Square root of one of $2,3,6$ exist in a prime field

Question

Using Gauss's law of Quadratic Resiprocity it is immediate that one of $2,3,6$ is a square in $\mathbb{F}_{p}$. I am looking for a solution which uses basic field theory only. I was thinking of considering the map $x\mapsto x^{2}$ its image has $\frac {p+1}{2}$ many elements, but I can't proceed after that.

Answer 1

There are two cosets in $\Bbb F_p^\times/(\Bbb F_p^\times)^2$ since $\Bbb F_p^\times$ is cyclic. If both of $2,3$ are in the nontrivial one...