# The Kaprekar Constant 495

In the previous post, I have mentioned that 2013 is not that special compared to other numbers.  Yes, there numbers that are extremely special and interesting.  One such number is 495. Why? To find out, let’s play this game.

Steps

1. Think of a 3-digit number with at least two different digits (leading zeroes are allowed).
2. Form the largest and the smallest number using the three digits.
3. Subtract the smaller from the larger number.
4. Repeat steps 2-3 as many times as you can and observe what happens.

If you are adventurous, you might want to do the steps before proceeding below.

Let’s try two examples.

Example 1

1. Number: 847
2. Largest: 874, Smallest: 478
3. Subtract: 874 – 478 = 396
4. Largest: 963, Smallest 369 (Step 2)
5. Subtract: 963 – 369 = 594 (Step 3)
6. Largest: 954, Smallest 459 (Step 2)
7. Subtract: 954  - 459 = 495 (Step 3)

Notice that repeating the process will still form 954 and 459 which will repeat number 6 and 7. The result is now pegged to 495.

Example 2

1. Number: 625
2. Largest: 652, Smallest: 256 (Step 2)
3. Subtract: 652 – 256 = 396 (Step 3)
4. Largest: 963, Smallest 369 (Step 2)
5. Subtract: 963 – 369 = 594 (Step 3)
6. Largest: 954, Smallest 459 (Step 2)
7. Subtract: 954  - 459 = 495 (Step 3)

Again, doing step 2 will return to number 6, and the result will still be 495.  Observe that the two cases above resulted to 495. Now, will the result be always 495?

Yes, the result will always be 495 as shown in the figure below (try several more numbers to verify). If you want an algebraic proof why this is so, read The Mystery of 495 Explained.

The number 495 is the 3-digit equivalent of the Kaprekar constant 6174. The Kaprekar constant is a ‘black hole’ of numbers when the calculation above is done. There are only two of such constants. For numbers with digits greater than four, the routine may terminate at one of several fixed values or may enter one of several loops instead (Wikiepdia).

The Kaprekar constant is named after Indian mathematician D.R. Kaprekar.

Reference: Wikipedia
Image Credit: Rdhettinger via Wikipedia