jmc

By Cauchy-Schwarz, $$(1^2+1^2+\cdots +1^2)(a_1^2+a_2^2+\cdots +a_{12}^2)\ge (a_1+a_2+\cdots +a_{12}{)}^2,$$so $$a_1^2+a_2^2+\cdots +a_{12}^2\ge \frac{1}{12}.$$Equality occurs when $a_i=\frac{1}{12}$ for all $i,$ so the minimum value is $\boxed{\frac{1}{12}}.$