The Loony Math of the Look-And-Say Sequence

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This weird, simple, math sequence is also known as the Audioactive Sequence, presumably to make it sound less goofy than 'speak-n'-spell.' It is that simple, and it is that funny.

When a million mathematicians take a million pieces of chalk to a million blackboards, we've already seen some the results. They're much crazier than Shakespeare's plays. But they can, on occasion, also be simpler. The Look-And-Say Sequence is an easy one to construct. Describe a sequence of numbers out loud, and write down your description. Start with one.



What do you see? You see one one.


What do you see? You see two ones.


What do you see? You see one two and one one.


Now you see one one, and then one two, and then two ones.


The sequence continues.

312211, 13112221, 1113213211, 31131211131221, . . . .

So far, to us non-math-heads, this seems obvious, useless, and maddening. But obvious, useless, and maddening is where mathematicians live, so they went to work on this sequence. First they showed that, when you start the sequence with '1,' you can never get above the digit '3,' anywhere.


Next they found that the nth term of the sequence was always proportional to (1.303577)^n. This was named Conway's Constant, after John Conway who studied the sequence, and it worked for any starting sequence except 22. Yes, they found that out.

Another mathematician, Roger Hargrave, grabbed the ball from Conway, and modified it. Now, instead of talking about how long the strings of ones and twos are, you describe how many of each number there is in the entire sequence. So, if you started with the term 321, the sequence will look like this:

321, 131211, 411312, 14311312, . . . .

Hargrave found out that his sequence generally caught itself up in an infinite loop. Eventually it would go back and forth between 23322114 and 32232114 forever. This has to be the way to survive a robot uprising. Just ask them to calculate the sequence and then destroy them at your leisure while they're caught in a loop. There's no telling how long Hargrave was caught in one.


Mathematicians have been playing with this silly, funny, and ridiculously easy sequence for decades, and they keep coming up with exceptions, rules, and new twists even now. It's tempting to admire the passion, the ingenuity, and the intellectual curiosity that creates sequences like these and finds the patterns within them. It's also tempting to tell them that they'd be better off burning their papers for warmth.

What I'm saying is, if you've ever been stood up for a date with a mathematician? This is what they were doing. You have to decide whether that's a comfort or not.


Image: Make Five and CDMI.

Via Wolfram Mathworld twice and The Math Book.