Cyclic Permutation of Kaprekar Number

Theorem

Let $n$ be a Kaprekar number of $k$ digits.

Let $m$ be an integer formed from a cyclic permutation of the digits of $n$.

Let $m$ be squared and the result split into $2$ parts, where the $2$nd part is of $k$ digits.

Let these two parts be added, in the way of operating on a Kaprekar number.

If the result is more than $k$ digits long, split that into $2$ parts, where the $2$nd part is of $k$ digits, and add the parts.

The result will be another cyclic permutation of the digits of $n$.

Proof

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Examples

972

$972$ is a cyclic permutation of the $3$-digit Kaprekar number $297$.

Thus we have:

and it is seen that $729$ is another cyclic permutation of $297$.

$\blacksquare$

2727

$2727$ is a cyclic permutation of the $4$-digit Kaprekar number $7272$.

Thus we have:

and it is seen that $7272$ is a (trivial) cyclic permutation of $7272$.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $297$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $297$