Top Answer
User Avatar
Wiki User
Answered 2007-11-23 19:51:42

This is a cracking question. [I'm assuming that 'integral side lengths' is intended also to mean integral positions. That is to say, the position vectors of the cubes' vertices should consist only of integer components...such that we are effectively drawing our cube in a 3D grid. If this was not a condition, the answer would be infinity.] So then: first, there are 53 = 125 1x1x1 cubes. Then 43 = 64 2x2x2cubes. 33 = 27 3x3x3 cubes. 23 = 8 4x4x4 cubes. And, stunningly, 13 = 1 5x5x5 cube. So far we have 225 cubes. That would not be interesting. But we have only counted the cubes with edges parallel to the edges of the main cube. Suppose there are some more cubes formed by diagonal lines? Since edge lengths and vertex positions both have to be integral, and we are working in 3 dimensions, we are actually looking for Pythagorean quadruples - integer solutions to a2 + b2 + c2 = d2 Up to now we have only used trivial solutions like this: 52 + 02 + 02 = 52 Our edges have only moved in one dimension. We might consider edges that move in two dimensions, using the smallest Pythagorean triple: 32 + 42 + 02 = 52 But diagonal edges of length five are clearly not going to fit. So introduce the third component, and we find this: 12 + 22 + 22 = 32 By using diagonal edges it is possible to constuct a few more valid 3x3x3 cubes. It turns out, I think, that four of these can be formed. So we have 229 cubes in total.

User Avatar

Your Answer


Still Have Questions?

Related Questions

How many 5x5x5 cubes are found in a 7x7x7 cube?

5x5x5 gives us 125 7x7x7 gives us 343 343/125 is 2.744

How many cubes in a 5x5x5 cube?

There would be a 125 cubes in a cube that is 5 units x 5 units x 5 units.

Are all cubes rectangles?

No because cubes do not have opposite sides that are different lengths. Cubes have all equal sides and all equal coners. A cube is not a rectangle.

A prism is completely filled with 3996 cubes that have edge lengths of 1/3 in.What is the volume of the prism?

650 ยฐC

What are all the types of rubik's cubes?

All I can think of are the 2x2x2, 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7, there might be a 8x8x8, Pyraminx, Megaminx, Octahedron, Teraminx, and the 1x1x1 :)

Why are number cubes called seven cubes?

They are not generally called seven cubes.

What are the different types of Rubik's cubes?

2x2x2 (Mini Cube)3x3x3 (The Classic)4x4x4 (Rubik's Revenge)5x5x5 (The Professor Cube)Rubik's MagicRubik's Twist Cube

What is the number sequence 1 8 27 64?

Cubes of digits 1 - 4n3. (nxnxn) 13 = 1, 23 = 8, 33 = 27...1x1x1 = 12x2x2 = 83x3x3 = 274x4x4 = 645x5x5 = 1256x6x6 = 216

Can a cube tessellate?

Sure cubes can tessellate. It's actually very easy to do so with cubes, as they would all have straight sides of even lengths from any angle.

Total volume of two cubes with the length of an edge of 5 ft?

The total volume of two cubes that each have edge lengths of 5 feet is: 250 cubic feet.

How many pieces are on a 7x7 Rubik's cube?

When c = the number that is the dimension of the cube (ex. this time, 7), you can find the number of cubes for any size of cube. c3 - (c - 2)3 So for 7, that is 7x7x7 - (5x5x5) 49x7 - 25x5 343 - 125 218 There are 218 cubies in a 7x7x7 Rubik's cube!

What is the probability of getting a sum of 3 when three number cubes are rolled?

If the number cubes are standard dice cubes, the odds of rolling 3 ones is 1 in 216.

How many number of squares are on two cubes?

There are 12 squares on 2 cubes

How do you calculate the number of cubes in a rubik cube?

You take the number of cubes on one side multiplayed by the number of sides. for example, in the standard rubik cube there are 9 cubes on each side and 6 sides. 9 times 6 is 54 so there are 54 cubes on a ribik cube

How could you find the number of cubes in one layer without counting?

For a rectangle, do the following:1. Count the number of cubes on one side of the retangle.2. Count the number of cubes on an adjacent side (not the opposite side)3. Multiple both numbers together. The result is your answer.

If a cube has an area of 49 cm2 what are the cubes dimensions?

If the surface area is 49 cm2 then the edge lengths are 2.858 cm

What is the volume of a rectangular prism that is constructed of inch cubes?

The answer is the number of inch cubes used in the construction.

What is the volume of two cubes with side lengths of 9 units?

Cube one's volume = 9*9*9 = 729 cubic units. As there are two cubes, their volumes added together = 1458 cubic units.

Why is 1729 a special number for Ramanujan?

According to Ramanujan, it's the lowest number that can be expressed in two different ways as the sum of two cubes (the cubes of 12 and 1; the cubes of 10 and 9). It 's the answer.

A cube has a volume of 75 cubic centimeter what are the cubes dimensions?

The cube root of 75 is about 4.217163327 cm which will be the lengths of the sides of the cube.

A box is in the shape of a cube the box has edge lengths of 5 inches how many unit cubes are neede to fill the box?

125 of them.

What is combined volume of 2 cubes one that has a edge that measure 5 inches and another one that is 3 inches stacked on top of each other?

5x5x5=125 and 3x3x3=27 add them to get the combined volume of 152 in^3

What are the prime cubes of the number 99?

The prime factors of 99 are 3 and 11. Their cubes are 27 and 1331.

How much cubes at the top of a rectangular prism?

There is no limit to the number of cubes which can be arranged on top of a rectangular prism.

How do you figure out what is the number of the number cube?

You cannot. And not all number cubes have the numbers 1-6 on them. For example, a doubling cube for backgammon.You cannot. And not all number cubes have the numbers 1-6 on them. For example, a doubling cube for backgammon.You cannot. And not all number cubes have the numbers 1-6 on them. For example, a doubling cube for backgammon.You cannot. And not all number cubes have the numbers 1-6 on them. For example, a doubling cube for backgammon.