Q:

Multiplication Homework

$7*7~$ is $~25$
$6*6~$ is $~18$
$9*9~$ is $~41$
When is this true?
Hint

 Think inside the box.

A:

Because $*$ means:

$\left\lceil\frac{n\times m}{2}\right\rceil$

A:

It is

 The maximum number of knights on a board of that size (i.e. width is the first number and height is the second number) where none of them can attack another. (Assuming it's free-for-all chess for some reason)
 And it's inside the "box" because the box is a chess board.
 
 It's also just how many black squares would be on a chess/checkers board, as @Sleafer pointed out. It turns out I over complicated it.

You would continue the sequence

 $$1*1=1\\2*2=4\text{ (or }2\text{ if counting squares)}\\3*3=5\\4*4=8\\5*5=13\\6*6=18\\7*7=25\\8*8=32\\9*9=41\\10*10=50$$
 

A:

These are true, 

 if you define $~x*y~$ as the number of black squares on a chessboard with $x$ rows and $y$ columns, where the lower left corner square is black.  

Equivalently,

 define $~x*y=\lceil xy/2 \rceil$, that is, as the product of $x$ and $y$, divided by $2$, and then rounded up to the next integer.

With this,  

 $7*7 ~=~ \lceil 49/2\rceil ~=~  \lceil 24.5 \rceil ~=~ 25$
 $6*6 ~=~ \lceil 36/2\rceil ~=~  \lceil 18~~~\rceil ~=~ 18$
 $9*9 ~=~ \lceil 81/2\rceil ~=~  \lceil 40.5 \rceil ~=~ 41$  

