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

(In reply to my solution revisited and revised by Ady TZIDON)

You claim that any n-digit number (n>1) can be converted to a (n-1)-digit number by adding a multiple of 7 to make it divisible by 10 and then reversing. This is not quite true.

Consider the number 99999 (five digits) adding 21 does produce a multiple of 10, namely 100020 -- a six digit number. Reversing produces 20001, a five-digit number. Fortunately, performing the operations a second time does reduce the digits to four.

For any number of digits there are numbers that do not reduce performing the sequence once. As long as the number of digits is greater than two, however, they all do reduce on the second pass.

Two digit numbers are a tougher breed, however. In the higher numbers the "glitch" only occurs when all the digits before the last two are 9's. The ten's digit, however, can be as low as 4 in some cases. And this affects two-digit numbers drastically.

This is where I bogged down. Other than individually evaluating each of the 98 numbers 2-99, I can't think of how to prove they will all reduce.
Edited on February 22, 2004, 11:55 pm

Posted by TomM on 2004-02-22 23:52:00