Explore Carry out these same steps with a few 3-digit numbers. What number will start repeating?

Question

Homi · Accepted Answer

Understand the Question

To perform the routine on any 3-digit number (with not all digits the same):
1. Arrange the digits in descending order to form the largest number ( $A$ ).
2. Arrange the digits in ascending order to form the smallest number ( $B$ ).
3. Subtract the smallest from the largest ( $C = A - B$ ).
4. Repeat the process with the digits of the resulting difference.
Continuing this process always leads to a single repeating number known as Kaprekar's constant for 3-digit numbers.

Step 1 · First Iteration

Choose the 3-digit number 123 with digits 1, 2, and 3. Diagram 1

Form the largest and smallest numbers $A = 321$ $B = 123$

Subtract $B$ from $A$

\begin{aligned} C &= A - B \\ &= 321 - 123 \\ &= 198 \end{aligned}

Step 2 · Second Iteration

Using the digits of 198 (1, 9, and 8) $A = 981$ $B = 189$

Subtract $B$ from $A$

\begin{aligned} C &= A - B \\ &= 981 - 189 \\ &= 792 \end{aligned}

Step 3 · Third Iteration

Using the digits of 792 (7, 9, and 2) $A = 972$ $B = 279$

Subtract $B$ from $A$

\begin{aligned} C &= A - B \\ &= 972 - 279 \\ &= 693 \end{aligned}

Step 4 · Fourth Iteration

Using the digits of 693 (6, 9, and 3) $A = 963$ $B = 369$

Subtract $B$ from $A$

\begin{aligned} C &= A - B \\ &= 963 - 369 \\ &= 594 \end{aligned}

Step 5 · Fifth Iteration

Using the digits of 594 (5, 9, and 4) $A = 954$ $B = 459$

Subtract $B$ from $A$

\begin{aligned} C &= A - B \\ &= 954 - 459 \\ &= 495 \end{aligned}

Step 6 · Sixth Iteration

Using the digits of 495 (4, 9, and 5) $A = 954$ $B = 459$

Subtract $B$ from $A$

\begin{aligned} C &= A - B \\ &= 954 - 459 \\ &= 495 \end{aligned}

The operation now continuously produces 495.

Answer

495

Common Mistakes

Identical Digits Exception: If all three digits are the same (like $111$ or $222$ ), the difference is $000$ , so the rule only applies to numbers with at least two distinct digits.
Leading Zeroes: When a difference results in a 2-digit number (e.g., $099$ ), make sure to include the leading zero as one of the three digits in the next step.

Question 4