Sorry, I don't care for that ~ symbol. I'll just use the word "reverse".

 

The following al-gore-ithm will convert any ***one-digit*** number N to 1:

 

If N is 3, add 7 and reverse the digits.

For all other values of one-digit whole number N:

LOOP: 

N=N+7

If a power of 10, less N, is a multiple of 7, add that multiple to N, reverse the digits to convert N to 1, exit.

Reverse the digits of N.

Repeat the above test.

If you can still can't convert N to 1, go back to LOOP.

E,g. for N=7:

7 -- > 14 --> 41 --> 48 --> 1000 (1000=48+[7*136]) --> 1

 

So if we can convert any whole number to a one-digit number, we can convert it to 1.

 

We can convert the rightmost digit of any whole number to 0 by adding the right combination of 7's. If that digit is 1, add 7 seven times. If it is 2, add 7 four times. etc. If the number was originally all 9's, this will add one more digit on the left, with all but two of the digits zero.

 

Once the rightmost digit is converted to zero, reverse the digits.

 

Then repeat the steps.  

 

Ultimately you will convert every digit but the leftmost digit to 0. 

 

Then reverse the digits, and you have a one-digit number. Convert the one-digit number to 1, using the method above.

 

E.g.  999999

 

add 7*3=21

1000020.

 

Reverse the digits.
200001

 

add 7*7=49

200050

 

Reverse the digits.

50002

 

Add 7*4=28

50030

 

Reverse the digits.

3005

 

add 7*5=35

3040

 

Reverse the digits.

403

 

add 7

410

 

Reverse the digits.

14

 

add 7*8=56

70

 

Reverse the digits.

7

 

Now convert 7 to 1.