#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 b2 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.
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