| Progressions
This month I started producing three puzzles per issue of Discover, not just one. Why? Because the chief editor heard that if there are different kinds of a puzzle on the same page, then there's more chance that you'll find something you like.
The book Gödel, Escher, Bach )"GEB") plays a very special role in my life. I first met the author Douglas Hofstadter in 1975 when I was an undergraduate at Stanford and he was a postdoctorate student studying artificial intelligence. He posted a flier announcing a class called "Gödel's Theorem and the Human Brain" that featured an image by Escher, and mentions of Bach's music.
 At the time I was studying music and mathematics. I had just read Nagel and Newman's thrilling book "Godel's Proof"; I was learning to play Bach's Art of Fugue and Musical Offering on piano, and I had been a long-time fan of Escher. So Hofstadter's flier seemed to good to be true. I visited his office and was delighted to find that he was not just some dilettante bandying names about, but a true kindred spirit.
I was thrilled by the illustration that Richard Downs created for this puzzle, which skillfully weaves Gödel, Escher and Bach into a single picture.
Gödel. The MU puzzle plays a central role in GEB. Of course Hofstadter could have chosen any three letters instead of M, I and U. He chose MU because it appears as a word in a famous Zen koan.
Escher. I've created a whole alphabet of tessellating letters, which I will feature as my April 2000 inversion on this site.
Bach. Larry Krakauer wrote me this interesting letter:
When I saw your February 2000 Bogglers column in Discover Magazine, I
immediately knew the answer to Item 3, question 2, "How many scales are possible
with whole and half steps in any order?".
This was because I recognized it as equivalent to a question I asked a number of
years ago when developing a new barcode: How many codes of a specified fixed
length can be made using either one-unit or two-unit wide alternating bars and
spaces? The resulting barcode, though not widely known, is denser than the
common codes in use today.
Here's my report on this work. It includes a proof by construction of the number of codes of a given length, and a proof by induction of the method of mapping between these codes and an integer. |
