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A truly original piece of work by Chris Pazsint. Read it with pleasure.
UNDOING THE REVERSE AND ADD PROCESS:
April 2005
Written by Chris Pazsint.

First thing you need to do, if your number begins with a '1' in L1 you need to look at R1. If R1 is not a '1' then the '1' in L1 is the result of the carry operation. Relabel it as carry and adjust the rest of the numbers on the left.

Now you need to go through the rest of the number pairs. The left number is either equal to the right or within one of it:

L=R L+1=R L-1=R

Don't worry about the center number if you have one quite yet.

You are going to need two or more colored pens or pencils for the following steps, if you are doing this on paper. If you are using Excel or another spread sheet program just follow the examples that are shown.

Under all of your numbers list all of the possible ways for two digits to add up to the number. Make sure there is enough room between these lists to add a little more to the right.

You also have to take in count that the original addition could have caused a carry. So in a different color or style list all ways to add up to each digit plus ten.

Now not only could that number cause a carry but you have to count in that the number on the left could have carried on to it. First add '+0' to your current list. Then list with either a third style or the style you used the first time all numbers that add up to the digit less 1. put '+1' to the right of these.

Example: For 5 you would list all the numbers that add up to 4, 0+4+1, 1+3+1 etc .

We have to count in carry on both side so we need to list all numbers that add up to the original +9 (+10 then -1) again adding '+1' to the end.

Example: 5, you would list 14 9+5+1

Now we have a list of all possible ways to get each digit now we need to start eliminating them.

If you have a carry column then you can eliminate all of the possibilities except for 0+0+1.

Remember whenever you remove something from the R column you must also remove it from the L side and vice versa. This is because add reverse is balanced around the center it is not possible to have a digit in L2 that could not be swapped into R2 that ends in the same result.

We will be working from one side of the number to the other working towards the center removing possible addition operations from the list.

Look at R1 there is no way for there to be a carry so you can eliminate all possibilities that have a +1 on the end. Adjust the L1 column as necessary.

Now look at L1 if you have a '1' in the Carry column then L1 must carry so you can eliminate all possibilities that do not. If you do not have a 1 in the Carry column then L1 did not cause a carry so you can eliminate anything that does. Adjust R1.

You should now have only one type of addition problem left in R1 and L1.

So now lets look at L2. We now know weather it caused a carry or not by looking what is left in L1. So eliminate anything that does not fit. Remember you must adjust R2.

Now we apply the same idea to R2 we need to eliminate any possibilities that do not fit with R1.

Continue with this method till you reach the center or you eliminate all possibilities for a number or you end up with a PAIR that no longer has possibilities that match. If you have a center it must fit with both the L and the R lists.

I originally came up with the idea of undoing reverse add when I was thinking about the possibility of using this process for encryption. At this point I don't know if I was successful at proving that it is a useable one way function or not.

Take the number 44044, it breaks up into 20 other numbers 2 of which can be broken up further 14003 and 13013. The number 13013 breaks down into 6 numbers while 14003 breaks into 60! I have yet to do through all 60 of these numbers to see if any break down further. One of the numbers I have tested 7546 breaks down into 30 numbers. That makes at least 118 possible origins for the number 44044.