Earthworm Sequence

“Earthworm” algebra was first discussed by Clifford A. Pickover in his bookComputers and the Imagination. While digging in his garden Clifford accidentally severed an earthworm. It is known that both parts of a severed earthworm generally continue growing, so Mr. Pickover did not worry to much about the fate of the earthworm. However, he devised a method of multiply-and-severe which he called “Earthworm algebra”. It works as follows: take any whole number A, a constant multiplier B and a number which will be the maximum number of digits allowed, C. Next, multiply A and B. Take the result and multiply it again by B. Repeat this process until the number of digits in the result get past C. Now, severe the result by truncating it to the its rightmost C digits. Multiply again by B and severe again to C digits, and so forth. It turns out that all earthworms (mathematically speaking) eventually enter a cycle, an infinite loop of repeating values. A simple example: lets take 2 as our first number, 3 as the multiplier and 2 for the maximum number of digits. The sequence will go as follows:

6 18 54 62 (severed 162) 86 58 74 22 66 98 94 82 46 38 14 42 26 78 34 2 6 18...

This particular combination gives a cycle of 20 different values. Some combinations generate “worms” that are thousands of values long. This simple procedure yields great complexity. The generated sequences seem to be almost random, but if you listen to them you’ll notice hidden patterns. I believe that aural inspection of these algorithms reveals information that would otherwise go unnoticed.