Problem:
Let ♣(x) denote the sum of the digits of the positive integer x. For example, ♣(8)=8 and ♣(123)=1+2+3=6. For how many two-digit values of x is ♣(♣(x))=3?
Answer Choices:
A. 3
B. 4
C. 6
D. 9
E. 10
Solution:
Let y=ϱ(x). Since x≤99, we have y≤18. Thus if ℓ(y)=3, then y=3 or y=12. The 3 values of x for which ϱ(x)=3 are 12,21 , and 30 , and the 7 values of x for which ϕ(x)=12 are 39,48,57,66,75,84, and 93 . There are 10​ values in all.
The problems on this page are the property of the MAA's American Mathematics Competitions