How many unit cubes fit inside a sphere inscribed in a 6 cm cube?

[recon]

I am having difficulty solving the following problem:

"A cube of edge 6 cm is divided into 216 unit cubes by planes parallel to the faces of the cube. A sphere of diameter 6 cm is inscribed in the large cube so that the faces of this cube are tangent to the sphere. What is the number of complete unit cubes contained in the sphere?"

Anyone care to point me in the right direction?

 

[phoenixthoth]

try doing it for a 6 unit square and 6 unit diameter circle first. maybe that will help.

 

[recon]

Hmmm, no replies...is the question really that hard? Or are you not helping me because you think it is homework? If that's the case, you're wrong. I'm just doing the problem out of interest.

So, please, please help me .

 

[matt grime]

It's not hard, it's just not that interesting for me cos it's fiddly.

here's a long winded way to do it:

for each cube, pick the furthest corner from the centre of the sphere, find the length from that corner to the centre of the sphere - not too hard, and see if its less than the radius of the sphere. by symmetry you only need to do it for something like 27 cubes. and clearly none on the surface of the larger cube will work, and that's 19 of the 27 got rid of straight away. of the remaining 8, only one, the furthest from the centre needs any consideration, really, and as sqrt 12 > 3 its furthest corner lies outside the sphere,

so the answer appears to be 7*8=56


that do you?