Expected Winnings in a Random-Walk Money Collection Game on 45 Stones

Forty-five stones are placed around a circle, each with $100 on it. A token is put on a uniformly random stone, and you collect the money on that stone. After that, you and an opponent alternate moves, and the opponent goes first. On a turn, the player flips a fair coin and shifts the token one stone clockwise on heads or one stone counterclockwise on tails, then collects whatever money remains on the stone landed on. Every flip is independent, and the game stops once all money has been collected. What is your expected total collection, counting the initial $100?

Hint: Look at one stone at a time Linearity of expectation reduces the problem to computing, for each of the other 44 stones, the probability that the first arrival at that stone occurs on one of your turns.

Hint: Unwrap the circle Track the token's net displacement on the integer line instead of its circular position. What can the number of moves already made tell you about that displacement?

Constraints and Clarifications

• Each stone's money is taken only once, by the player whose move first brings the token onto it; landing on an already emptied stone yields nothing.
• The $100 on the starting stone is yours.
• The opponent takes moves 1, 3, 5, and so on; you take moves 2, 4, 6, and so on.
• Provide an exact value and a dollar approximation.

What a Strong Answer Covers

• Reducing the expected total to a sum of probabilities for individual stones.
• A first-passage argument for when and from which side the token first reaches each stone.
• The connection between who collects a stone and the parity of the first move to reach it, including the role of the circle having an odd number of stones.
• An exact total with sanity checks, for example consistency with the total money available and a small circle checked by hand.

Follow-up Questions

1. How would the result differ for a circle with an even number of stones?
2. Would you prefer to move first, and how much would that change your expected total?
3. What is the probability that a given stone is the last one collected?

Overview

A random-walk puzzle where a token steps one position clockwise or counterclockwise around a circle of 45 stones, each holding $100, while two players alternate coin-flip moves and collect money from stones they reach first. It asks for your expected winnings when the opponent moves first, testing linearity of expectation and first-passage reasoning.