The 1089 Phenomenon

Take any three-digit number in which the first and last digits are different.  Reverse the digits to get a new number, so now you have two numbers, and each one is the reverse of the other.  Subtract the smaller from the larger to get a third three-digit number.  (If subtracting gives you a two digit number, then please treat it as a three-digit number whose first digit is zero.)  Now add this number to its reverse.  The result will be 1089.

Why?

Example:

571-175 = 396,
396+693 = 1089

Here's why it works. Let the digits of the original number be ABC, and assume without loss of generality that A is larger than C. Now consider the subtraction,

 ABC
-CBA
----
 DEF

When you do the subtraction, you find that you can't take A from C because A is larger than C, so you need to "borrow".  So the ones digit, F, is 10+C-A.

Since you borrowed from the 10's column, and you have B minus B in that column, so you have to borrow again. The result must be 9 in the 10's column, so E=9.

Since you borrowed from the 100's column, the result in that column must be A-C-1

It's interesting to observe that D + F = 9.  Do you see why?  D=A-C-1, and F=10+C-A, so D+F = A-C-1+10+C-A = 9.  This will come up later, so it's good to be clear about this point.

Now, let's go on to the addition of DEF to FED.  Did you remember that E is 9?  I'll just write it that way:

 D9F
-F9D
----
1XYZ

Did you also remember that F+D=9?  So Z must be 9, too.

 D9F
-F9D
----
1XY

And there's no carry out of the one's place, so Y must be 8, but then there's a carry into the 100's place.  Here's what we have so far.

 1
 D9F
-F9D
----
1X89

I wrote a little "1" over the hundreds column to show the carry out of the tens place.  We'll use, again, the fact that D+F=9, and with the carry, the result becomes 10, so the final result is

 1
 D9F
-F9D
----


Hopefully you're convinced that the answer is always 1089, no matter what ABC we start with.  In case you have lingering doubts, or if you would like more ammunition to show someone else, let's do this again, algebraically. The number, ABC, is equal to 100A+10B+C.  Therefore the difference ABC-CBA can be represented this way:

ABC-CBA = 100A+10B+C - (100C+10B+A)
= 100(A-C) + (C-A)
= 100(A-C-1) + 100 + (C-A)
= 100(A-C-1) + 90 + (10-(A-C))
= DEF, where D=A-C-1, E=9, and F=10+C-A

Note that D+F = A-C-1+10+C-A = 9. This fact will help us when we add DEF to its reverse. I'll show it to you algebraically:

DEF+FED = 100D + 10E + F + (100F+10E+D)
= 100(D+F) + 20E + (D+F)
= 100*9 + 20*9 + 9
= 1089

Internet References

Amazon.com: 1089 and all that, by David Acheson

Related pages in this website

The webmaster and author of the Math Help site is Graeme McRae.
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