Kaprekar's Constant and the Road to 6174

Definition Kaprekar’s constant is the number $6174$, named after the Indian mathematician D. R. Kaprekar, who devised the routine that leads to it in 1949. The routine works in base ten with exactly four digit places. Write the number with four digits, adding leading zeros if needed, so that $5$ becomes $0005$. Arrange the four digits in decreasing order to get the largest number they can make and in increasing order to get the smallest, and subtract the smallest from the largest. Write the result with four digits and repeat. The theorem says that every start from $0000$ to $9999$ whose four digits are not all equal reaches $6174$ after at most seven subtractions, and that from then on every subtraction gives $6174$ again. The ten starts with four equal digits, such as $7777$, are excluded, because both arrangements are the same number and the routine falls to $0000$ and stays there. The theorem is proved by checking all $9990$ allowed starts, which is possible because there are only finitely many. Seven is the longest route, and most starts need fewer rounds.

Definition Kaprekar’s routine on four digits is the rule $K$ that sends a number $n$ from $0$ to $9999$, written with four digit places, to the largest arrangement of its four digits minus the smallest arrangement, again written with four places. Suppose the digits of $n$ in decreasing order are $a \ge b \ge c \ge d$. The largest arrangement has $a$ in the thousands place and $d$ in the units place, the smallest has them the other way round, and $b$ and $c$ trade the hundreds and tens places. So

The result depends only on the outer gap $a - d$ and the inner gap $b - c$. When the digits are not all equal, the outer gap is a whole number from $1$ to $9$ and the inner gap is a whole number from $0$ up to the outer gap, which gives $54$ possible pairs of gaps. Different pairs give different results, so after one round every allowed start has become one of just $54$ numbers.

Definition A fixed point of a rule $f$, which sends each allowed input to an output, is an input $x$ with $f(x) = x$. When a value is fed back into the rule over and over, reaching a fixed point means that every later round returns the same value. A fixed point does not by itself say which starts lead to it. That has to be shown separately, and some repeated rules never settle at a fixed point and run around a loop instead.

Definition A leading zero is a $0$ written in front of the first nonzero digit of a number so that the number fills a fixed number of digit places. It does not change the value, since $0999$ and $999$ are the same number, but it does change the list of digits, and that list is what Kaprekar’s routine sorts. The routine always uses four places, so $999$ counts as the digits $0$, $9$, $9$ and $9$, and $5$ counts as $0$, $0$, $0$ and $5$. A zero digit sorts like any other digit, so it stands at the front of the smallest arrangement, as $3087$ gives $0378$. The gap formula $999(a - d) + 90(b - c)$ is less than $1000$ only when $a - d = 1$ and $b - c = 0$, so for an allowed start the only result that begins with a zero is $0999$.