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Question

A 10cm x 10cm x 10cm cube is cut into 1cm x 1cm x 1cm cubes. Some of these cubes are glued together to form the largest possible cube which looks solid from any point on the outside, but is hollow inside. What is the number of smaller cubes left over?

Answer 1

Well, first off you will have 1000 (10 * 10 * 10) 1cm cubes if you divide up the larger cube the way you described.

The largest cube you could form would be one of the same dimensions you started with, or 10 x 10 x 10. If it is hollow inside, then there has to be a layer of cubes at least one cube thick around all sides. Therefore, there is an 8 x 8 x 8 cube missing from the inside.

So there must be 8 * 8 * 8 = 512 cubes left over out of the original 1000 you started with.

good luck!

Answer 2

Birdwoman 1 is right. Here's the scheme:

1. Two vertical faces have dimension 10 x 10. Hence, a total of 200 cubes are used to make them.

2. Since the two vertical faces above contribute a row of 10 blocks 1 block wide each toward the face of each of the two remaining vertical sides, then the second pair each require 10 x 8 = 80 cubes to complete them. So together they use 160 cubes.

3. Now, imagine you are looking straight down at the cube from above. Then the horizontal faces each have cubes contributed by the vertical faces. Each of the vertical faces is 1 cube thick and we must account for this in both directions on each horizontal face. Therefore, we reduce each edge of both horizontal faces by two. So they need only 64 cubes to complete each of them: (10 - 2)(10 - 2) = 64. So together they use 2 x 64 = 128 cubes.

Now, all we have to do is add these all up:

200 + 160 + 128 = 488

1000 - 488 = 512

So 512 = 8³ smaller cubes would be left over.

Answer 3

10 x 10 x10 = 1000 so there would be 1000 1cm cubes

A cube has six sides, so that would mean 1000 / 6 = 166.667 average number of cubes per side, to give a rough estimate.

The cube would consist of 2 large sides, 2 medium sides & two small sides. I know that sounds odd, but the sides cannot be regarded as the same size because otherwise the 1cm cubes on each edge would be double counted.

Thus

13 x 13 = 169
13 x 13 = 169
11 x 13 = 143
11 x 13 = 143
11 x 11 = 121
11 x 11 = 121

This gives a total of 866, so I reckon there would be 134 cubes left over.

Answer 4

Well Said is right. Using Birdwoman's technique gives 1000 - (13^3 - 11^3) = 134 cubes left over, and this agrees with the result from Well Said's alternative technique.

14^3 - 12^3 = 1016; so you cannot make a larger cube.

Answer 5

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