1. ## square problem

The rows of chairs in section A of an auditorium are arranged in a square with as many chairs in each row as the total number of rows. Section B is also arranged in a similar manner. If section A has 23 more chairs than section B, how many chairs are in section A?

It has to be 144 but why?

2. Hi Sarah,

The problem is to find a where $a^2 - b^2 = 23$. So you have 1 equation and two unknowns and this is a real problem! The only way I can see to solve it is to try some numbers. You suggested 144 i.e. $a = 12^2.$
So $b^2 = 144 - 23 = 121$ and this is $11^2.$

I hope this helps

3. Following on s_ingram's hint, $a^2- b^2= (a- b)(a+ b)= 23$.

Since a and b are "numbers of chairs" they must be positive integers and the only way to factor 23 in positive integers is 1*23.

Solve a- b= 1, a+ b= 23.

4. Originally Posted by HallsofIvy
Following on s_ingram's hint, $a^2- b^2= (a- b)(a+ b)= 23$.

Since a and b are "numbers of chairs" they must be positive integers and the only way to factor 23 in positive integers is 1*23.

Solve a- b= 1, a+ b= 23.
Thanks Ivy! I spotted it now. And thanks s_ingram too!

5. Hello, Sarah!

Knowing a fact about squares, I "eyeballed" the problem.

Two consecutive squares always differ by an odd number.
. . $(n+1)^2 - n^2 \:=\:2n+1$

Since the difference is 23, we have: . $2n+1 \:=\: 23 \quad\Rightarrow\quad n = 11$

Therefore, the squares must be: . $11^2$ and $12^2.$