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Collatz Conjecture

It is named after the mathematician Lothar Collatz, who introduced the idea in 1937. This is also
known as the 4 – 2 – 1 conjecture, as the end result in any case comes out to be 4 – 2 – 1.

If number is even Divide by 2

Choose a Repeat 4 – 2 – 1
number

Multiply by 3
If number is odd
and add 1
Let us take a number 11 (say).
Here, number = 11 (odd)
Step 1: 3 × 11 + 1 = 34 (even) Step 2: 34 ÷ 2 = 17 (odd) Step 3: 3 × 17 + 1 = 52 (even)

Step 4: 52 ÷ 2 = 26 (even) Step 5: 26 ÷ 2 = 13 (odd) Step 6: 3 × 13 + 1 = 40 (even)
Step 7: 40 ÷ 2 = 20 (even) Step 8: 20 ÷ 2 = 10 (even) Step 9: 10 ÷ 2 = 5 (odd)
Step 10: 3 × 5 + 1 = 16 (even) Step 11: 16 ÷ 2 = 8 (even) Step 12: 8 ÷ 2 = 4 (even)
Step 13: 4 ÷ 2 = 2 (even) Step 14: 2 ÷ 2 = 1 (stop)
Kaprekar Constant for 3-digit Numbers

D.R. Kaprekar was a famous number lover from the state of Maharashtra, India. He has contributed
to different unique numbers and constants in mathematics. The Kaprekar constant of 3-digit
number is 495. The algorithm to find this constant is as follows:
1. Take any three-digit number with at least two digits different.
2. Arrange the digits in ascending and then in descending order.
3. Subtract the smaller number from the larger number.

4. Repeat the process until you reach the constant.
At the end, you will get 495.
For example, let us begin with 100. Arranging the digits in ascending order gives 001.
Subtracting the smaller number from the larger number, we get the following steps, which finally
yields to 495.
Step 1: 100 – 001 = 099 Step 2: 990 – 099 = 891 Step 3: 981 – 189 = 792
Step 4: 972 – 279 = 693 Step 5: 963 – 369 = 594
Step 6: 954 – 459 = 495 (Kaprekar constant)

Kaprekar Constant for 4-digit Numbers

Like the Kaprekar constant for 3-digit numbers, there is a famous constant for 4-digit numbers
given by Kaprekar. The four-digit Kaprekar constant is 6174. Let us understand the algorithm.
1. Take any four-digit number with at least two digits different.
2. Arrange the digits in ascending and then in descending order to get two four-digit numbers,
adding leading zeros if necessary.
3. Subtract the smaller number from the larger number.
4. Repeat the process until you reach the constant.

89 Number Play