In Havil's 2003 book Gamma he states that Hardy offered up his chair in Oxford to whoever could prove that the Euler-Mascheroni constant $\gamma$ is irrational.
I'm almost positive I had heard a source of this prior to Havil.
But what is the earliest source of this? Did Hardy really make the offer, and if so was it serious or just as a statement meant to attest to the difficulty of such a question?
Nick R points to Mathworld as indicating that there is no written reference for this quote.
So then I'll change my question - what was Havil's source? His book was quite solid otherwise.