SOLUTION: If I add 14 to my house number, the sum is a perfect square. If I subtract 14 from my house number, the sum is also a perfect square. What is my house number?

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Question 1142356: If I add 14 to my house number, the sum is a perfect square. If I subtract 14 from my
house number, the sum is also a perfect square. What is my house number?
Found 2 solutions by math_helper, greenestamps:
Answer by math_helper(2461) (Show Source):
You can put this solution on YOUR website!

house number = 50
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the work out:

Subtracting:
n>m; n,m integers
28 is 1*28, 2*14, or 4*7
Case 1:
n-m = 1
n+m = 28
(adding): 2n = 29 non-integer solution, no good
Case 2:
n-m = 2
n+m = 14
2n = 16 --> n=8 --> m=6 this solution works
Case 3:
n-m = 4
n+m = 7
2n = 11 non-integer solution, no good
Using case 2, and :
--> x = 50
Check using :
(ok)

Answer by greenestamps(13198) (Show Source):
You can put this solution on YOUR website!

Here is an unorthodox solution method that is fast if you understand it....

The house number is 14 less than one perfect square and 14 more than another, so it is halfway between two perfect squares whose difference is 28.

Now use the fact that the difference between consecutive perfect squares n^2 and (n+1)^2 is 2n+1. That difference is clearly always odd. So the two perfect squares in this problem are not consecutive perfect squares.

So what about squares of numbers that differ by 2? 28 = 13+15. 13 = 2(6)+1 is the difference between 6^2 and 7^2, and 15 = 2(7)+1 is the difference between 7^2 and 8^2. And so 28 is the difference between 6^2 and 8^2.

So the two perfect squares in the problem are 6^2=36 and 8^2 = 64; the house number is halfway between 36 and 64, so it is 50.