Consider a wooden 5x5x5 block. A termite wishes to eat the block in the following way

1) It starts with an external, central 1x1x1 cube of any face.
2) The termite eats it, and the heads towards a neighboring 1x1x1 cube.
3) It repeats step 2 until it can't go any further.

Determine:

a) A possible path for the termite to follow, in order to eat every single 1x1x1 cube.

b) If it's possible for the termite to eat all 1x1x1 cubes, knowing it ate the central internal cube last.

For part (a), the termite could start in the middle cube of the top face, then go to a neighboring cube on that face and proceed in a clockwise spiral, ending up at a corner cube on the top face. The termite then goes straight down to what was the second level from the top and spirals inward ending up at the center, at which point he goes down to the center of what had been the middle level. This level is done spiralling outward like the top, and the next level is again done spiralling inward like the second level from the top. The bottom is done spiralling outward from the center again.

For (b) we can only assume that eating the central internal cube last allows for the fact that that cube is going to be unsupported at a certain point in the progress of the eating, and therefore no longer in position.

A way for the termite to proceed in this instance could be to spiral out along the top face as before, and then go down to a corner of the second level. This time instead of spiralling in to the center, go from left to right on one row of what was the second level, then right to left, left to right, right to left and finally left to right, ending up at the diagonally opposite corner of what originally was the second level from the top.

Then proceed down to what was the middle level, at that corner, and spiral in, but don't consume the center. Instead go down to the next level at a position adjacent to the center cube. Go out toward the edge (away from the center) and then proceed clockwise around the outside. When the last cube on the outer edge is done, come back to the set surrounding the central cube of this originally 4th level and complete its consumption in a U-path, finishing on a cube that's diagonally adjacent to the center cube, so that now, on the 3rd and 4th levels the center cube is still intact.

Proceed down to the bottom level. Go away from the center again, which places you one unit away from a corner. Proceed to that near corner and then continue to proceed around the outer cubes of that level until again forced to go one in, and reverse direction around the central cube. What's left is a column of the center cubes of the mid to bottom levels. Eat upward so that the central cube of the large cube is last. At some point the column will probably topple, but keep eating anyway.

Posted by Charlie on 2003-05-09 04:40:45