Kaprekar's Routine: A Curious Number Game

Have you ever played with numbers and found a surprising pattern? One such fascinating pattern is hidden within Kaprekar's Routine, named after the Indian mathematician D.R. Kaprekar. It's a simple game that, for most four-digit numbers, always leads to the same result: 6174.

Let's dive in and see how it works!

The Rules of the Game

1. Pick a four-digit number with at least two different digits. (Numbers like 1111, 2222, etc., won't work).
2. Arrange the digits to form the largest possible number.
3. Arrange the same digits to form the smallest possible number.
4. Subtract the smallest number from the largest number.
5. Repeat steps 2-4 with the new number you get.

You'll be amazed at what happens!

Example 1: Starting with 3524

Let's try with the number 3524:

• Step 1: Our number is 3524. It has different digits.
• Step 2: Largest number: 5432
• Step 3: Smallest number: 2345
• Step 4: Subtract: 5432 - 2345 = 3087

Now, we repeat the process with 3087:

• Step 2: Largest number: 8730
• Step 3: Smallest number: 0378 (remember to include leading zeros to keep it a four-digit number)
• Step 4: Subtract: 8730 - 0378 = 8352

Repeat with 8352:

• Step 2: Largest number: 8532
• Step 3: Smallest number: 2358
• Step 4: Subtract: 8532 - 2358 = 6174

And there it is! We reached 6174.

Example 2: Starting with 1987

Let's try another one with 1987:

• Step 1: Our number is 1987.
• Step 2: Largest number: 9871
• Step 3: Smallest number: 1789
• Step 4: Subtract: 9871 - 1789 = 8082

Repeat with 8082:

• Step 2: Largest number: 8820
• Step 3: Smallest number: 0288
• Step 4: Subtract: 8820 - 0288 = 8532

Repeat with 8532:

• Step 2: Largest number: 8532
• Step 3: Smallest number: 2358
• Step 4: Subtract: 8532 - 2358 = 6174

Again, we arrived at 6174!

The Magic of 6174

This number, 6174, is known as Kaprekar's Constant. For almost any four-digit number (with at least two different digits), if you keep applying Kaprekar's routine, you will eventually reach 6174. Once you reach 6174, the next step will always be:

• Largest: 7641
• Smallest: 1467
• Subtract: 7641 - 1467 = 6174

It's a loop!

Kaprekar's routine is a wonderful example of how simple arithmetic operations can lead to unexpected and beautiful mathematical constants. Try it with your own four-digit numbers and see the magic unfold!