$Q.M.-A.M.$ inequality give us $\sqrt{\frac{\sum_{i=1}^{n}{\Big( |a_{2i-1}-a_{2i}|\Big)^2}}{n}}\geq \frac{\sum_{i=1}^{n}{|a_{2i-1}-a_{2i}|}}{n}$ From here we have $\sum_{i=1}^{n}{|a_{2i-1}-a_{2i}|} =n^2$ So we get $(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2\geq n^3$ and the rest is to show that equality case not occur. Since $a_1-a_2<a_3-a_4<a_5-a_6$ so there exist at least two different value among $|a_1-a_2|,|a_3-a_4|,|a_5-a_6|$ So equality not occur and we are done.

Let $n\in \mathbb{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}$, find the minimum value of $(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2$