Well, now I’ve learned something!  I thought that Gollum called the first riddle a “chestnut” because that might be an idiom for “easy to crack”.  I have been educated.  OED tells us:

 6. slang. A story that has been told before, a ‘venerable’ joke. Hence, in extended use, anything trite, stale, or too often repeated.

• 05.032 Chestnuts, chestnuts,
• 05.032 Chestnuts, chestnuts,
• 05.041 This he thought a dreadfully easy chestnut,

“chestnut | chesnut, n. and adj.” OED Online. Oxford University Press, March 2016. Web. 2 June 2016.