ADAC and HE Puzzles from GEB

By Susam Pal on 16 May 2009

I have been reading Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter for a few weeks. The book follows an interesting format of alternating between chapters and dialogues between imaginary charaters. 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. The text shows the transcript of only one end of the call, only what Achilles speaks. The Tortoise is at the far end of the call. The sentences spoken by the Tortoise at the other end are not present in the text. This makes the reading experience very interesting as we keep guessing about what is going on at the other end.

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 puzzle 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 about these puzzles, this is a good time to pause and think about them. There are spoilers ahead.

It didn't take much time for me to solve the puzzle because I cheated with the word list file available on Linux distributions. Here is what I found with my Debian system (Debian 5.0).

$ grep 'adac' /usr/share/dict/words
headache
headache's
headaches
$ grep '^he.*he$' /usr/share/dict/words
headache
heartache

The answers to both puzzles seem to be "HEADACHE". Take another look at the last sentence in the dialogue above. 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: https://www.wikiart.org/en/m-c-escher/mosaic-ii. We can see how the black animals form the figure (the positive space) and how the background or ground (the negative space) beautifully fits all the white animals.

The figure and ground in hints make sense now. The first puzzle has "ADAC" in the question. Let us consider "ADAC" as the figure or the positive space. If we remove "ADAC" from "HEADACHE", we are left with the ground or negative space, which consists of "HE" and "HE" at the beginning and the end of the word. The figure is used in the first puzzle and the ground is used in the second puzzle

By the way, what is the first answer from the Tortoise that Achilles finds very ingenious but degenerate? I believe, it is "HE" as this word both begins and ends with "HE".

The funny thing about this dialogue is that both of them asked two puzzles to each other without knowing that the answers to them were the same. This is exactly what happened when a friend of mine and I challenged each other with combinatorics puzzles. See this blog post for the story: Combinatorial Coincidence.