The mysterious number 495

There are certain numbers that have certain characteristics.   Until now, many of these numbers become source of fascinations of mathematicians and non-mathematicians alike. Many mathematicians have spent their entire lives solving the mysteries of some numbers.  In this post, we are going to talk about the mysterious number 495.
Let’s have  a game.
  1. Think of a 3-digit number where not all the digits are the same.
  2. Arrange the digits in descending order and ascending order from left to right. This will give you two 3-digit numbers — one the smallest number that you an form from the digits, and the other the largest.
  3. Subtract the smaller number from the larger number.
  4. Go back to step 2.
If you repeat this process over and over again, you will end up with 495!
You don’t believe it? Let’s have 3 examples:
Example 1
  1. Our number is 592.
  2. Arranging it in descending and ascending number, we have 952 and 259. 
  3. Subtracting 259 from 592 we get 693.
  4. Repeating steps 2-3,  we have 963 -369 = 594
  5. Repeating steps 2-3 again, we have 954 – 459 = 495
  6. Repeating steps 2-3 again, we have 954 – 459 = 495
Notice: If the difference contains the digits 9, 5, and 4, you will end up with 495 in each repitition.
Example 2 
  1. Our number is 466
  2. Arranging in ascending and descending we have 664 and 466.
  3. Subtracting  466 from 664 we have 198.
  4. Repeating steps 2-3, we have 981 – 189 = 792.
  5. Repeating steps 2-3, we have 972 – 279 = 693. And this is the same as the number 3 in Example 1 above, which will end up in 495.
Example 3
  1. Let’s have 990. 
  2. Arranging the digits in ascending and descending, we have 990 and 099 or simply 99.
  3. Subtracting, we have 990 – 99 = 891.
  4. Repeating steps 2-3, we have 981 – 189 =792. And this is the same as difference in Example 2 step 4. This means that it will also end up in 495!
The number 495 is the Kaprekar constant for three digit numbers. Yes, if you choose any 3-digit number, you will end up in 495 (try other numbers).  The more famous Kaprekar constant is 6174 for 4-digit numbers.
In the next post, we are going to investigate the mystery behind 495.
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