1. ## Ant crossing

A set of 12 rods, each 1 metre long, is arranged so that the rods form the edges of a cube. Two corners, A and B, are picked with AB the diagonal of a face of the cube.

An ant starts at A and walks along the rods from one corner to the next, never changing direction while on any rod. The ant's goal is to reach corner B. A path is any route taken
by the ant in travelling from A to B.

(a) What is the length of the shortest path, and how many such shortest paths are there?

(b) What are the possible lengths of paths, starting at A and finishing at B, for which the ant does not visit any vertex more than once (including A and B)?

(c) How many di erent possible paths of greatest length are there in (b)?

(d) Can the ant travel from A to B by passing through every other vertex exactly twice before arriving at B without revisting A? Give brief reasons for your answer.

I have a solution for a) and b), which I did by just experimenting and counting the ways. Not sure about them though, and wasnt able to work out the next 2.

a) 2 metres, and 2 ways

b) I got 2, 4 or 6 metres

2. a)2m 2 ways is correct

b)2,4,6 is correct as well

c)Counting these would be very hard especially with a length of 6. So one much do it mathmatically. The ant's first move has 3 choices. His next move has 2 choices, and same with all the rest of his moves, until the last one at which time he has only one move which is to end at b.

6 moves have these many choices:
$(3)(2)(2)(2)(2)(1)=48$
There are 3 choices times 2 paths times 2 paths times 2 paths times 2 paths and ending at b which there is only one path left.
(im not 100% sure on this but i am fairly certain)

d)I cant do it...There might be some way. Sorry.
EDIT: I doubt there is a way for d to work ive tried it alot.