Problem Solving

If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of s^2 – r^2 – q^2?

-20
0
8
12
16

OA is [spoiler] 8. The key to doing this problem is to just write the whole equation out and then test every case. My question is wouldnt this take way way too long to test each case, factor it out and then see if it matches? Any shortcuts or cues? Thanks [/spoiler]

well i used diff approach
so the set of three consecutive numbers can have unit digits likes this:
0,2,4 or 2,4,6 or 4,6,8, or 6,8,0 or 8,0,2
square the unit digits of q,r,s and subtract
case 0,2,4: 6-4-0=2
case 2,4,6: 6-6-4=6
case 4,6,8:4-6-6=2
case 6,8,0:0-4-6=0
case 8,0,2:4-0-4=0
u will never get a number ending with 8
and it took less than a minute.

thanks for the advice. I'm still stuck in high school algebra mode where I always try to find out the "theory" behind every problem by writing out the algebraic expressions, I rarely substitute in numbers.

Is it faster to plug in numbers? My first instinct is to always write out the theoretical equation first?

abcdefg wrote:thanks for the advice. I'm still stuck in high school algebra mode where I always try to find out the "theory" behind every problem by writing out the algebraic expressions, I rarely substitute in numbers.

Is it faster to plug in numbers? My first instinct is to always write out the theoretical equation first?

well I did not substitute numbers. IMO that's the theory behind the question. You see only those can be the combinations for three consecutive even integers.
In general, plugging in numbers is faster than using algebra but u need to practice a lot to make it work for you.

well I did not substitute numbers. IMO that's the theory behind the question. You see only those can be the combinations for three consecutive even integers.
In general, plugging in numbers is faster than using algebra but u need to practice a lot to make it work for you.

Well plugging in numbers is definitely faster, but you need to satisfy all the given conditions and at times its a good decision to solve a few questions by basic algebra rather than plugging

But everything depends on the situation and the type of question.

It's really more "art than science" when it comes to knowing when to plug in numbers. Sometimes it's a lot faster than using algebra. Sometimes it's a total pain and takes forever, when doing it with algebra would've been simple and elegant. It's just something that comes with practice.

Personally, unless the algebraic equation just jumps out at me from the outset, I go to plugging in by default. If that isn't working, I'll change mid-problem and try to work out the algebra or find some sort of number property behind the question. It's usually pretty easy to tell when plugging in will work, once you actually start the problem. This problem had really easy calculations, so I was confident from the get-go that plugging in was the best way to do it. If there had been funky calculations (decimals, roots, whatever), I would've leaned more towards the algebra approach.

Like I said, more art than science...