Problem 1174

An interesting measure of optimality is the number of letters required to write the word associated with the solution. As a solution, one could choose

$$\left[\left[\left[a_{1}, a_{2}\right], a_{3}\right], a_{4}\right]=a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{3}^{-1} a_{4} a_{3} a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3}^{-1} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{4}^{-1}$$

which contains 22 symbols, but better is

$$\left[\left[a_{1}, a_{2}\right],\left[a_{3}, a_{4}\right]\right]=a_{1} a_{2} a_{1}^{-1} a_{2}^{-1} a_{3} a_{4} a_{3}^{-1} a_{4}^{-1} a_{2} a_{1} a_{2}^{-1} a_{1}^{-1} a_{4} a_{3} a_{4}^{-1} a_{3}^{-1}$$

which contains only 16.