Problem:
In a parlor game the "magician" asks one of the participants to think of a three-digit number , where and represent digits in base in the order indicated. Then the magician asks this person to form the numbers and , to add these five numbers, and to reveal their sum, . If told the value of , the magician can identify the original number, . Play the role of the magician and determine if .
Solution:
Adding (abc) to , and observing that each of the digits and appears exactly twice in each column, we are led to the equation
In view of this, the problem can be resolved by searching for an integral multiple of , say , which is larger than (since ) and less than (since is a three-digit number), so that is satisfied. That is, the sum of the digits of must be . If this holds, then is the answer to the problem.
the search is limited to and . Of these only works. Specifically, and .
The problems on this page are the property of the MAA's American Mathematics Competitions