An alternative phrasing:
In the Atlas Mountains, one can meet two species of Sphinxes: True Sphinxes and Hollow Sphinxes. Each True Sphinx has a secret number, and it will truthfully answer any question about it, provided it can do so in less than a hundred words. A Hollow Sphinx, however, just enjoys wasting people's time : it has no secret number but will pretend it does, and answers questions so as to never contradict itself and maintain uncertainty in the asker's mind.
While you can be sure you are speaking to a True Sphinx (for example, by guessing it's number), you can never be sure that you are speaking to a Hollow Sphinx - it might be a True Sphinx whose number is very large. In fact, no-one has been able to determine whether any Hollow Sphinxes still exist.
I wonder if your explanation will make sense to people who didn't understand the original problem... but it's awesome!
Followup to: What's a "natural number"?
While thinking about how to make machines understand the concept of "integers", I accidentally derived a tiny little math result that I haven't seen before. Not sure if it'll be helpful to anyone, but here goes:
You're allowed to invent an arbitrary scheme for encoding integers as strings of bits. Whatever encoding you invent, I can give you an infinite input stream of bits that will make your decoder hang and never give a definite answer like "yes, this is an integer with such-and-such value" or "no, this isn't a valid encoding of any integer".
To clarify, let's work through an example. Consider an unary encoding: 0 is 0, 1 is 10, 2 is 110, 3 is 1110, etc. In this case, if we feed the decoder an infinite sequence of 1's, it will remain forever undecided as to the integer's value. The result says we can find such pathological inputs for any other encoding system, not just unary.
The proof is obvious. (If it isn't obvious to you, work it out!) But it seems to strike at the heart of the issue why we can't naively explain to computers what a "standard integer" is, what a "terminating computation" is, etc. Namely, if you try to define an integer as some observable interface (get first bit, get last bit, get CRC, etc.), then you inevitably invite some "nonstandard integers" into your system.
This idea must be already well-known and have some standard name, any pointers would be welcome!