ADAC and HE puzzles from GEB

I have been reading Gödel, Escher, Bach: An Eternal Golden Braid by since the last Monday. The book alternates between chapters and dialogues. In the words of the author:

The long and the short of it is that I eventually decided - but this took many months - that the optimal structure would be a strict alternation between chapters and dialogues. Once that was clear, then I had the joyous task of trying to pinpoint the most crucial ideas that I wanted to get across to my readers and then somehow embodying them in both the form and the content of fanciful, often punning dialogues between Achilles and the Tortoise (plus a few new friends).

After the second chapter (Chapter II: Meaning and Form in Mathematics) there is a dialogue between Achilles and the Tortoise on telephone. The title of the dialogue is Sonata for Unaccompanied Achilles. Achilles is the only speaker, since it is a transcript of one end of a telephone call. The Tortoise is at the far end of the call. The sentences spoken by the Tortoise at the other end are not present. This makes it very interesting as we keep guessing what the Tortoise might have spoken.

It starts in this manner.

Achilles: Hello, this is Achilles.
Achilles: Oh, hello, Mr. T. How are you?
Achilles: A torticollis? Oh, I'm sorry to hear it. Do you have any idea what caused it?

As the dialogue proceeds, they share a few puzzles. Here is the first one from the Tortoise.

Achilles: A word with the letters 'A', 'D', 'A', 'C' consecutively inside it … Hmm … What about "abracadabra"?
Achilles: True, "ADAC" occurs backwards, not forwards, in that word.
Achilles: Hours and hours? It sounds like I'm in for a long puzzle, then. Where did you hear this infernal riddle?

Here is the second one from Achilles.

Achilles: Say, I once heard a word puzzle a little bit like this one. Do you want to hear it? Or would it just drive you further into distraction?
Achilles: I agree - can't do any harm. here it is: What's a word that begins with the letters "HE" and also ends with "HE"?
Achilles: Very ingenious - but that's almost cheating. It's certainly not what I meant!
Achilles: Of course you're right - it fulfills the conditions, but it's a sort of "degenerate" solution. There's another solution which I had in mind.
Achilles: That's exactly it! How did you come up with it so fast?
Achilles: So here's a case where having a headache actually might have helped you, rather than hindering you. Excellent! But I'm still in the dark on your "ADAC" puzzle.

If you want to think on these puzzles, don't read further as there are spoilers below.

It didn't take much time for me to solve the puzzle because I cheated with the word list file available in Debian 5.0.

Here is the output of my cheating.

susam@nifty:~$grep adac /usr/share/dict/words headache headache's headaches susam@nifty:~$ grep ^he.*he\$ /usr/share/dict/words
heartache


So, the answers to both puzzles seem to be 'HEADACHE'. Read the last sentence in the dialogue I have shown above, again. It makes sense now as Achilles says that having a headache might have helped the Tortoise.

Later in the dialogue the Tortoise offers 'figure' and 'ground' as hints to the 'ADAC' puzzle.

Achilles: Well, normally I don't like hints, but all right. What's your hint?
Achilles: I don't know what you mean by "figure" and "ground" in this case.
Achilles: Certainly I know Mosaic II! I know ALL of Escher's works. After all, he's my favorite artist. In any case, I've got a print of Mosaic II hanging on my wall, in plain view from here.
Achilles: Yes, I see all the black animals.
Achilles: Yes, I also see how their "negative" space - what's left out - defines the white animals.
Achilles: So THAT's what you mean by "figure" and "ground". But what does that have to do with the "ADAC" puzzle?
Achilles: Oh, this is too tricky to me. I think I'M starting to get a headache.

The famous painting discussed in the dialogue can be found here: http://www.worldofescher.com/gallery/A30L.html. One can see how the black animals form the figure or the positive space and how the background or ground or negative space beautifully fits all the white animals.

I was unable to use this hint to solve the puzzle. But after cheating and finding the answer I could make sense of the hint and understand how 'figure' and 'ground' lead to 'HEADACHE'. The first puzzle has 'ADAC' in the question. Let us consider 'ADAC' as the figure or the positive space. Now, if we remove 'ADAC' from 'HEADACHE', we are left with the ground or negative space, which consists of 'HE' in the beginning of the word and 'HE' in the end of the word. The figure is used to make the question in the first puzzle. The ground is used to make the question in the second puzzle.

An interesting question is: What was the first answer from the Tortoise that Achilles found very ingenious but degenerate? I believe, it is 'HE' as this word begins with 'HE' and also ends with 'HE'.

The funny thing is that both of them asked two puzzles to each other without knowing that the answers to them were same. This is exactly what happened when a colleague of mine and I challenged each other with combinatorics puzzles. I wrote a blog post on this here: Combinatorial coincidence.

Paritosh said:

Interesting! Now I wanna read this book.

TKD said:

Thanks man. Started reading GEB last Saturday and I could figure out the 'he' and 'headache' ones but did not realize headache was also the response to tha adac puzzle.

Cheers from Argentina,
TKD

rb said:

Thank you for posting this!