The Idea Behind the 1089 Math Trick

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.

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