The game Instant Insanity is played with four cubes. Each face of a cube is colored with one of four colors. The object of the game is to stack the cubes one on top of another so that on each long face of the resulting shape, all four colors show up exactly once. Note that there are four such faces, which we will describe as top, bottom, front, and back. The picture above shows the four cubes placed together in a way that you can see the top and front faces.
In this assignment we will see how to determine if it is possible to win a given game and, if so, how to win.
To begin, we must make copies of an instant insanity game. On poster board or index cards, draw four copies of the following pattern without the numbers inside the squares. Each copy will be cut out, folded, and taped to make a cube. Make each square 1 inch by 1 inch.
The following table describes three different instant insanity games. Pick one of the games, and color your four patterns according to the color scheme in the table. To do this, y stands for yellow, b for blue, r for red, and g for green. Color the six squares of a cube, as numbered in the pattern above, as the colors are listed. For example, if you choose game 1, then the six faces of the first cube will be colored, in order, yellow, yellow, blue, red, yellow, green. After coloring the faces, cut out the pattern, fold along the edges, and tape together so that the colored faces are on the outside of the cube. Alternatively, if you click on the game number, you will see a page showing colored diagrams telling you how to build the four cubes. If you print this page with a color printer, you can simply fold the diagrams to get your cubes.
It would be a good idea to make marks on your cubes to indicate which game you are making. It will also help later on if you label your cubes numbers 1, 2, 3, and 4. Of the games below, one cannot be won.
| Game | Cube 1 | Cube 2 | Cube 3 | Cube 4 |
|---|---|---|---|---|
| 1 | y, y, b, r, y, g | b, b, r, g, y, r | b, b, r, y, g, r | y, g, g, r, b, y |
| 2 | g, b, y, r, b, r | y, g, b, y, r, b | r, r, g, b, g, y | b, b, g, b, y, y |
| 3 | g, b, r, b, r, y | g, r, g, r, y, b | g, r, g, r, y, b | r, g, r, g, b, y |
After you have made your game, play around with it and try to win. You may or may not succeed quickly, and if you have a game that cannot be won, you probably won't be able to tell if you cannot win.
To help explain what we will do in the following problems, here is another game, which with we will illustrate the graph-theoretic ideas used to solve the game.
The key to the game is to consider opposing sides of the cube. For example, considering the front and back faces, each of the colors must occur once on each face. If we try to put a given color on the front, then the color on the back is determined by the pattern of colors on the given cube. We will record, in a useful way, this information.
Problem 1. Draw a graph with four vertices, labeled b, g, y, r, for the four colors used in the game, or colored with the four colors. If two colors are on opposing sides of a cube, draw an edge between the colors, and label the edge according to which cube the colors are on. You should have three edges for each cube. For example, the graph for the game above is
Problem 2. Find two subgraphs of the given graph each with the following properties: (i) each graph has two edges hitting every vertex, (ii) each graph uses exactly one edge for each cube, (iii) the same edge is used at most once in the two graphs. An example of this for the graph above is
As a hint to do this, first find as many subgraphs as you can satisfying the first two properties. You may be able to find more than two such graphs. If you cannot find two graphs with the three properties, then you have a game that cannot be won.
Problem 3. To solve the game, the first graph represents the front and back faces, and the second graph represents the top and bottom faces. Align the cubes according to which edges you have in your two graphs. There is some choice on how to do this; for example, on the game above, cube 1 has the front and back faces blue and green, and the top and bottom faces red and yellow. However, we do not yet know whether blue is on the front or back. You may have to play around a little to arrange your cubes appropriately. Write down your solution, if one exists, as in the following example, which represents the game described above.
| Cube | Front | Back | Top | Bottom |
|---|---|---|---|---|
| 1 | yellow | red | green | blue |
| 2 | red | green | yellow | red |
| 3 | green | blue | blue | yellow |
| 4 | blue | yellow | red | green |
