38.

We are given that: $$A=\frac {B+31+64+20}{4}=\frac {B+115}{4}$$ and

$$B=\frac {A+18+54+38}{4}=\frac{A+110}{4}$$

$$4A=B+115$$ Subbing in the second equation gives

$$4A=\frac {A+110}{4}+115$$ so $$16A=A+110+460=A+570$$.

Therefore $$A=\frac {570}{15}=38$$.

*This problem is taken from the UKMT Mathematical Challenges.*