Arthur took a sheet of paper and drew a careful sketch. Then following his plan, he took two pieces of wood and sawed and chiselled and bevelled and sanded until the two pieces fitted snugly together in the form a of a cube.

Gwenlyn looked at the finished product admiringly. The top and bottom of the cube were not cut, but each vertical face appeared to interlock in exactly the same way (see the adjacent diagram).

‘I don’t know how you put it together,’ she said, ‘but one thing’s for sure, you won’t be getting it apart again in a hurry!’

‘Not so!’ laughed Arthur, and presto! The cube was in two pieces again. How were the pieces fitted together?

Hint 1

Do the hieroglyphicson teh side faces have any common pattern in their structure? Any symmetry?

Hint 2

We must assume that there is a solution. If the two pieces are to come apart, systematically check out the possible directions in which the pieces could be moved.

Solution

The solution to this problem is aggravatingly simple! The bottom half of the cube is illustrated. The top portion is crafted to fit on top of this, so that the two pieces form the cube. The two pieces slide apart diagonally.

Is it possible to logically arrive at this solution? Consider the possible ways in which the two pieces might come apart. From the face intersections it is clear that there can be no vertical movement. Hence any movement must be in a horizontal direction. Again looking at the face intersections, any horizontal movement parallel to an edge of the base is also out of the question – the face intersections prohibit such movement. Hence any movement must be horizontal, but in some direction other than parallel to an edge of the base. This leads us to consider the correct possibility: movement which is horizontal, and in the direction of one of the base diagonals.

Extensions

A regular tetrahedron (triangular based pyramid) can be cut into two identical pieces having a common square face. See if you can adapt the cube problem to this case.

Hint 1

Hint 2

Solution

Extensions