“Alf Todd,’ said Ukridge, soaring to an impressive burst of imagery, ‘has about as much chance as a one-armed blind man in a dark room trying to shove a pound of melted butter into a wild-cat’s ear with a red-hot needle.”

This is attributed, with an expression of doubt as to its correctness, in Mathematics, Our Great Heritage: Essays on the Nature and Cultural Significance of Mathematics (1948) by William Leonard Schaaf, p. 163; also attributed in Pi in the Sky : Counting, Thinking and Being (1992) by John D. Barrow. There are a number of similar expressions to this with various attributions, but the earliest published variants seem to be quotations of Lord Bowen:
When I hear of an 'equity' in a case like this, I am reminded of a blind man in a dark room — looking for a black hat — which isn't there.
Lord Bowen, as quoted in "Pie Powder", Being Dust from the Law Courts, Collected and Recollected on the Western Circuit, by a Circuit Tramp (1911) by John Alderson Foote; this seems to be the earliest account of any similar expression. It is mentioned by the author that this expression has become misquoted as a "black cat" rather than "black hat."
An earlier example with "hat" as a learned judge is said to have defined the metaphysician, namely, as a blind man looking for a black hat in a dark room, the hat in question not being there Edinburgh Medical Journal, Volume 3 (1898)
With his obscure and uncertain speculations as to the intimate nature and causes of things, the philosopher is likened to a 'blind man in a dark room looking for a black cat that is not there.'
William James, himself apparently quoting someone else's expression, in Some Problems of Philosophy : A Beginning of an Introduction to Philosophy (1911) Ch. 1 : Philosophy and its Critics
A blind man in a dark room seeking for a black cat — which is not there.
A definition of metaphysics attributed to Lord Bowen, as quoted in Science from an Easy Chair (1913) by Edwin Ray Lankester, p. 99
A blind man in a dark room looking for a black cat which isn't there.
A definition of metaphysics attributed to Lord Balfour, as quoted in God in Our Work: Religious Addresses (1949) by Richard Stafford Cripps, p. 72
A philosopher is a blind man in a dark room looking for a black cat that isn't there. A theologian is the man who finds it.
H. L. Mencken, as quoted in Peter's Quotations : Ideas for Our Time (1977) by Laurence J. Peter, p. 427
A metaphysician is like a blind man in a dark room, looking for a black cat — which isn't there.
Variant published in Smiles and Chuckles (1952) by B. Hagspiel
Misattributed