Kaprekar's Process for 2-Digit Numbers
Jump to navigation
Jump to search
Theorem
Kaprekar's process, when applied to a non-repdigit $2$-digit positive integer leads to the cycle:
- $09 \to 81 \to 63 \to 27 \to 45 \to 09$
Note that it is important to retain the leading zero on the $9$, or the process trivially terminates in $0$.
Proof
Let $n \in \Z_{>0}$ be a $2$-digit positive integer.
Without loss of generality, let $n$ be expressed in decimal notation as:
- $n = 10 x + y$
where:
- $x > y$
- $0 \le x \le 9, 0 \le y \le 9$
The reversal of $n$ is $10 y + x$.
We have:
\(\ds 10x + y - 10 y - x\) | \(=\) | \(\ds 9 \left({x - y}\right)\) | where $1 \le x - y \le 9$ |
Thus after the first iteration of Kaprekar's process, $n$ is one of:
- $09$
- $18$
- $27$
- $36$
- $45$
or their reversals.
Applying Kaprekar's process to $09$ gives:
\(\ds 09\) | \(=\) | \(\ds 90 - 09\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 81\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds 81 - 18\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 63\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds 63 - 36\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 27\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds 72 - 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 45\) | ||||||||||||
\(\ds \) | \(\to\) | \(\ds 45 - 54\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 09\) |
All those multiples of $9$ can be seen to be in (or end up in) this loop.
Hence the result.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $63$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $63$