u/frogkabobs

This is a slightly modified problem from r/askmath (if you go searching for it, you’ll find my answer, so don’t spoil yourself).

Two players play a game as follows. There are n spots labeled 0 to n-1 in sequence around a circle, and both players start at 0. They alternate turns, starting with player 1, where a turn consists of flipping a coin to determine whether to move to the left or to the right one spot. Each non-zero spot awards 1 point to the first player to reach it, and the game ends when all spots have been visited. What is the expected (signed) point difference between player 1 and player 2?

EDIT: I should clarify that players move independently of each other, not as a group.

Let ~ be the equivalence relation on ℂ³ generated by z~f_jk(z) for 1≤j,k≤3, where f_jk(z)_j is the reflection of z_j over z_k, and z and f_jk(z) agree on all other indices. Let S be the equivalence class of (1+i,1,i). Compute a closed form for Σ_((a,b,c) in S) 1/|a(b-a)(c-a)|⁴.

The graph Gₙ consists of vertices (x,y) for integers 1≤x≤n and 1≤y≤3 and edges between (x,y) and (x',y') iff x=x' or both |x-x'|=1 and y=y'. Find and prove a closed form expression for the number of Hamiltonian paths (paths visiting each vertex exactly once) from (1,1) to (n,3) in Gₙ.

Let M be a connected topological manifold (second countability assumed), and U⊂M a proper open subset. Show that there exists a subset A⊂U with empty interior such that every connected component of M-A contains exactly one connected component of M-U.

Bonus: What’s the largest generalization you can prove?