2, 4, 6, 8, never too late…6, 1, 7, 4, why, oh why?

You may have encountered this numerical magic trick at a party or in a classroom. It starts simply: think of a four-digit number where at least two digits are different. Rearrange those digits to create the largest possible number, then rearrange again to make the smallest number, and subtract the latter from the former. Write down the answer and repeat the process with the result, and keep going.

After several turns, you will hit 6174. If you were performing this as a trick, you might reveal this number from a sealed envelope prepared before your volunteer even spoke, guaranteeing a round of applause.

No matter which valid four-digit number you choose, the process, known as Kaprekar’s Routine, always homes in on 6174. Once it arrives, it stays there forever. If you carry out the routine on 6174 itself (7641 – 1467 = 6174), you get 6174 again. It is a mathematical fixed point. This short video shows the process in action.

It feels like numerical sleight of hand, but the routine hides a rigid structure. The first clue to its predictability lies in what the process ignores. Kaprekar’s routine discards the original order of your digits immediately.

Whether you begin with 7263, 3276, or 2763, the routine sees them all as the same set of digits. They all become 7632 – 2367 = 5265. From there, the path is set:

• 5265 becomes 6552 – 2556 = 3996

• 3996 becomes 9963 – 3699 = 6264

• 6264 becomes 6642 – 2466 = 4176

• 4176 becomes 7641 – 1467 = 6174.

The routine effectively forgets the starting number and remembers only the relationship between its digits. This is why any number made of 1, 4, 6, and 7 reaches the finish line in a single step.

Think of Kaprekar’s routine as a landscape. Since there are a finite number of ways to arrange four digits, the process cannot wander endlessly. Instead, it behaves like a marble rolling down a funnel. No matter what point you release the marble from the top of the funnel, it’s always going to fall out of the hole at the bottom, mathematically the hole at the bottom of the funnel is an attractor. In mathematics, 6174 is an attractor.

The only way to escape is to choose a number with identical digits (like 7777), which results in zero, a different kind of dead end, like forgetting about the funnel altogether and simply dropping the marble on to the floor.

For three-digit numbers, the attractor is 495. Pick any three-digit number where the digits are different, and they will always end up at 495. 736? This one might be more appropriate for a mathematical trick for children.

763 – 367 = 396

963 – 369 = 594

954 – 459 = 495