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$7*7~$ is $~25$
$6*6~$ is $~18$
$9*9~$ is $~41$

When is this true?

Hint

Think inside the box.

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10  
When Common Core says it is. – crayzeedude Mar 3 at 19:34
2  
@crayzeedude Common Core is just a standard for what should be taught at what level - it's pretty unobjectionable. The problem is with parents and teachers who make misinterpretations viral rather than accepting change. – Deusovi Mar 4 at 1:22
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@Deusovi And I completely get that. I'm mostly just poking fun at some of the ridiculous stuff that parents of elementary schools post. – crayzeedude Mar 4 at 1:56
1  
@Deusovi Common core is more than just a standard. It is also a prescription of how exactly something should be learned. Case in point: Math addition. They want to force kids to understand 7 + 7 as 7 + (3 + 4) = 10 + 4. That is not the first step in learning addition. That is a higher level concept that some kids intuitively get, others don't until much much later. – Καrτhικ Mar 4 at 20:57
    
@Καrτhικ "Force kids to understand" - isn't that just called teaching? You make it sound like knowledge is something bad. (Oh, and it's not the first step - it's taught after addition.) If you want to continue this conversation, we should probably move to chat. – Deusovi Mar 4 at 23:42

Because $*$ means:

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

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I'm not understanding this one. Could you please elaborate? – LN6595 Mar 3 at 22:54
4  
@LN6595 ceil means ceiling which is the mathematical term for rounding up. So what he does is multiply the two numbers, divide it by two and round that number up – Ivo Beckers Mar 3 at 23:16

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

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

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Or simply the number of black squares. – Sleafar Mar 3 at 20:26
2  
@Sleafar No, it's more than the number of black squares, if you place it properly. I'll make a .gif showing 1x1 up to 10x10. – Artyer Mar 3 at 20:28
    
For any n > 2 the number should be the same, at least it is for all you posted. – Sleafar Mar 3 at 20:32
2  
@Sleafar It turns out I was counting an inverted board's squares – Artyer Mar 3 at 20:38
2  
If you put 4 knights on a 2x2 board, none of them attack each other. – Kiran Linsuain Mar 4 at 0:16

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,

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

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14  
Wouldn't $\lceil\frac{xy}{2}\rceil$ work just as well and be marginally simpler? – frodoskywalker Mar 3 at 20:04
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This was my thought too. – Improve Mar 3 at 20:57
    
@frodoskywalker: yes, the two formulas are equivalent; your formulation is simpler to parse; I'll update my answer. – Gamow Mar 4 at 10:05

My take on it is you would change the counting base from decimal (base 10), so 7 * 7 = 49 in decimal, which is 25 in base 22 ((22 * 2) + 5).

Thus 6 * 6 = 36 in b10, which is 18 in b28, and 9 * 9 = 81 in b10, which is 41 in b20.

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What's the system for which base to use? – charfellow Mar 4 at 21:44

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