Question

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 $N=3194.$

Answer

Answer: 358

Concepts:digits place value system of equations

Difficulty rating: 1830

Small Hint:

First include the original number and sum all six permutations

Big Hint:

If $s=a+b+c,$ express the original number in terms of $s$ and $N$

Solution:

Across all six permutations, each digit occurs twice in each place, so their total is $222(a+b+c).$ Put $s=a+b+c$ and let the original number be $M.$ Since the other five sum to $3194,$ $M=222s-3194.$ Because $100\leq M\leq999,$ we need $15\leq s\leq18.$ Testing these four values gives $M=136,$ $M=358,$ $M=580,$ and $M=802,$ respectively. Only $358$ has digit sum equal to its assumed value, namely $16.$ Therefore the original number is $358.$

1986 AIME Problem 10

Answer: 358

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 10 in Other Years