Explore Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got.

Question

Homi · Accepted Answer

Understand the Question

Step 1 · First Iteration Starting with 1234

Let the starting $4$ -digit number be $1234$ . Diagram 1

Largest number from digits: $A = 4321$

Smallest number from digits: $B = 1234$

Subtract $B$ from $A$ :

\begin{aligned} C &= A - B \\ &= 4321 - 1234 \\ &= 3087 \end{aligned}

Step 2 · Second Iteration with 3087

Using the digits of $3087$ :

Largest number: $A = 8730$

Smallest number: $B = 0378$

Subtract $B$ from $A$ :

\begin{aligned} C &= A - B \\ &= 8730 - 0378 \\ &= 8352 \end{aligned}

Step 3 · Third Iteration with 8352

Using the digits of $8352$ :

Largest number: $A = 8532$

Smallest number: $B = 2358$

Subtract $B$ from $A$ :

\begin{aligned} C &= A - B \\ &= 8532 - 2358 \\ &= 6174 \end{aligned}

Step 4 · Fourth Iteration with 6174

Using the digits of $6174$ :

Largest number: $A = 7641$

Smallest number: $B = 1467$

Subtract $B$ from $A$ :

\begin{aligned} C &= A - B \\ &= 7641 - 1467 \\ &= 6174 \end{aligned}

Repeating the process on $6174$ will always yield $6174$ .

Answer

Carrying out these steps always reaches $6174$ (the Kaprekar constant), which then repeats indefinitely.

Common Mistakes

Omitting Leading Zeros: Forgetting to use leading zero(s) when arranging digits to form the smallest $4$ -digit number (e.g. for digits $3, 0, 8, 7$ , the smallest $4$ -digit combination is $0378$ , not $3078$ ).
Numbers with All Identical Digits: Trying this routine with numbers like $1111$ or $2222$ results in $0000$ . The routine requires at least two distinct digits.

Question 3