The well-known ABRACADABRA problem states (see D. Williams, "Probability with martingales", for example):
a monkey is typing letters A-Z randomly and independently of each other, each letter with probability $1/26$. What is the expected amount of time when the monkey first types ABRACADABRA ?
I am well aware (and there are tons of references on the internet) of the martingale approach to this problem, leading to the expected time of $26^{11} + 26^4 + 26$.
While the connection with martingales is undoubtedly elegant and provides a streamlined answer to this question, I'm still wondering, however, if there are other techniques one can use here ?
For instance, for a similar problem with coin tosses, such as waiting for a particular pattern with heads and tails, one can work out things just by hand (using simple conditioning), if the pattern is short. With longer patterns, that type of brute force approach does not work well.
Although I'm mainly interested in non-martingale based approaches to this type of problems, sharing your insights/intuition on the connection with martingales would also be very welcome.