1. ## 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?

2. 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

3. 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

4. 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
$\displaystyle a + b = x^2$

$\displaystyle a + c = y^2$

$\displaystyle b + c = z^2$

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

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

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

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

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

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.
The content structure consists of a user posting a mathematical problem about arranging numbers in a triangle so that the sums of the numbers in each line form square numbers. The user asks for help, and another user provides a solution with an example and a general formula for finding such numbers.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. ## 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?

2. 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

3. 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

4. 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
$\displaystyle a + b = x^2$

$\displaystyle a + c = y^2$

$\displaystyle b + c = z^2$

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

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

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

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

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

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.
user
The content structure consists of a user posting a mathematical problem about arranging numbers in a triangle so that the sums of the numbers in each line form square numbers. The user asks for help, and another user provides a solution with an example and a general formula for finding such numbers.

TYPE: SINGLE_QUESTION_WITH_ANSWER