I came across an article in Wikipedia about Dudeney numbers. These are numbers whose digit sum add up to their cube root:

    1 =  1 x  1 x  1   ;   1 = 1
  512 =  8 x  8 x  8   ;   8 = 5 + 1 + 2
 4913 = 17 x 17 x 17   ;  17 = 4 + 9 + 1 + 3
 5832 = 18 x 18 x 18   ;  18 = 5 + 8 + 3 + 2
17576 = 26 x 26 x 26   ;  26 = 1 + 7 + 5 + 7 + 6
19683 = 27 x 27 x 27   ;  27 = 1 + 9 + 6 + 8 + 3

The wiki page went on to proclaim that those are the only six such numbers. Somebody on the talk page asked where the proof was.

I poked on Google and didn’t find anything, so I wondered if I could just prove it myself. Here’s what I came up with off the cuff. Perhaps others would find it interesting.

(more…)

(UPDATE 26-Jun-2009: A fuller development of the Arecibo Ascii code is now available at http://hostilefork.com/uscii/)

Every programmer who knows English is aware of the ASCII code, which declares that 65 means “A” and 66 means “B”, etc. Yet there is nothing intrinsically “A-like” about the number 65 (binary: 1000001), nor anything B-like about the number 66 (binary: 1000010). To see that, just imagine living in the 1800s and this fell from the sky on a piece of paper:

1000001100001010000101000001

Even if you knew it was supposed to represent text, I think it would be impossible to read that as “ABBA” with any degree of confidence. You might be able to get a clue that 7-bit sections were significant if you had a large body of data and realized they were always multiples of seven in length, but any single signal like this would not be enough. You’d be in an especially bad position if you didn’t know anything about alphabetical order (which isn’t a strict prerequisite of being able to read or write English successfully)!

To address this, I created something called “Arecibo ASCII”. It’s named after the infamous Arecibo message—a binary sequence transmitted into space that tried to explain some things about humanity. The goal was to make as few assumptions about the receiving aliens as possible…only that they had an understanding of physics and math (and obviously, the ability to detect electromagnetic waves).

When Carl Sagan and Frank Drake composed the message, they took it to Richard Feynman without explaining to him what it was. They figured if Feynman couldn’t decode it—given his upper hand of already knowing Earth science—then the aliens wouldn’t have a chance! Luckily, Feynman got pretty much all of it.

In the spirit of that test, I’ll send you an “Arecibo ASCII” message before I tell you how it works!

11111​11111​11111​11111​11111​11111​11111​01111​11111​11111​11111​11111​11111​11111​10111​11111​11111​11111​11111​11111​11111​11011​11111​11111​11111​11111​11111​11111​11101​11111​11111​11111​11111​11111​11111​11110​10001​10001​10001​11111​10001​10001​10001​00000​00000​00111​01000​11000​11000​10111​00000​00000​00011​11100​00011​10000​01111​10000​10000​10011​11100​10000​10000​10100​01000​00000​01000​00000​01000​01000​01000​01000​01100​00100​00100​00100​00100​00100​01110​00000​00000​00111​01000​11111​11000​00111​10000​00000​00000​00000​00000​00000​00000​00011​11110​00010​00011​11010​00010​00010​00000​00000​00000​11101​00011​00011​00010​11100​00000​00000​10110​11001​10000​10000​10000​00100​00100​00100​10101​00110​00101​00100​10111​11111​11111​11111​11111​11111​11111​11011​11111​11111​11111​11111​11111​11111​11101​11111​11111​11111​11111​11111​11111​11110​11111​11111​11111​11111​11111​11111​11111​01111​11111​11111​11111​11111​11111​11111​10111​11111​11111​11111​11111​11111​11111​11011​11111​11111​11111​11111​11111​11111​1110​

Want to test your alien-codebreaking-savvy? See if you can figure out what that says before you read the rest of the article! Otherwise, just read on as I spill the beans…

(more…)

If you’ve ever looked at a hi-fi component stereo system you’ve probably seen a graphic equalizer. Traditional analog ones look like this one sold by Crutchfield:

Each slider lets you raise or lower the volume of certain sounds in your music. If you know that a flute is in a frequency range of 250 Hz - 2500 Hz, you could make the flute part louder by raising the sliders that corresponded to that range. (As you see from this acoustics chart, frequency ranges for instruments do overlap—so raising a flute can possibly also affect other things like Oboes or vocals.)

Cheaper stereo systems usually only offer “bass” and “treble” knobs. Yet nearly every MP3 player has a digital EQ. Even iTunes has one, if you click Show Equalizer on the View menu:

But here’s a problem: because graphic equalizers break your signals into pieces and then put them together again, they introduce some distortion. There’s a simple way to get a sense that this distortion exists: you can set all the sliders to “flat”… run the equalizer algorithm… and see that you don’t get the same data out that you put in.

As a student back in 1996, I came up with a way you can design a digital graphic equalizer so the algorithm will (nearly) exactly output the original signal when sliders are flat, even after running your audio through a barrage of filtering mathematics. Practically speaking, you would want to bypass the equalizer in that case for performance reasons. Yet there is an alluring elegance to an approach which would converge upon the original signal if you applied it with flat levels… and it strongly suggests you’d have overall less distortion when the sliders weren’t at zero. (This turns out to be true!)

I employ only two digital filters: a specially matched highpass and lowpass. Recursively applying these filters on downsampled versions of the data creates the exact logarithmic bands that are found in graphic equalizers. The process additionally leads to a phenomenon known as “perfect reconstruction”. I explain the details below.

(more…)