Let the mean and the variance of 20 observations $$x_{1}, x_{2}, \ldots, x_{20}$$ be 15 and 9 , respectively. For $$\alpha \in \mathbf{R}$$, if the mean of $$\left(x_{1}+\alpha\right)^{2},\left(x_{2}+\alpha\right)^{2}, \ldots,\left(x_{20}+\alpha\right)^{2}$$ is 178 , then the square of the maximum value of $$\alpha$$ is equal to ________.

Let $$a_{1}, a_{2}, a_{3}, \ldots$$ be an A.P. If $$\sum\limits_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4$$, then $$4 a_{2}$$ is equal to _________.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $$\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}$$, in the increasing powers of $$\frac{1}{\sqrt[4]{3}}$$ be $$\sqrt[4]{6}: 1$$. If the sixth term from the beginning is $$\frac{\alpha}{\sqrt[4]{3}}$$, then $$\alpha$$ is equal to _________.

The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________.