Problem:
An ordered pair of non-negative integers is called "simple" if the addition in base requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to .
Solution:
Since there is no carrying in the addition, the ones column must add to , the tens column to , the hundreds column to and the thousands column to for each simple ordered pair of non-negative integers summing to . To get a single digit as the sum of two digits, there are ways:
Thus the number of simple ordered pairs of non-negative integers that sum to is .
The problems on this page are the property of the MAA's American Mathematics Competitions