Progress

The hint we get from Achilles is the following:

Very well. Perhaps by then I’ll have thought of the right answer to YOUR puzzle, using your figure-ground hint, relating it to MY puzzle.

Hmm, due to the hint I think the solution might be “HEADAIC”?

You know, headache, but spelled wrong? That would fit with “figure” and “ground”, by which the turtle was referring to “Mosaic II” by M. C. Escher, and it would also relate to Achilles’ puzzle, the solution to which was “HEADACHE”.

A couple of weeks later, revisiting the puzzle …

Hm, okay. I don’t think “HEADAIC” is a satisfactory answer here.

Staring at the dialogue for a while.

Oh wait, the letters even have to be consecutive, so “HEADAIC” wouldn’t be a valid answer even if we allowed incorrect spellings.

Hmm …

Oh my lord. Of course! I am an idiot sometimes …

My Solution

The solution to the ADAC puzzle is, naturally, also “HE ADAC HE”. And what’s more, this points back at the theme of the entire chapter, as we could either see “HE…HE” as “figure” or as “ground” of the word “HEADACHE”, with “ADAC” being the respective other.

So Achilles’ and the Tortoise’s puzzles each give us one complementary half of the picture, which we can then fit together to make the whole.

Beautiful.