**Tesseract: Cube to the Fourth** Edwin Abbott's Flatland inspired many later works that are worth tracking down:

*Sphereland* (Dionys Berger, 1965),

*The Planiverse* (A. K. Dewdney, 1984), and most recently,

*Flatterland* (Ian Stewart, 2001).

**1.** For a three-dimensional cube, the farthest distance from one corner to another is √3 feet. To determine the farthest distance, first consider the answer for a one-foot square. For a one-foot square, the farthest distance is the diagonal: √1

^{2} + 1

^{2} , or √2. For a cube, the farthest distance is also the diagonal, which is the hypotenuse of the right triangle whose legs are one side of the cube (one foot) and the diagonal of one square face: √1

^{2} + √2

^{2} = √3.

**2.** To build a four-dimensional hypercube, make two copies of a three-dimensional cube. Lay one cube flat on the ground and raise the other in the fourth dimension the distance of one side length. This doubles the 8 corners of the original cube for a total of 16 corners. Connect each corner of the bottom cube to the corresponding corner of the top cube, for a total of 32 edges. The completed hypercube has 8 cubical hyperfaces: one each top and bottom and 6 around the sides. If the edge length is one foot, then the farthest distance from one corner to another is √

__4 = 2__ feet. To determine the farthest distance, consider the answer for a cube. For a 1x1x1 cube, as described above, the farthest distance is the diagonal, or √3. For a hypercube, the farthest distance is its diagonal, which is the hypotenuse of the right triangle whose legs are one side of the hypercube (one foot) and the diagonal of one cubical face (√3): √1

^{2} + √3

^{2} = √4.

**4-D TICKTACKTOE** **1.** X has two ticktacktoes, whereas O has four ticktacktoes.

**2.** X can complete a ticktacktoe in six squares (red X's), whereas O can complete a ticktacktoe in eight squares (blue O's). Note that there is one square in which both players can play to complete a ticktacktoe.

**3.** There are 15 potential ticktacktoes that include the red square. There are eight possible ticktacktoes in a standard 3x3 board, and there are 49 possible ticktacktoes in a 3x3x3 board. In a 3x3x3 hypercubical board, as rendered in our problem, there are 272 possible ticktacktoes. The breakdown: 72 possible ticktacktoes contained solely within each 3x3 board (nine times the standard eight in a 3x3 board), 72 possible ticktacktoes that form vertical lines straight through each 3x3x3 cube (nine cells in each square times the eight possible configurations), and 128 possible ticktacktoes that progress sequentially across all three levels of a cube (16 possible ticktacktoes in each 3x3x3 cube times eight possible configurations).

**4.** Ticktacktoe on a conventional 3x3 board can end in a draw, but every arrangement of X's and O's on a 3x3 cubical board must contain at least one ticktacktoe. From this we can conclude that ticktacktoe on a 3x3x3 hypercubical board also cannot end in a draw, because cubical boards make up the hypercubical board.

**Hypercross** **1.** Figures c and e cannot be folded into cubes; the shaded squares collide.

**2.** Eleven ways:

**3.** Figures b and d cannot be folded into hypercubes; the shaded cubes collide.

**4.** There are 261 ways to unfold a hypercube.

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