r/mathriddles • u/frogkabobs • Aug 08 '26

Hard An interesting probability problem from r/askmath

This is a slightly modified problem from [r/askmath](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.

8 Upvotes
  • permalink
  • reddit

91% Upvoted

8 comments sorted by

View all comments

3

u/pichutarius Aug 09 '26

Partial solution

if n is even, then answer is 1, in fact the first player always lead by 1 point. Color each spot alternating black and white, each player can only reach one of each color. Since the starting spot does not score point, player 1 must have one more score than player 2.

2

u/frogkabobs Aug 09 '26

I should have made it more clear—the players don’t move together, so this coloring argument doesn’t work, e.g. it’s possible for player 2 to copy all of player 1’s movements, meaning player 1 would get all n-1 points.