# Making numbers add up to square numbers

• Nov 18th 2007, 09:35 AM
Markdevs
Making numbers add up to square numbers
Hi everyone,

I'm stuck on this question and it's driving me crazy!!

Basically you have to add numbers in a line of a triangle that will add up to a square number.

e.g. a
b c

a&b have to add up to a square, as well as a&c and b&c.

Any help would be appreciated.

Can anyone help?
• Nov 18th 2007, 09:36 AM
Jhevon
Quote:

Originally Posted by Markdevs
Hi everyone,

I'm stuck on this question and it's driving me crazy!!

Basically you have to add numbers in a triangle that will add up to a square number.

Can anyone help?

you may want to be a little more specific than that
• Nov 18th 2007, 09:56 AM
topsquark
Quote:

Originally Posted by Markdevs
Hi everyone,

I'm stuck on this question and it's driving me crazy!!

Basically you have to add numbers in a line of a triangle that will add up to a square number.

e.g. a
b c

a&b have to add up to a square, as well as a&c and b&c.

Any help would be appreciated.

Can anyone help?

Do you mean like
Code:

  -1 2    2
Or is there some restriction on the numbers a, b, c?

-Dan
• Nov 18th 2007, 10:17 AM
topsquark
Quote:

Originally Posted by Markdevs
Hi everyone,

I'm stuck on this question and it's driving me crazy!!

Basically you have to add numbers in a line of a triangle that will add up to a square number.

e.g. a
b c

a&b have to add up to a square, as well as a&c and b&c.

Any help would be appreciated.

Can anyone help?

Code:

  a b    c
such that
$a + b = x^2$

$a + c = y^2$

$b + c = z^2$

Subtracting the first two equations gives me
$a - c = x^2 - y^2$

Adding this equation to the third equation give me
$2b = x^2 - y^2 + z^2$

Thus
$b = \frac{x^2 - y^2 + z^2}{2}$

$c = z^2 - \frac{x^2 - y^2 + z^2}{2} = \frac{-x^2 + y^2 + z^2}{2}$

$a = x^2 - \frac{x^2 - y^2 + z^2}{2} = \frac{x^2 + y^2 - z^2}{2}$

You can pick any numbers x, y, z that you like.

For example, letting x = 1, y = 2, and z = 3 gives
Code:

  -2  3    6
-Dan

EDIT: I can show that at least one of a, b, and c must be negative.