A Number Trick

Select a three digit number in which the first and the last digit differ by at least two. Construct a second number by reversing the order of digits in the first. Form a third number by taking the difference of the first two. Reverse the order of digits in the third number to construct a fourth, and add the third and fourth number. The result is the number 1089.

Example: Select the number 732. Note: 7 - 2 = 5 > 2, making this number a valid choice. Next, construct the number 237 by reversing the order of digits. Take the difference:

From 495, construct the number 594, and add:

Proof: Let the originally selected number be represented by

where a, b, c, are integers between 0 and 9 inclusive, and occupy the hundreds, tens, and units places, respectively. We may assume, without loss of generality, that a > c. Now, construct a new number by reversing the order of digits

Take the difference

and simplify to

99(a - c) = 99

4.

where = a - c. We would like to represent this number, algebraically, as a number in base ten; i.e., with a form similar to that shown in 1. We begin by rewriting 99 as

99 = (10 . 9) + (1 . 9)

5.

Since 2 9 by hypothesis, we know that 18 9 81 so that we may write

9 = 10 + v

6.

where and v are integers between 0 and 9 inclusive, and occupy the tens and units places, respectively. Thus, the Right Hand Side of eq. 5 may be rewritten as

(10 ^. 9) + (1 ^. 9) = [10 ^. (10 + v)] + [1 ^. (10 + v)]

= 100 + 10( + v) + v

7.

If the integer + v does not exceed a single digit, then the number,

100 + 10( + v) + v

is the number sought. Let us find all possible values of + v for 2 9:

	9	+ v
2	18	9
3	27	9
4	36	9
5	45	9
6	54	9
7	63	9
8	72	9
9	81	9

Thus, + v does not exceed 9 (in fact, it equals 9 in all cases). We conclude that the number, 100 + 10( + v) + v, is the one we sought, and correctly represents the difference, 99, as a number in base ten, with occupying the hundreds place; ( + v), the tens place; and v, the units place. The number obtained by reversing the order of its digits is then

100v + 10( + v) +

8.

and the sum of these last two numbers is

100( + v) + 20( + v) + ( + v) = 121( + v)

9.

But, for all values of a considered, ( + v)= 9, so that 121( + v) = 1089.