PuzzleMall ~ 3-D Posers

Math Puzzle Mall

3-D Posers

1. The picture shows a triangular pyramid with a belt around its middle. Albert, the Man of the (previous) Century, asks you to place the numbers 1 through 7 at the nodes so that all groups of three, i.e. the three up-sloping edges and the two triangles have the same sum.

Of course, this problem will be easier to work on a flat surface. What follows is a "Schlegel" diagram of the pyramid (Figure 1a). To make such a diagram for any solid, you spread the base and press the rest flat. The base must be large enough to contain the rest of the parts. Sparky is seen going to work on squashing a couple of cubes down to flat figures.

2. Use the Schlegel diagram for the cube (below) to solve the next task. Each of the six faces of the cube has four corners. Number the corners with 1 through 8 so that all the face sums, for example, A + B + C + D and C + D + E + F, are equal.

3. Sketch a Schlegel diagram of the octahedron. In the photo below, this is the solid in the front row center. The octahedron has eight triangular faces and six nodes.

4. For this poser, imagine that you are a salesman. Here is a Schlegel diagram for the icosahedron which appears at the back right of the photo above. Imagine the diagram is a map of the towns you must visit on your sales route. Can you make a continuous path that visits all 12 vertices (towns) and does not retrace any edge (road)? Can you make a path that accomplishes that and returns to its starting point?

5. Complete a magic cube. What is a magic cube? It is a three-dimensional version of the magic square. An n-order magic square has n by n unit squares filled with numbers so that each row, column, and the two main diagonals have matching sums. An example magic square is shown. Its magic sum is 51.

Complete the magic cube with the numbers from 8 through 34 at the nodes in the picture below, so that each row, column, vertical column, and the four main diagonals (one of which is suggested by the yellow struts in the photo) have the same magic sum. What will that sum be? A flat grid is provided below the photo.

In the grid below, a typical row is labeled EKG, a column ADJ, a vertical column DEF, and a main diagonal JKL.

Solution

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