Look and Say Sequence

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.

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.

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.

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.

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)

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 :

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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)