in just a moment's thought.
Any ideas?

If I’ve understood it well, you want to play some ‘mathematical game’ to introduce/teach a pupil something. I thought of trying to estimate a ‘difficult’ number (as fast as possible) (like $\sqrt[3]{1.5}$) in order to introduce the use of derivatives.
In this example, define $f(x)=\sqrt[3]{x}$.
We know that $f(1)=\sqrt[3]{1}=$1 and that $f’(1)=\frac{1}{3\sqrt[3]{x^2}}$, so a good estimate would be 
$$\sqrt[3]{\frac{3}{2}}\approx f(1)+\frac{1}{2}\cdot f’(1)=1+\frac{1}{6}=1.1666\cdots$$
