Frequently Asked Questions About Kaprekar’s Constant
Short answers to the questions people ask most often about Kaprekar’s constant, 6174, and the routine that produces it. For longer explanations, follow the links to the deeper pages.
What is Kaprekar’s constant?
Kaprekar’s constant is the number 6174. For any four-digit number with at least two different digits, repeatedly rearranging the digits in descending order, then ascending order, and subtracting the smaller from the larger will always reach 6174 in at most seven iterations. Once you arrive, the routine produces 6174 forever.
See how the routine works for the full step-by-step explanation.
Who discovered Kaprekar’s constant?
The Indian mathematician Dattatreya Ramchandra Kaprekar (1905–1986) discovered the property in 1949 and published it in 1955 in the journal Scripta Mathematica. He was a schoolteacher in Devlali, India, and worked on recreational mathematics largely outside the academic mainstream.
Read more on the history page.
Why doesn’t the routine work for 1111 or 2222?
Numbers with all four digits the same are called repdigits. For a repdigit, the descending and ascending arrangements are identical, so subtracting them gives 0000 immediately. The routine collapses. This is why Kaprekar’s constant requires a starting number with at least two different digits.
Is there a 3-digit version of Kaprekar’s constant?
Yes. For three-digit numbers with at least two different digits, Kaprekar’s routine always reaches 495 in at most six iterations. Try the 495 solver.
What is the maximum number of steps needed to reach 6174?
Seven iterations. No matter which four-digit starting number you choose (provided it has at least two different digits), you will reach 6174 within seven steps. The starting number 9831 is one example that requires the full seven; see the examples page for a full walkthrough.
Does Kaprekar’s routine work in other number bases?
Yes. Analogous fixed points exist in other bases. For example, in base 4, numbers of the form 3021 serve as fixed points of the Kaprekar map. The behaviour varies by base and by digit length — sometimes a single constant, sometimes a cycle of values.
Is there a 5-digit Kaprekar constant?
No. The five-digit case of Kaprekar’s routine does not converge to a single number. Instead it enters cycles. It has been formally proven that only the three-digit (495) and four-digit (6174) cases produce a single non-trivial Kaprekar constant. Six, eight, and nine-digit numbers have two constants each. Most other lengths produce cycles rather than fixed points.
Why is every result in Kaprekar’s routine a multiple of 9?
Rearranging a number’s digits does not change its digit sum, and two numbers with the same digit sum leave the same remainder when divided by 9. Their difference is therefore always a multiple of 9. This holds at every step of the routine. Read more on the mathematics page.
What is the connection between 495 and 6174?
Both are Kaprekar constants — 495 for three-digit numbers, 6174 for four-digit numbers. They are the only two non-trivial single fixed points of Kaprekar’s routine across all digit lengths. They behave the same way: every qualifying starting number converges to them within a small number of iterations.
Is Kaprekar’s constant useful for anything practical?
Not directly. Kaprekar’s routine is a classic example of a simple iterative process with surprising convergence behaviour, and it is often used in teaching recursive functions, fixed-point analysis, and basic computational mathematics. Its value is pedagogical and recreational rather than applied.
Why does 6174 reproduce itself under the routine?
Because 7641 − 1467 = 6174. The descending arrangement of 6174’s digits is 7641; the ascending arrangement is 1467; and their difference is 6174 itself. This makes 6174 a fixed point of the Kaprekar map — a value that maps to itself. Once the routine reaches a fixed point, it stays there.
Can I use this tool in teaching?
Yes, freely. The interactive solver and the three-digit 495 solver are designed to be used in classrooms and demonstrations. Attribution is appreciated but not required.
Still have questions?
The full explanation is on How It Works, with worked examples and the full mathematical treatment.