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# what is the maximum possible number of identical pieces a cube can be cut into by 10 cuts?

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### Mohammed

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## [Solved] What is the maximum number of pieces into which a cube can b

Given:  A cube that has been applied 13 cuts.  Concept: By cutting the cube for the first time, it will be divided into 2 parts.  By cutting

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## What is the maximum number of pieces into which a cube can be cut by 13 cuts?

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## Detailed Solution

Given:

A cube that has been applied 13 cuts.

Concept:

By cutting the cube for the first time, it will be divided into 2 parts.

By cutting the cube along the perpendicular, it will be divided into 4 parts.

Again, by cutting the cube along the perpendicular, it will be divided into 8 parts.

⇒ With 3 cuts, there will be a total of 8 cubes.

⇒ It can be concluded that 'n' cuts made along one plane will have 'n + 1' pieces.

Calculation:

To obtain equal pieces, cuts must be made parallel to 3 planes.

Let's assume, a, b, and c cut made along three planes such that there will be (a + 1)(b + 1)(c + 1) = m pieces & a + b + c = 13.

⇒ Several combinations can be made but 'm' will be maximum in (4, 4, 5) cuts.

∴ The required number of pieces = m = (4 + 1)(4 + 1)(5 + 1) ⇒ 5 × 5 × 6 ⇒ 150

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## What is the maximum number of identical pieces a cube can be cut into by 13 cuts?

What is the maximum number of identical pieces a cube can be cut into by 13 cuts?

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What is the maximum number of identical pieces a cube can be cut into by 13 cuts?

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What is the maximum number of identical pieces a cube can be cut into by 13 cuts?

First cut will divide the cube in 2 parts. The second cut along perpendicular direction will double the number of parts to 4, now the 3rd cut will also be in perpendicular direction which will double the 4 parts to 8. So totally 8 cubes can be formed with 3 cuts. All the 3 direction should be perpendicular to each other.

Note that 1 cut = 2 parts of cube, 2 cut = 3 parts and so on.

x cuts made along one plane results in x + 1 pieces.

Logic behind this is, divide the given number of cuts in 3 equal parts and put that many cuts on each perpendicular direction.

For obtaining identical pieces, cuts must be made parallel to 3 planes of the cube.

Let we made x, y, z cuts along all three planes such that (x+1)(y+1)(z+1) = t is maximum & x + y + z = 13.

Different combinations can be possible but t will be maximum in (4, 4, 5) cuts.

So maximum pieces = (4 + 1) * (4 + 1) * (5 + 1) = 5 * 5 * 6 = 150.

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