Quick Math question..Please help!?

Question

A 4x4x4 cube contains 64 identical small cubes that exactly fill the larger cube. How many of these small cubes touch a side or bottom of the large cube?

I have a hard time picturing this, so could someone please briefly explain it??

Answer 1

well the cube has 6 sides ever cube on these sides touches a side however some of the cubes belond to more than one side. Firstly I will begin counting those that belong to only one side; these are the 4 middle ones of each side:

6(4) = 24

next there are 12 edges all the cubes on which touch a side and have not already been counted however the cube on each vertex is on more than one edge so for now I will leave them out. The remaining ones are 2 on each edge:

12(2) = 24

Finally there are 8 edges none of the cubes on these have yet been counted and none belong to more than one edge:

8(1) = 8

add these all together:

24 + 24 + 8 = 56 (I believe this is your answer)

however I have just realised a much better method would have been to realise only the inner 2by2by2 cube of cubes do not touch any side. The rest all do. There only 8 cubes don't touch a side so the answer is:

64 - 8 = 56 (the same as last time so it must be correct - and with a much better method)

Answer 2

answer

= 4*4*4 - 2*2*3

= 64 - 12

= 52, choice C.

Answer 3

the only cubes that do not touch the sides or top or bottom are those that are completely inside the cube

so try to imagine a 3x3x3 cube first...here, all the cubes except the very inner most one has a face on the outside...another way to think of this is that the cubes on end and the cubes on the opposite ends touch the outside...but only the cubes that are interior to those outer cubes do not touch the outside (this would be so much easier with a pic)

so...in a 4x4x4...the only cubes that dont touch the outside are the ones that make up a 2x2x2 cube on the inside...there are 8 cubes in this inner 2x2x2, so there are 64-8=56 cubes touching the outside

hope this makes sense