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Adjacent Square Sum II (Posted on 2013-06-10) Difficulty: 2 of 5
In the six rows of numbers below, each of the pairs adds up to 25. Now 25 happens to be a perfect square.

Fill in the blanks with a third number (a different number in each row) so that the sums of any two numbers on any row is a perfect square.
+---+---+---+
| 1 |24 |   |
+---+---+---+
| 2 |23 |   |
+---+---+---+
| 3 |22 |   |
+---+---+---+
| 4 |21 |   |
+---+---+---+
| 5 |20 |   |
+---+---+---+
| 6 |19 |   | 
+---+---+---+

No Solution Yet Submitted by K Sengupta    
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Comments: ( Back to comment list | You must be logged in to post comments.)
More generalizations of the algebra | Comment 4 of 5 |
(In reply to The algebra by Jer)

It's not all Pythagorean triple, just the ones like (5,12,13) and (7,40,41)...(2a+1, 2a+2a, 2a+2a+1)

For pairs that add to (2a+1), call them n and (2a+1)-n
you can add the quantity n+n(-4a-4a-1)+(4a^4+8a+4a)
which yields the squares
n+n(-4a-4a)+(4a^4+8a+4a) = (n-(2a+2a))
and
n+n(-4a-4a-2)+(4a^4+8a+8a+4a+1) = (n-(2a+2a+1))
  Posted by Jer on 2013-06-10 15:28:26

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