G. H. Hardy said he thought on paper ('with my pen'). He wrote everything out (in his invariably admirable handwriting), scrapping and copying whenev… - John Edensor Littlewood

[For an unproved Lemma] I had what looked like a promising idea for this, but it was fallacious. In the middle of a three week holiday - mathematics completely below the horizon - the idea came again out of the blue when I was in bed. I forgot it had been rejected as fallacious, and this forgetting did the trick; because this time I noticed that it did prove something, and, by what was nearly, or quite, automatic writing, a proof got written down which deduced the Lemma from the something. ... I did experience automatic writing again. ... my pencil wrote down the formula [omitted] for no reason at all, and almost unattended by consciousness. On the face of it the formula has no apparent connexion with the problem, but it turns out to be an essential key to the proof. ... as if my subconsciousness knew the thing all the time, and finally got impatient.

Improbabilities are apt to be overestimated. It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 10⁶ : 1. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely. ... There must exist a collection of well-authenticated coincidences, and I regret that I am not better acquainted with them. ... I sometimes ask the question: what is the most remarkable coincidence you have experienced, and is it, for the most remarkable one, remarkable? (With a lifetime to choose from, 10⁶ : 1 is a mere trifle.) ... Eddington once told me that information about a new (newly visible, not necessarily unknown) comet was received by an Observatory in misprinted form; they looked at the place indicated (no doubt sweeping a square degree or so), and saw a new comet. ...

It is possible for a mathematician to be 'too strong' for a given occasion. He forces through, where another might be driven to a different, and possibly more fruitful, approach. (So a rock climber might force a dreadful crack, instead of finding a subtle and delicate route.)