## Transporting wax

A skilled artist once made a magnificent wax structure made of two wax cubes stacked on top of each other. The height of this structure was equal to the height of the wax cube from which these two cubes were carved out.

He decided to melt the remaining unused wax, mould it into a rectangular block, pack it in a rectangular wooden box and transport it to a warehouse a few miles away. He hired a carpenter to make the box. However, much to the dismay of the artist, the carpenter made a rectangular box that wasn't big enough to pack the unused wax.

The rectangular block of wax he could fit in the box was as tall as the wax structure he had made. Its length was equal to the height of one wax cube in the structure and its breadth was equal to the height of the other cube. How many trips did he have to make to the warehouse?

[SOLVED]

#### Vikram Agrawal solved this puzzle:

Let x be the height of the original cube. So, its volume is x3. Let y and z be heights of the two smaller cubes. So, y + z = x.

So, the volume of the rectangular box is xyz. Let n be the required number of trips to the warehouse.

x3 − y3 − z3 = nxyz
⇔ (y + z)3 − y3 − z3 = n(y + z)yz
⇔ 3yz(y + z) = n(y + z)yz
⇔ n = 3

#### Mike Terry solved this puzzle:

3.

I just visualise a cube disected into two smaller cubes plus three rectangular blocks of the dimensions given. The two cubes are situated in opposite corners of the larger cube. The same could be seen with a little algebra, but it's interesting that an exact disection works.

#### Karthik Krishnamurthy solved this puzzle:

Let the height of the original cube be 'x'. and the height of the two carved out cubes be 'y' and 'z'. Converting the last paragraph of the problem to an equation we get:

n = (x3 − y3 − z3) / (xyz)  — (1)

Since the two carved out cubes are stacked on top of each other and the height of the structure is equal to the height of the original cube,

x = (y + z)  — (2)

Substituting (2) in (1), we get:

n = (y + z)3 − y3 − z3 / (y + z)(yz)
= 3(zy2 + yz2) / (zy2 + yz2)
= 3

#### Rajkumar V. solved this puzzle:

Let the volume of two cubes A and B be X3 and Y3 cubic units respectively.

The volume of the bigger cube whose height is same as the height of the structure we get by stacking the cubes A and B is (X + Y)3.

So, the amount of unused wax is m = 3(X2Y + XY3).

The volume of the wooden box is n = (X + Y)XY = (X2Y + XY3).

So, the number of trips required is m / n = 3.

### Credit

This is an original puzzle from cotpi.