The book I referred to above quotes Carroll's 1896 Preface as follows:
"Enquiries have been so often addressed to me, as to whether any answer to the Hatter's Riddle can be imagined, that I may as well put on record here what seems to me to be a fairly appropriate answer, viz. 'Because it can produce a few notes, though they are
very flat; and it is never put with the wrong end in front!' This, however, is merely an afterthought: the Riddle, as originally invented, had no answer at all."
(I like Thufir's version better!)
But Huxley also mentions some solutions proposed by others:
Sam Loyd, the famous puzzler, suggested, "Because they both should be made to shut up." He also came up with another good one: "Because Poe wrote on both."
A. Cyril Pearson suggested, "Because it slopes with a flap."
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After writing the above, I then went to Google, and found
this page, which could have saved me a lot of time!!