Mathcounts Question

[trim creek]

OK so after the first move the ant is at a distance 1 from A. Let's look at the moves that come after : (and let's remark that the only point with a greater distance than 1 to point A is the bottom, let's name it B, and let's name C the square formed by all the other points )
Second move :
If the ant goes toward B (p=1/3), then the 3rd move will bring it alongside C anyway, so distance will be 1.
If the ant stays on C (p=2/3), then the third move could be :
Third move :
-Going back to A (p=1/3), so this one doesn't work
-Staying on C (not closer or farther from A)(p=1/3), so not what we look for
-Going to B (p=1/3), which is our solution.

At the end of the day, the probability to finish in B is $2/3*1/3=2/9$

[obsidian vale]

Tigre_Rampant

[subtle path]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[hazy monolith]

holy yap

[hazy monolith]

the only vertex a distance greater than 1 from A is the one at the bottom, so you need to find all of the paths of length 3 that lead to that vertex

this problem has a lot of symmetry so use that to your advantage