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196 AND OTHER LYCHREL NUMBERS

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This is a break down of "undoing add reversing" process using the number 17,509,097,067 as an example.

First off the number in the first place on the left is a 1. This means that this number is most likely the result of carry.

Rewrite the number listing the individual places like so:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  

We need to list all possibilities for X+Y that would result in the digit in each place. For 1R this would be for X+Y=7.

7+0, 6+1, 5+2, 4+3 All equal 7

We also need to cover if 1R would generate a carry so we need all possibilities for X+Y = 17.

9+8=17

So in Column 1R we have:

1R
7
7+0
6+1
5+2
4+3
9+8

The reason I have 9+8 Bolded is to set it out as resulting in a carry.

In R2 it is a little different we need X+Y=6 and X+Y=16. We also need to account for the possibility that 1R causes a carry so we need X+Y+1=6 and X+Y+1=16:

2R  
6  
6+0 0
5+1 0
4+2 0
3+3 0
9+7 0
8+8 0
5+0 1
4+1 1
3+2 1
6+9 1
7+8 1

The Splitting off of the carry value is to make thing more readable. After this is done for all places you should have something like this:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    

There are 23,579,476,910 possible ways to combine all of these columns. We don't need to go through all of them to find if this number is the result of add reverse. All that is needed is a bit of logic.

Looking at the Carry column in comparison to 1R it is safe to say that the 1 digit is the result of a carry. So first off we eliminate anything in 1L that does not result in a carry giving:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    

Next we have to eliminate 0+1+0 since its carry and anything in1L and 1R that is not listed in both. In this step you can ignore the carry column:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    

So if it was an add reverse it was either 9xxxxxxxx8 or 8xxxxxxxx9

Now lets work on 2R we know that it has to use carry so we can eliminate anything that does not:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    

Now we eliminate anything that does not match up in 2L and 2R:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    

Looking back at 1L we can see that what ever was in 2L resulted did not include a carry so we can eliminate anything from 2L and 2R that does:

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    
 

Ok now continue with 3L and 3R. We know because of what is left in 2L that it didn't carry and from 2R that it didn't need a carry. So what we have left is 0+0.

CARRY   1L   2L   3L   4L   5L   5R   4R   3R   2R   1R  
1   7   5   0   9   0   9   7   0   6   7  
0+1 0 7+0 0 5+0 0 0+0 0 9+0 0 0+0 0 9+0 0 7+0 0 0+0 0 6+0 0 7+0 0
0+0 1 6+1 0 4+1 0 9+1 0 8+1 0 9+1 0 8+1 0 6+1 0 9+1 0 5+1 0 6+1 0
    5+2 0 3+2 0 8+2 0 7+2 0 8+2 0 7+2 0 5+2 0 8+2 0 4+2 0 5+2 0
    4+3 0 6+9 0 7+3 0 6+3 0 7+3 0 6+3 0 4+3 0 7+3 0 3+3 0 4+3 0
    9+8 0 7+8 0 6+4 0 5+4 0 6+4 0 5+4 0 9+8 0 6+4 0 9+7 0 9+8 0
    6+0 1 4+0 1 5+5 0 8+0 1 5+5 0 8+0 1 6+0 1 5+5 0 8+8 0  
    5+1 1 3+1 1 9+0 1 7+1 1 9+0 1 7+1 1 5+1 1 9+0 1 5+0 1  
    4+2 1 2+2 1 8+1 1 6+2 1 8+1 1 6+2 1 4+2 1 8+1 1 4+1 1  
    3+3 1 9+5 1 7+2 1 5+3 1 7+2 1 5+3 1 3+3 1 7+2 1 3+2 1  
    9+7 1 8+6 1 6+3 1 4+4 1 6+3 1 4+4 1 9+7 1 6+3 1 6+9 1  
    8+8 1 7+7 1 5+4 1 9+9 1 5+4 1 9+9 1 8+8 1 5+4 1 7+8 1    

In column 4L and 4R there is 9, 7. After add reverse the only possible combination for an L R pair is X and X or X+1 and X. Now 5L and 5R is an example of something else. It looks like it X=9, so X+1=10 Which would work in locations other then the center but that would only work is X+Y = 19 the largest value for X+Y is 18, 9+9. If that was the case for the center two digits it would be 99 or 98.