Suppose A writes the series 1, 5, 11, 19, 29, and B says, "now I can go on," writing down the number 41. B might have calculated the formula a(sub)n = n^2 + n - 1 in his head, or he might simply have recognized a pattern in an intuitive fashion. In knowing how to continue the series 2, 4, 6, 8, we probably do not need to consult an algebraic formula in our heads. Knowing a formula seems a symptom of being able to go on, suggesting that there must be something explicable about the vague idea of "being able to go on."
The sentence "B can continue the series," can mean a number of things: that B knows the formula, that B has done this before, that B has not forgotten the formula, etc. While "B knows the formula" can often mean the same thing as "B can continue the series," under certain circumstances it might not. We should not conclude that "B can continue the series" describes a special state of affairs that stands behind the individual instances of knowing the formula, imagining further terms, and so on.
At the end of game sixty-four, Wittgenstein considers a variety of cases in which we say "I can " and then fail to do what we said we could, which he links with the various cases in which we talk about "forgetting" and "trying." Because "can" is in the present tense, we think of being able to do something as a mental mechanism that is present in us. This is similar to the notion that reading is a mental mechanism. Wittgenstein shows us that there is a family of different activities that we might refer to as "reading," ranging from skimming through a book silently, to painfully spelling out each letter. On the other hand, if the beginner has already memorized the passage he is being told to read, and then speaks the passage perfectly from memory, we say that he is not reading.
Suppose someone who is being trained to read occasionally pronounces random words, and sometimes these words are coincidentally the words on the page he is looking at. His teacher might claim he is not reading, because it is mere accident that he says the correct words. However, if the student goes on to pronounce some further words correctly, the teacher might slowly conclude that he has learned to read. However, we cannot point to any specific moment and say the student began reading there. There is no clear line between reading and not reading. Wittgenstein considers the suggestion that a person is reading if he derives spoken words from the page according to the rule provided by the alphabet. But suppose he reads a "b" for every "a," a "c" for every "b," and so on. He is still deriving a rule from the alphabet, and is still reading.
Wittgenstein stresses that these examples are not meant to show us the inessential features of reading, deriving, and so on, so that we can see through them and grasp the essential features. His method is purely descriptive: he is not suggesting there is something essential behind these inessential features.
Wittgenstein addresses the topic of reading precisely because it seems like a case in which there is a mental mechanism at work. We can think of humans as reading machines, analogous to a pianola, where there is an internal mechanism that turns written words into sounds. It even seems that the alphabet gives us a standard rule to transform these written symbols into sounds.