John Conway analyzed the Look and Said Sequence after it was introduced by one of his students at a party.

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## Who is John Conway?

John Conway was an English mathematician who was active in the theory of finite groups, number theory, knot theory and many other theories.

He was born and raised in Liverpool and spent half his career at the University of Cambridge. After moving to the USA, he spent his remaining career at Princeton University. In 2020, at 82, he died due to covid complications.

The Sequence is called Morris Number Sequence, named after cryptographer Robert Morris.

## Look and say Sequence can be Defined as: –

The Look and Say Sequence can be defined as an integer sequence that begins with a single digit in which the next term is obtained by describing the previous term.

The look-and-say Sequence begins as follows:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, …

## Understand the Look and Say Sequence with an example:

Let us understand the Sequence with the help of an example: –

If Starting with 1, the Sequence would be defined by “1,

one 1,

two 1s,

one 2 one 1,” etc.,

and the result is 1, 11, 21, 1211, 111221

To generate a member (number) of the Sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit.

As: –

- First, 1 is read as “one 1” or 11 as we advance.
- Now, 11 is read as “two 1s” or 21 going forward
- Now, 21 is read as “one 2, one 1” or 1211 as we advance
- Now, 1211 is read as “one 1, one 2, two 1s” or 111221 further
- 111221 is read as “three 1s, two 2s, one 1” or 312211.

And the Sequence goes on like that…………….

In other words, as the name of the Sequence itself suggests, Look And Say, you have to look at the number and read(say)it.

- As 1 can be read as “one 1,” or we can say 11 (basically, we have to write the words or what we read in numeric form), the next value of the Sequence is 11.
- 11 can be read as “two 1s” or 21, so the next value of the Sequence is 21.
- 21 can be read as “one 2, one 1” or 1211, so the next value of the Sequence is 1211.
- 1211 can be read as “one 1, one 2, two 1 “or 111221, so the next value of the Sequence is 111221.
- 111221 can be read as “three 1s, two 2s, one 1” or 312211, so the next value of the Sequence is 312211.

According to the example: –

The first few values of the Look and Say Sequence are: –

1, 11, 21, 1211, 111221, 312211……

## Properties of Look and Say Sequence

- The look and say Sequence grows indefinitely.
- No digit other than 1,2, or 3 appears in the Sequence.
- If started with any digit d from 0 to 9, then d will remain indefinitely as the last digit of the Sequence. For any d other than 1, the Sequence starts as follows:

d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …

## How is the Look and Say Sequence Generated?

- Nth term is generated by reading (n-1)’ th term.
- The first term is “1.”
- The second term is “11”, generated by reading the first term as “One 1” (There is one 1 in the previous term)
- The third term is “21”, generated by reading the second term as “Two 1.”
- The fourth term is “1211”, generated by reading the third term as “One 2 One 1.”

And so on ………………………………………

## Questions Related to the Look and Say Sequence

Now, we will solve questions related to the Look and Say Sequence: –

- A positive integer n is given to you. You have to find the nth term of the look and say Sequence.

**Case 1 **= If n is 3, which means

Input = 3

The output will be, or the nth term of the Sequence will be 21

**Case 2 =** If n is 5, which means

Input = 7

The output or the nth term of the Sequence will be 111221.

## Practice

- A positive integer n = 4 is given to you; what will be the nth term of the look and say Sequence?