Permalink Submitted by Anonymous on April 23, 2016

Firstly, not all numbers have reversible components, e.g. 125.
Secondly, this solution applies only when the components have three digits, although the number itself could be as high as 1998 (= 999 + 999).
Let m be the number you want to decompose into its reversible components.
Let x and y be the components, then
x + y = m
The difference between two, three-digit numbers is evenly divisible by 99. So,
x – y = 99k
where k is some integer 0 … 9. It is the difference between the first and last digit of the larger component, as described in the above article.
Now,
y = m – x
x = 99k + y
On substituting m – x for y in the second equation and then adding x to both sides gives
2x = 99k + m
Or
x = (99k + m) / 2
The solution can be found by trying zero and even numbered values of k when m is even and odd numbered values m is odd.
There can be more than one solution to the problem. 685, for example has 3 solutions, namely when k = 1, 3 and 5. This means 685 has four reversible components: 392, 491, 590 and their respective reverses.

## Decomposing the sum of a reversible pair

Firstly, not all numbers have reversible components, e.g. 125.

Secondly, this solution applies only when the components have three digits, although the number itself could be as high as 1998 (= 999 + 999).

Let m be the number you want to decompose into its reversible components.

Let x and y be the components, then

x + y = m

The difference between two, three-digit numbers is evenly divisible by 99. So,

x – y = 99k

where k is some integer 0 … 9. It is the difference between the first and last digit of the larger component, as described in the above article.

Now,

y = m – x

x = 99k + y

On substituting m – x for y in the second equation and then adding x to both sides gives

2x = 99k + m

Or

x = (99k + m) / 2

The solution can be found by trying zero and even numbered values of k when m is even and odd numbered values m is odd.

There can be more than one solution to the problem. 685, for example has 3 solutions, namely when k = 1, 3 and 5. This means 685 has four reversible components: 392, 491, 590 and their respective reverses.

Michael Greaney