<p>16. On a square gameboard that is divided into n rows of n squares each, k of these squares lie along the boundary of the gameboard. Which of the following is a possible value for k?</p>
<p>(A) 10
(B) 25
(C) 34
(D) 42
(E) 52</p>
<p>---- I REALLY don't get the question. What is "a square gameboard that is divided into n rows of n squares each"? +_+</p>
<p>A square game board has the same # of rows and columns because it is a square. So the total outside (k) could be 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, or 52. Therefore the answer is E) 54. Just think about the perimeter.</p>
<p>Let’s represent the number of outside squares on a square game board mathetatically. The number of outside squares on each side is equal to n. However, there are four corners each shared by two sides, so it’s important to avoid counting them twice. Therefore the total number of outside squares is equal to 4n - 4. 4n - 4 must be a multiple of four, and only (E) 52 is a multiple of four.</p>
<p>^Amphear’s is probably the most efficient solution.</p>
<p>An alternate approach might help in 3D.</p>
<p>On the plane:
In order to isolate an outer layer of the nxn square we can remove the core -
(n-2)x(n-2) square:
n^2 - (n-2)^2 = 4(n-1)
For n=14
196 - 144 = 52.</p>
<p>To calculate how many unit blocks are in an outer layer of nxnxn cube we remove the core -
(n-2)x(n-2)x(n-2) cube:
n^3 - (n-2)^3.
For n=4
4^3 - 2^3 = 56.</p>
<p>I would just plug it into my calc as Ans+4 (since that is the trend) and hit enter until I got to the answer. That would likely be the fastest approach.</p>
<p>^It seems to be the fastest, but not the most obvious way.</p>
<p>Yeah, I am a bit unorthodox when it comes to math and science, but if you think a little more abstractly it can be pretty quick. That is just my natural tendency for whatever reason. I am wired incorrectly or something. ;)</p>
<p>Thanks you guys so much. Wow, I think this question is difficult because of its wordiness.</p>
<p>Here’s a variation on the old SAT question I was finally able to locate.</p>
<p>A large solid cube is assembled by gluing together 125 identical unpainted small cubic blocks. All six faces of the large cube are then painted red. How many small cubic blocks have red paint on them?</p>
<p>From post #4:
</p>
<p>The large cube is 5x5x5, so the core is a 3x3x3 cube,
and there are 5^3 - 3^3 = 125-27 = 98 blocks with red paint on them.</p>
<p>^ Wow. At first I decided to go with Amphear’s method but yours seems applicable to many more cases 
you’re so devoted, thanks so much <3<3</p>