Look and Say Sequence

Background

The look-and-say sequence is a concealed and mysterious topic of mathematics. But don’t think of it as just a sequence. It’s more than a sequence. This sequence has a unique and mysterious characteristic that is really difficult to understand and solve. This sequence is also known as the Morris Number Sequence. Robert Morris was a famous cryptographer and he asked a puzzle: “What is the next number in the sequence 1, 11, 21, 1211, 111221?” Can any of you solve this? I think, not too easily. You can try your best to solve it.

Look and Say Conway Sequence

The sequence

\[1, 11, 21, 1211, 111221, \ldots\]

is known as the look-and-say sequence. Why this type of name for the sequence? It is the key point about this sequence. Just try to think of the sequence from a different view of your mind. You may get the answer.

Definition

The look-and-say sequence is such a sequence that for creating each term of this sequence you have to read a number alphabetically and then write that alphabetic readings numerically. You can take any number as a starting number, and then follow this rule to produce next numbers.

Rules for Creating the Sequence

The rules are as follows:

Take any number you like.
Pronounce the number correctly. As for the number 1, It must be read as "one 1," where “one” represents the total digits of that number.
Now write the "one 1" numerically. So we get the number 11.
Next we can read 11 as "two 1." So the next number of this sequence is 21.
Apply the same method for finding the next number of the sequence.
The starting number may be any number you want.
This sequence grows indefinitely.
Only for same digits of number like 22, if you apply the above method , this will return the same result 22 instead of a sequence.
Only the digits 1, 2, and 3 are the members of this sequence. There must not be any other number.

The Graph of Look and Say Conway Sequence

\[\]

Conway’s Cosmological Theorem

Every sequence eventually splits (“decays”) into a sequence of “atomic elements,” which are finite subsequences that never again interact with their neighbors. There are 92 elements containing the digits 1, 2, and 3 only, which John Conway named after the natural chemical elements. There are also two “transuranic” elements for each digit other than 1, 2, and 3. (From Wikipedia)

The First Some Numbers of Look and Say Conway Sequence

If we start from 1 as the first digit, the sequence will be as follows:\[\]

1

11

21

1211

111221

312211

13112221

1113213211

31131211131221

13211311123113112211

11131221133112132113212221

3113112221232112111312211312113211

1321132132111213122112311311222113111221131221

11131221131211131231121113112221121321132132211331222113112211

311311222113111231131112132112311321322112111312211312111322212311322113212221

Python Code for Generating Look and Say Conway Sequence

Python Code :

def lookandsay(number):
    result = ""

    repeat = number[0]
    number = number[1:]+" "
    times = 1

    for actual in number:
        if actual != repeat:
            result += str(times)+repeat
            times = 1
            repeat = actual
        else:
            times += 1

    return result

num = "1"

for i in range(10):
    print (num)
    num = lookandsay(num)

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