Among 6 nails, 3 are blue and 3 are red. Hang the painting on these nails so that the painting falls when a blue nail is removed, or when two red nails are removed, but not if only one red nail is removed.
Let $a_{1}, a_{2}, a_{3}$ be the blue nails and $a_{4}, a_{5}, a_{6}$ the red ones. We know that $a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{3}^{-1}$ works to meet the condition for the blue nails, and similarly $a_{4} a_{5} a_{6} a_{4}^{-1} a_{5}^{-1} a_{6}^{-1}$ for the red ones. Denoting $A$ and $B$ as these two words, and following the rule $\mathrm{d}^{\prime}$ inversion previously stated, $A B A^{-1} B^{-1}$ is a sequence that works, since as soon as one of the two conditions is met, one of the two uppercase letters reduces to 1.