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Year and Perfect Square Puzzle (Posted on 2015-08-22) Difficulty: 3 of 5
Find all possible pairs (A,B) of positive integers such that:
A2 + B = 2016 and, A + B is a perfect square.

*** As an extra challenge, solve this puzzle without using a computer program aided method.

No Solution Yet Submitted by K Sengupta    
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Solution my way Comment 2 of 2 |

 My way (D2):

Given   (i)  a^2+b=2016

and      (ii)  a+b=n^2

we get  a*(a-1)= 2016- n^2      (ii)-(i)

since   both  a*(a-1)  and 2016 are  even    n   must be even as well

clearly n is in the 2 to 44 range

Checking for what even n in this range
(2016- n^2)/2 equals a triangular number

We get  (a, b, n)=(32, 992,3 2)  as a solution

Tools used: Calculator and a list of triangular numbers.

 


  Posted by Ady TZIDON on 2015-08-23 02:38:13
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