Page 85 - Ganit Kaushal

3.13 THE COLLATZ CONJECTURE (An Unsolved Mystery!)

Look at the sequences (patterns of numbers) below.

(a) 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
(b) 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

(c) 21, 64, 32, 16, 8, 4, 2, 1
(d) 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Can you identify the pattern? The same pattern rule is applied in all the sequences:

Yeah! Why not? Well, the pattern rule is as follows:

1. Start with any natural number.
2. If the number is even, take half of it; if the number is odd, multiply it by 3 and add 1.

3. Repeat it till you get 1.
Mathematically, this sequence {T }, where T is its nth term, can be written as follows:
n
n
 T n , if T is even

T n+1 =  2 n ; n = 1, 2, 3, 4, ……; T can be any number of your choice.
1
 3 T +1, if T is odd

n
n
Observe that all four sequences above eventually reached the number 1. In 1937, the German mathematician
Lothar Collatz conjectured that:

‘The sequence will always reach 1, regardless of the whole number you start with.’

The sequence is known as the ‘Collatz Sequence’.

In mathematics, a conjecture is a statement that is believed to be true based on observations and evidence but
has not yet been proven or disproven. The Collatz conjecture has not been proved or disproved yet. Even
today, despite many mathematicians working on it, it remains an unsolved problem as to whether Collatz’s
conjecture is true!

® Students are advised to make the ‘Collatz sequence’ by taking the starting numbers of their own
choice and observe whether they end up at 1 or not.

Exercise 3.5

1. Write an example for each of the scenarios given below, whenever possible. (NCERT)
(a) 5-digit + 5-digit to give a 6-digit sum

(b) 5-digit + 5-digit to give 18,500
(c) 5-digit – 5-digit to give a difference less than 56,503

(d) 5-digit – 3-digit to give a 4-digit difference
(e) 5-digit – 4-digit to give a 4-digit difference

(f) 5-digit – 5-digit to give a 3-digit difference

Number Play 83