<ol>
<li>A large solid cube is assembled by gluing together
identical unpainted small cubic blocks. All six faces
of the large cube are then painted red. If exactly 27 of
the small cubic blocks that make up the large cube have
no red paint on them, how many small cubic blocks
make up the large cube?
answer is 125 please explain</li>
</ol>
<p>Suppose the side length is s. If you visualize the cube and all the painted/unpainted faces, you will see that the unpainted blocks will be contained in a cube of side length s-2. Since 27 = 3^3 (e.g. the “unpainted” cube has side length 3), we can say that s-2 = 3, or s = 5. Therefore there are 5^3, or 125 blocks contained in the large cube.</p>
<p>Questions of this form can be solved quite rapidly by realizing that a border layer of bricks creates a relationship between the number of non-border bricks and the number of total bricks such that the latter number will be the perfect cube appearing two terms after the former number in a list of increasing perfect cubes (8, 27, 64, 125, 216…).</p>
<p>So, for instance, if the question specified 64 unpainted blocks, the answer is quickly recognizable as 216.</p>