Two players Alice and Bob play the following game consisting of $n-1$ turns. Initially the segment $[0,1]$ is given. Alice and Bob then alternate breaking one segment into two pieces. After all turns have passed, there are $n$ pieces. What is the maximum number of triangles Alice can guarantee forming?
If you cannot see the answer for the total can you at least give the answer for the case $n=4$ or higher because I'm lost.
Anything would help.
Thanks.