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The "look and say" sequence

It has the appeal of patterns that seem almost visible and yet slip away for a moment longer. There is no hidden algebra here: just learn to look at the row with the correct rhythm.

Observe the sequence: \[
1,\ 11,\ 21,\ 1211,\ 111221,\ 312211,\ 13112221
\] What is the next term?

Hints

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  1. It is not added or multiplied: each term is obtained by reading the previous one.
  2. The reading is done by groups of consecutive equal figures.

Solution

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Answer: the next term is 1113213211. The rule is to describe the previous term by counting groups of consecutive equal digits.
For example, 13112221 is read as:

  • a 1 -> 11
  • a 3 -> 13
  • two 1 -> 21
  • three 2 -> 32
  • a 1 -> 11 Concatenating: 11 13 21 32 11 => 1113213211. This sequence is known as look-and-say (or Morris sequence).

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