In the previous post, we have learned about the **1089 math trick**. In this post, we are going to learn why the math trick works. Recall that the math trick has the following steps:

1.) Write a 3-digit number where the digits are decreasing.

2.) Reverse the digits of the number.

3.) Subtract the result from (2) from the original number in (1).

A 3-digit number, write the digits abc has value 100a + 10b + c. For example, the number with digits 347 has value 100(3) + 10(4) + 7.

In Step 1, we write the number 100a + 10b + c where a >= b and b >= c.

In Step 2, if we reverse the digits, it will become 100c + 10b + a.

In Step 3, we subtract the numbers in Step 2 from Step 1:

(100a + 10b + c) – (100c + 10b + a) = 99a – 99c = 99(a – c)

Since a > c, a – c is at least 2 and at most 9. So, the possible values of 99(a – c) are

99(2) = 198

99(3) = 297

99(4) = 396

99(5) = 495

99(6) = 594

99(7) = 693.

These are the only possible values of 99(a – c) and all of these numbers added to itself equals 1089 a shown below.

198 + 891 = 1089

297 + 792 = 1089

396 + 693 = 1089

495 + 594 = 1089

594 + 495 = 1089

693 + 369 = 1089.