The number of values of $\mathrm{a}\in \mathrm{\mathbb{N}}$ such that the variance of 3, 7, 12 a, 43 - a is a natural number is:

Mean =13
Variance $=\frac{9+49+144+{\mathrm{a}}^2+(43-\mathrm{a}{)}^2}{5}-{13}^2\in \mathrm{N}$
$\Rightarrow \frac{2{\mathrm{a}}^2-\mathrm{a}+1}{5}\in \mathrm{N}$
$\Rightarrow$ 2a² - a + 1 - 5n = 0 must have solution as natural numbers its D = 40n - 7 always has 3 at unit place
$\Rightarrow$D can't be perfect square
So, a can't be integer.