Consider the beginning of the following sequence:
Could you find the next term and the formation rule of the sequence?
Before you continue reading, you should stop reading and think about this problem for a moment.
Let us start with the first term:
"1", What do you see? The answer is "one 1", you can put it differently as "11".
So the next term of the sequence is "11". What do you see now? "two 1s" that is "21".
This term is read as "one 2 and one 1", that is "1211", the fourth term of the sequence.
Now "1211" is read as "one 1 one 2 and two 1s", that is "111221". If you read this term is "three 1, two 2s and one 1", or "312211", the sixth term of the sequence.
Have you grasp the idea of the formation rule of the sequence? If so you can say the eight term of the sequence is "1113213211".
For this reason this sequence is known as look and say sequence and it was introduced by the British mathematician John Conway in 1986.
You might think that this is a mathematical joke. Well, it started like a joke, but we can say some interesting things about this sequence from the mathematical point of view. We can ask different questions, among them:
- Besides the numbers 1, 2 and 3, Do other digits appear in the sequence?
- What is the growth of length of the general term of the sequence? This means the following, consider the n-th term of this sequence and denote by Dn the number of digits that has. For example: D1=1, D2=2, D3=2, D4=4, D5=6 and so on.
We would like to know the growth rate of Dn, i.e.
The answer of the first question is negative. It can be proved that besides the digits: 1, 2 and 3. no other digits appear in the terms of the sequence.
The answer of the second sequence was also given by Conway. This limit exist and today is called the Conway constant and its value is approximately 1.303577269034... It is the only positive real root of the following polynomial of degree 71, called the Conway's polynomial:
The roots of this polynomial can be seen in the figure as blue dots.
A more sophisticated analysis of this sequence can be done.
References:
//en.wikipedia.org/wiki/Look-and-say_sequence
//oeis.org/A005150
//oeis.org/A137275
//mathworld.wolfram.com/LookandSaySequence.html
J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16.
The image of the roots of the polynomial was done by myself using Mathematica and the polynomial formula with LaTex.
1, 11, 21 Riddle is a tricky riddle and here is the answer. It’s a mathematical Puzzle in which you need to guess the next sequence number. This kind of riddles has a pattern in it which you have to find out.
Question: 1, 11, 21, 1211, 111221, 312211. What is the next number in the sequence?
Answer: The answer to “1 11 21 Riddle” is “13112221.”
The next sequence after the 312211 will be 13112221. The pattern in this puzzle is:
- In each number one is increasing and decreasing (once places).
- If you look the 2 number in Tens place its also increasing after each sequence.
- Then see the third line after the 2 the one is also increasing and decreasing.
- At last we will add one as the one is added every time as with the 2.
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This is the popular say-series. Each new term is obtained by 'speaking out' the number of times each number occurs. Like 1122 has 2 1s and 2 2s. → Next term is 2122.
11 ≡ 2-1's ≡ 21
21 ≡ 1-2's 1-1's ≡ 1211
1211 ≡ 1-1's 1-2's 2-1's ≡ 111221
111221 ≡ 3-1's 2-2's 1-1's ≡ 312211
∴ The missing term is 312211 & correct alternative is 1.
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Casadi S.
I can only get the first two. I can't figure out the rest.
2 Answers By Expert Tutors
You have stumbled upon a series that I used to give my students as a fun challenge! If you really are stuck, I'll give the answer below, but don't look if you still want to give it a shot! The first number is 1, so say "one." Now, to form the next number, tell me what I have in the previous one---"one one"---so write 11. Then, the next consists of "two ones", so write 21. The next consists of "one two, one one", so write 1211, etc. Keep reading the previous number in terms of blocks of any given digit, and you'll have your sequence. Very smart-alecky type of recursive sequence, and it always elicits a groan from the audience once they hear the answer.
Rusty P. answered • 09/06/13
Math can be Fun, or at least tolerated