Why is $12\cdot 12=144$ but $8 \cdot 16=128$? (I was asked by a co-worker) Why is $12\cdot 12=144$ but $8 \cdot 16=128$? a friend thinks it should be the same because $12+12=24$ and $8+16=24$. He said it is only $4$ that is moved to the other side. Please give some sort of reason so I can inform him. thanks
 A: Intuitive answer: Let's look at a simple case of $2\times2$ and $1\times3$. Those two definitely don't equal, in fact, the second will always be smaller than the first. 
I highly recommend what Arthur suggested in the comments - in fact, you may even visualise why this is true by taking the bottom $4$ rows as collumns and cutting away the parts that "sticks out" - a $4\times4$ square (I would draw this, but my MathJax skills are sadly not up to par here.
Edit: Well I tried anyway. Take $4\times 4$ and $2\times 6$
 \begin{matrix}
        \cdot & \cdot & \cdot & \cdot \\
        \cdot & \cdot & \cdot & \cdot \\
        \color{red}{\cdot} & \color{red}{\cdot} &\color{red}{\cdot} &\color{red}{\cdot} \\
        \color{red}{\cdot} & \color{red}{\cdot}&\color{red}{\cdot} & \color{red}{\cdot} \\
        \end{matrix}
\begin{matrix}
        \cdot & \cdot & \cdot & \cdot & \color{red}{\cdot} & \color{red}{\cdot} \\
        \cdot & \cdot & \cdot & \cdot & \color{red}{\cdot}&\color{red}{\cdot} \\
         &  &  &  & \color{fuchsia}{\cdot} & \color{fuchsia}{\cdot} \\
         & &  & & \color{fuchsia}{\cdot} & \color{fuchsia}{\cdot}\\
        \end{matrix}
Mathematical answer: For $b >0$ $$ (a-b)(a+b)=a^2 - b^2 < a ^2$$
A: Suppose you have two numbers $x,y$ whose sum is $24$ and whose product is $144$. That is,
$$x+y=24,\\
x\,y=144.$$
Just knowing the sum and product, we can figure out what $x$ and $y$ must be. Subtracting $x$ from both sides in the first equation isolates $y$: $~~~~y=24-x.$ Substituting this expression for $y$ into the second equation yields:
$$x\,(24-x)=144\\
\Leftrightarrow 24x-x^2=144\\
\Leftrightarrow 0=x^2-24x+144=(x-12)^2\\
\Leftrightarrow x=12.$$
From $y=24-x$, we find $y=24-12=12.$
In general, a pair of numbers is determined by their sum and product. If you believed your friend who thinks that a pair's sum determines its product, then you end up with the following contradiction: $$\text{Suppose two different pairs of numbers have the same sum.}\\
\text{If they have the same sum, they also have the same product.}\\
\text{But if two pairs of numbers have both the same sum and product, the pairs are actually identical.}\\
\text{Since two things cannot be both different and the same, contradiction.}\\
$$
A: 
because $12+12=24$ and $8+16=24$.

Tell your friend that $0+24$ is also $24$... :-)


He said it is only $4$ that is moved to the other side.

Then tell him that the difference between $128$ and $144$ is also “only” $4^2=16$... :-)
