The 1089 trick and why it works

The trick step by step

1. Choose 723 (first and last digits differ by 4).
2. Reverse it: 327.
3. Subtract: 723 − 327 = 396.
4. Reverse the result: 693.
5. Add: 396 + 693 = 1089.

Try any other valid start, for example 841: 841 − 148 = 693, and 693 + 396 = 1089.

Why the answer is always 1089

Write the number as abc with a > c. Its reverse is cba, and the difference is:

The middle digit cancels. The difference is always a multiple of 99: 198, 297, 396, 495, 594, 693, 792 or 891. Every one of these has 9 in the middle, and its first and last digits add to 9. Adding such a number to its reverse gives 100 × 9 + 10 × 18 + 9:

If the first and last digits differ by exactly 1, the difference is 099. Keep the leading zero, reverse to 990, and the sum is still 1089. Dropping the zero gives 99 + 99 = 198, which is why magicians ask for a difference of at least 2.

The link with 495

In Kaprekar’s three-digit routine, the digits are sorted before subtracting, so the first result is also 99 × (largest digit − smallest digit). That is why the first step of the 495 calculator always produces one of 099, 198, …, 891, the same list as in the 1089 trick. Kaprekar keeps sorting and subtracting and converges to 495; the trick instead reverses and adds, landing on 1089.

Following the arithmetic with borrowing

It helps to see why the middle digit of the difference is always 9. In 723 − 327 the units column needs a borrow, because 3 is smaller than 7. That borrow leaves the tens column computing 1 − 2, which needs another borrow and becomes 11 − 2 = 9. Whenever the first digit is larger than the last, this double borrow happens, so the tens digit of the difference is always 9.

The hundreds and units then carry the rest of the difference: their digits add up to 9. When you add the difference to its reverse, the units give 9, the tens give 18, and the hundreds give 9 plus the carried 1, which produces 1089.

Using it in the classroom

The trick is an excellent starting point for algebra. Students first enjoy the surprise, then try to break it with unusual inputs, and finally write abc as 100a + 10b + c to explain what they have seen. The same algebra leads straight into Kaprekar’s three-digit routine and to why every number reaches 6174. A full lesson plan for the four-digit routine is in teaching 6174.

A few more curiosities

• 1089 × 9 = 9801, which is 1089 reversed.
• 1089 = 33², and its digital root is 9, as for every result above. See digital roots.
• With four digits the trick no longer gives a single answer: depending on the digits, the result is 990, 9999, 10890 or 10989.