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/
ady
my solution
