Free notebooks were distributed equally among the children of a class. The number of notebooks each child got was $\frac{1}{8}\text{th}$ of the number of children. Had the number of children been half, each child would have got $16$ notebooks. How many notebooks were distributed in total?

1. Let the total number of children in the class be $N$.
* Number of notebooks each child received $=\frac{N}{8}$
* Total notebooks distributed $=N\times \frac{N}{8}=\frac{N^2}{8}$

2. According to the second condition, if the number of children was half ($\frac{N}{2}$), each child gets $16$ notebooks:
* Total notebooks distributed $=\frac{N}{2}\times 16=8N$

3. Equate the two expressions for total notebooks:

$$\frac{N^2}{8}=8N$$

$$N^2=64N⟹N=64(\text{since }N\ne 0)$$

4. Find the total number of notebooks distributed:

$$\text{Total notebooks}=8N=8\times 64=512$$