Test yourself no. 2?

Question

A large cube is constructed from 125 smaller cubes. Number of smaller cubes whose faces touch the faces of exactly four other cubes is

a)24 b)36 c)48 d)64 e)81

Answer 1

Visually construct a cube and determine which parts of the cube fit the criteria of having only 4 faces touching other faces of the smaller cubes. We can classify the smaller cubes into 4 different parts.

The smaller cube that is totally encased in the bigger cube has all 6 of its faces touching other smaller cubes.

The smaller cube that makes up the face of the larger cube (but not an edge or corner) has 5 faces touching other smaller cubes (the 4 surrounding it and the 1 "affixed" to the inner cubes)

The smaller cube that makes up the corner of the larger cube has 3 faces touching the smaller cubes.

Finally the smaller cube that makes up the edge (but not a corner) of the larger cube has exactly 4 faces touching the faces of other smaller cubes.

Knowing this we simply need to count the number of edge pieces in a cube consisting of 125 smaller cubes.

A cube consisting of 125 smaller cubes is a 5x5x5 cube with each layer consisting of 25 smaller cubes in a 5x5 square. Note that 5*5*5 = 125.

What's left is to simply count how many pieces in on an edge qualify as an edge piece (remember, corners do not count) and multiplying it by the number of edges that a cube has,

So each side of the cube consists of 5 pieces, subtract 2 for the corners to get 3 edge pieces per edge. A cube consists of 12 edges (4 on the top layer, 4 on the bottom layer, 4 going around the middle)

3*12 = 36
The answer is b) 36

Answer 2

The large cube should be made from 5 x 5 x 5 smaller cubes.

Those on the edges minus those at the corners will have 4 faces touching the other four cubes.

1 cube has only 12 edges, Right ?

so 12 edges x 3 cubes = 36 cubes

b) 36

Answer 3

36

Answer 4

The large cube has side of 5 (because 5^3 = 125);

Only those cubes at the edges of the large cube but not at the vertices touch exactly four other cubes.

There are 12 edges, and only 3 (=5-2) of these cubes on each edge

=> 3*12 = 48. (c)

Answer 5

b. all of the cubes at the cube face interfaces not including the corners

12 cube face interfaces * 3 cubes = 36

Answer 6

I say roman king is right
16 edges times 3 cubes in each edge makes 48

Answer 7

81

Answer 8

a)24

Answer 9

b...36