#math trivia for #August25: More counting: #237 has the form abc where b^2 = a+c. How many three-digit numbers are like this?

— Burt Kaliski Jr. (@modulomathy) August 26, 2013

Answer: 16. Similar to the solution to #236, there are at least two ways to count:

- By
*a*‘s, seeing how many*c*‘s yield a value of*a*+*c*that is square:- (
*a*must be at least 1) *a*= 1:*c*= 0, 3, 8 (3 answers)*a*= 2:*c*= 2, 7 (2 answers)*a*= 3:*c*= 1, 6 (2 answers)*a*= 4:*c*= 0, 5 (2 answers)*a*= 5:*c*= 4 (1 answer)*a*= 6:*c*= 3 (1 answer)*a*= 7:*c*= 2 (1 answer)*a*= 8:*c*= 1, 8 (2 answers)*a*= 9:*c*= 0, 7 (2 answers)

- (
- By
*b*‘s, seeing how many ways there are to represent*b*^{2}as a sum of digits:*b*= 0: none (a must be at least 1)*b*= 1: (*a*,*c*) = (1, 0) (1 answer)*b*= 2: (*a*,*c*) = (1, 3), (2, 2), (3, 1) (3 answers)*b*= 3: (*a*,*c*) = (1, 8), (2, 7), …, (9, 0) (9 answers)*b*= 4: (*a*,*c*) = (7, 9), (8, 8), (9, 7) (3 answers)

Either way, there are 16 answers.

For the reader: Show the result by counting a third way.