Question 10
In a parlor game, the magician asks one of the participants to think of a three-digit number $(abc),$ where $a,$ $b,$ and $c$ represent base-$10$ digits in the indicated order. The magician then asks this person to form the numbers $(acb),$ $(bca),$ $(bac),$ $(cab),$ and $(cba),$ to add these five numbers, and to reveal their sum $N.$ If told $N,$ the magician can identify the original number $(abc).$ Play the role of the magician and determine $(abc)$ if $N=3194.$