I just got this book of number brain teasers and this one is sort of interesting and giving me some difficulty.
Ask Joe to think of any positive integer. Tell Joe to scramble the digits of this integer to obtain another number and then subtract the smaller of the two from the larger. If the difference consists of at least two digits ask Joe to tell you all but one of these digits including all zeros, you can provide the missing digit! Explain.
So I believe I've made some progress on this problem because I know that any integer can be decomposed as follows:
$n = a_{0} + a_{1}*10 + a_{2}*100 \space + \space ... \space + \space a_{n}*10^n$
$n = 9*(a_{1} + 11*a_{2} + 111*a_{3} + 111…1*a{n}) \space + \space (a_{0} + a_{1} + \space … \space + \space a_{n})$
So we could let $a_{n} = s_{n} - t_{n}$ where $s$ was the larger integer. So that's my thoughts on the problem so far, but I know I'm far from a solution. Any help on understanding this problem would be appreciated!