Identifying the two three digit numbers from a set of single digit numbers I know the two numbers have to start with 7 & 6 respectively and end with 2 or 3 (though I don't know which of these two numbers will go to the number with 7 and 6). That leaves with 4 and 5 as the middle number.
Is "Trial and Error" the only way to figure out the answer to this problem or is there a "Method" to solve this problem.
Any help will be greatly appreciated since I have to teach this to my 3rd grader son.
Question:
Two three-digit numbers are made of 2,3,4,5,6 and 7. Their product is the largest that is possible. What are these two numbers?
 A: You want the "hundreds product" to be as large as possible, so you start off with:


*

*$7ab\times6cd$


You want the "tens product" to be as large as possible, so you continue with either one of:


*

*$75b\times64d$

*$74b\times65d$


You want the "ones product" to be as large as possible, so you end up with either one of:


*

*$753\times642$

*$752\times643$

*$743\times652$

*$742\times653$



Then, you explain to your 3rd grader son:


*

*The sum of each pair of numbers in the list above is equal ($1395$)

*Therefore, the closer they are to each other, the higher their product is


So you simply choose the last option in that list, which is $742\times653=484526$.
A: I assume, you can only use each number once.
Lets suppose, just for a moment, that maybe you could use the midpoints.
and remember that $742 = 7\cdot 100 + 4\cdot 10 + 2$
$7\cdot 100 + 4.5\cdot 10 +2.5 = 747.5$
$6\cdot 100 + 4.5\cdot 10 +2.5 = 647.5$
Now, we need to click one of those number up by 5 and the other down by 5, because we can't use the midpoint.
$(747.5 + 5)(674.5 - 5) < (747.5 - 5)(674.5 + 5)$  Multiply it out if it isn't obvious.  
Clicking up the smaller number has more power, because the amount you click it up is multiplied by a bigger number.
$(742)(653)$ is your answer.
A: It is basically just the Greedy algorithm. You want to assign a place to the digits starting with the largest digit and work down to the smallest digit. If you take any two digits then 7*6 gives the biggest value. So those two values go in the two hundreds places. 
unknown 1 = 7??
unknown 2 = 6??
Now is 5*60 bigger or is 5*70 bigger? So the 5 is the tenths digit with the second unknown, not the first unknown. 
unknown 1 = 7??
unknown 2 = 65?
Now for the first unknown pick 4 to go into the hundredths digit.
unknown 1 = 74?
unknown 2 = 65?
Then using same logic
unknown 1 = 74?
unknown 2 = 653
and finally
unknown 1 = 742
unknown 2 = 653
