Correlation between multiplied numbers? I do not have a strong math background, but I'm curious as to what this pattern is from a mathematical standpoint.
I was curious how many minutes there were in a day, so I said "24*6=144, add a 0, 1440."  Then it immediately struck me that 12*12=144, 6 is half of 12, and 12 is half of 24.  So, I checked to make sure it worked in other circumstances:
4*16=64
8*8=64
11*44=484
22*22=484
9*36=324
18*18=324
So what exactly is going on here from a logical standpoint to create that pattern?  Thanks in advance for satiating my curiosity!
 A: Think about multiplication as having piles of rocks. $6*24$ represents 6 piles of 24 rocks. 
Now what happens if you split each pile in two halves? The number of rocks in your pile half, but the number of piles double. Thus, you get 12 piles of 12 rocks, or $12*12$. 
You didn't change the number of rocks, you only rearrange them in a different way.
Thus you have $6*24=12*12$ rocks.
And the same holds with any numbers. If you split each pile in halves, the number of rocks in piles half, and the number of piles double. But in total you have the same number of rocks... Thus, if you multiply two numbers, half one and double the other, you get the same product. [Note that this intuitive explanation works for whole numbers, but can also be made to work easily for fractions].
A: I'm not sure exactly what you are asking, but I can't think of a more logic pattern than:
There are $24$ hours to a day, and $60$ minutes to an hour, thus
$$\text{minutes/day} = 24*60$$

Since $24*6 = (24/2)*(6*2)$, your logic was faulty.
A: You are recombining the prime factors of the numbers.
64 => 2*2*2*2*2*2
So any split of these numbers will result in the total 64.
Same with the other numbers, you are just rearranging the prime factors of the numbers to form different combinations which will end up as the same product.
A: You are doing $$(kx)\frac { x }{ k } $$ where $k=2$ 
