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

Consider the following table:

10^n ( 10^n )mod7

10 3

100 2

1000 6

10000 4 100000 5

1000000 1

10000000 3 etc

Clearly for any n-digit number we can add 7 for so many times to get O at the end.

Reversing the n-digit number ending with a zero- we get an (n-1)digit number.

Repeating this routine n-1 times we get a one digit number say m . m is not a zero.

For m=1 STOP, WE'RE DONE.

For m=3 add 7 1 time, inverse - you get ONE.

For m=5 add 7 5 timeS ,inverse - you get FOUR.

For m=4 add 7 8 timeS ,inverse - you get SIX .

For m=6 add 7 2 timeS ,inverse - you get TWO.

For m=2 add 7 14 timeS ,inverse - you get ONE.

For m=8 add 7 6 timeS ,inverse - you get FIVE.

SPECIAL CASE: For m=7 add 7 - 1 time only ,inverse - you get 41 THEN add 7 7 timeS and inverse- you get NINE,

For m=9 add 7 3 timeS ,inverse - you get THREE,

following the links we always get ONE

.

q.e.d/

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