You have a 3x3x3 cube of tofu thought of consisting of 27 in

You have a 3x3x3 cube of tofu (thought of consisting of 27 individual cubes). A worm wants to sample every individual cube of the 27 cubes. The worm wants to start at some cube on the outside, and return to the outside at some other cube. The worm wants to visit every cube exactly once. After sampling a cube , the worm can proceed to any cube that is adjacent (i.e. shares a face with the present cube – sharing an edge, or a corner , is NOT good enough!)

Investigate:

[4]        1. A)Can the worm do it?

B)Can he start at some cube and finish at an adjacent cube?

[4]        2. assuming that the worm can do it SOME way - can we ask for more restrictive solutions, i.e. Can any cube be the “finishing” cube, when a certain cube is given as a starting cube?

[4]        3. What are the necessary and sufficient conditions for “there is a path starting at cube a and ending at cube b” (a, b not necessarily on the outside)

Hint: consider coloring the cubes “3D – chess – board style”…

Solution

4)A) yes the worm can do it as there are 27 cubes with each 3*3*3

hence if it starts from 1 end it will reach the other end where it will find the other adjacent cube.

B) yes he can start at some cube and finish at adjacent as because moving in cube will reach to the other end where the adjacent cube is there.