What sentence convinced Russell that Wittgenstein was not a "complete idiot"?

Question

On several occasions I've come across Russell's account of his decision that Wittgenstein was not a "complete idiot", but I've never been able to figure out what Wittgenstein wrote to him or what that one sentence was. Can anyone tell me?

Here's a quote from Russell's Collected Papers:

[Wittgenstein] was queer, and his notions seemed to me odd, so that for a whole term I could not make up my mind whether he was a man of genius or merely an eccentric. At the end of his first term at Cambridge he came to me and said: “Will you please tell me whether I am a complete idiot or not?” I replied, “My dear fellow, I don’t know. Why are you asking me?” He said, “Because, if I am a complete idiot, I shall become an aeronaut; but, if not, I shall become a philosopher.” I told him to write me something during the vacation on some philosophical subject and I would then tell him whether he was complete idiot or not. At the beginning of the following term he brought me the fulfillment of this suggestion. After reading only one sentence, I said to him: “No, you must not become an aeronaut.” And he didn’t.

Answer 1

Welcome TCP

Russell doesn't so far as I'm aware tell us what the subject was of the 'something' he asked or invited Wittgenstein to write - perhaps he left the topic entirely to Wittenstein. Nor does Wittgenstein provide information from his side.

Some slight conjectural indication of what Wittgenstein wrote on is provided by the subject-matter of Russell's 1911 lectures and research writings, which informed the first deep intellectual engagements of the two men. In 1911 Russell was lecturing on the foundations of mathematics and was beginning to work out the ideas that later emerged in the logic and epistemology of 'Logical Atomism'. As well, Wittgenstein was critical at this time of Russell's theory of types, which he had arrived at around 1908. All this describes a big circle of possibilities for the contents of Wittgenstein's essay or notes; and we can't rule out as indicated above that Wittgenstein struck out on a topic of this own, though given his intellectual intimacy with Russell at the time I think it likely that what he wrote was germane to Russell's work - probably critically.

Given the paucity of information I doubt if we'll ever know what the relevant sentence was. I suspect Russell was not speaking literally in any case. Confronted with extremely distinguished work I have known several academics say something like, 'I only had to read one sentence to realise that this was an outstanding essay'. This generally meant, 'Any of several sentences I read would have convinced me of the essay's distinction'. No specific sentence was being referred to.