OT - Squaring sums puzzle

[QuikSand]

Okay, a little more math-heavy than many of our puzzles... but it might be worth some discussion.

- - - - -

In the six rows of numbers below, I have filled in two out of three numbers. You will notice that in each case, the two numbers add up to 25, which happens to be a perfect square (a number whoch can be reached by miltiplying an integer time itself).

Your task is to fill in the thirs number for each row, such that on each row, the sum of any two numbers is a perfect square.

Here are the six number sets:

1, 24, ___
2, 23, ___
3, 22, ___
4, 21, ___
5, 20, ___
6, 19, ___


There is a pattern here -- you might be able to solve it by brute force or trial and error, but catching the basic pattern is probably the best path to the solution. An intuitive explanation of the pattern involved is worth extra credit.

[ice4277]

I tried but the thinky part in my head hurts :(

[EagleFan]

1, 24, 120 squares of 11,12
2, 23, 98 squares of 10,11
3, 22, 78 ... 9,10
4, 21, 60 8,9
5, 20, 44 7,8
6, 19, 30 6,7

[EagleFan]

Either that or 2/3. :D

[Bee]

My money's on 2/3rds...

[albionmoonlight]

What we know:

List= A,B,C

A+B = 25 (or A = 25-B)
A+C =Y^2 (or 25-B + C = Y^2)
B+C = Z^2

So . . .

C = Y^2 + B - 25
And
C = Z^2 - B

Or Z^2 - B = Y^2 + B -25
Z^2 = Y^2 +2B -25

Now, we know that in the first set of numbers, B = 24, so the equation becomes Z^2 =Y^2 +23.

In the second set of numbers, B= 23, so the equation becomes Z^2 = Y^2 +21.

Etc . . .

You can combine this with the fact that the perfect squares are increasing odd numbers apart from each other (i.e., the difference between 1 and 4 is 3; the difference between 4 and 9 is 5; the difference between 9 and 16 is 7; etc.) to find the answers given above.

[QuikSand]

Originally posted by albionmoonlight
You can combine this with the fact that the perfect squares are increasing odd numbers apart from each other (i.e., the difference between 1 and 4 is 3; the difference between 4 and 9 is 5; the difference between 9 and 16 is 7; etc.) to find the answers given above.

Yes - this is the "pattern" that I thought made the puzzle most intuitively understandable.

Well done all around.

(Too easy, grumble grumble...)