In honour of Robert Morris (1932–2011)

[Tiberius]

1, 11, 21, 1211, 111221

What is the next number in the sequence?

Answer:

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312211

Post if you understand how

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[Darwinian]

Best guess..

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312211

Three ones, two two's and one one...

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[Anymouse]

1, 11, 21, 1211, 111221

What is the next number in the sequence?

Answer:

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312211

Post if you understand how

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Robert Morris, cryptographer (my specialty in the Navy, repairing cryptographic equipment.)

The reasoning for the answer is hidden below.

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This string, which he defined, is called a "look & say" sequence. It is also used sometimes on intelligence tests and as "one time pad" cryptographic keys.

The next numbers in the sequence become immediately apparent as you read the elements:

start with one element "1," then read them to create the next element by writing down exactly what you read. . . .
The first element is 1.
So the next is "one 1," or 11.
Then "two 1's" or 21.
Then "one 2 and one 1," or 1211.
Then "one 1 and one 2 and two 1s," or 111221.
&c.
Therefore the next element is "three 1s, two 2s, and one 1," or 312211

This can be done with any initial element. For example: "2"
The next is "One 2," or 12.
Then "One 1 and One 2," or 1112.
Then "Three 1's and One 2," or 312 (et cetera)

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Comin' at ya like a ton of speeding guano . . . James.