You have a simple (base-ten, whole number) calculator which can perform only two operations: visually reversing a number, and adding seven.

Prove that you can use this calcluator to convert any number to 1.

*Notation: use ~ to denote reversal, as in*

~53 = 35

Lemma 1 Any digit can be converted into 1 by a chain of only two operations( add7 and reverse).

Proof:

1 =1 0 operations needed

2 4add inv 1add inv 7 operations needed

3 1add inv 2 operations needed

4 8add inv 2add inv 4add inv 1add inv 19 oper,

5 5add inv 8add inv 2add inv 4add inv 1add inv 21 oper

6 2add inv 4add inv 1add inv

7 4add inv 1add inv 2add inv 4add inv 1add inv

17 operations

8 6add inv 5add inv 8add inv 2add inv 4add inv 1add inv 23 oper

9 3add inv 1add inv only 6 operations

Lemma 2, Any n-digit number can be reduced into into another (n-1)-digit number by a chain of only two operations( add7 and reverse).

Proof:

For any number an appropriate multiple of 7 can be added to make the sum divisible by 10.

If the last digit of the original number was 1-

add 7*7,if the last digit of the original number was 2 add 4*7, if 3-1*7 4-5*7 5-5*7 6-2*7 7-9*7

8-6*7,9-3*7.

or:If the last digit of the original number was k-

add t*7 such that k+7*t=0 mod 10.

Applying both lemmas as needed will allow the requested transformation, clearly not optimal as far as the length of the procedure is concerned.

e.g.

4753==>4760==>674==>730==>37==>73==>80==>8==>50==>5==>40==>4==>60==>6==>20==>2==>30==>3==>10==>1

0R:4753==>4760==>674==>730==>37==>100==>1

ady