# Bell’s Casino Problem

2015-08-16T18:56:21Z

A new casino has opened up in town named “Bell’s Casino”. They are offering a coin game. The game works as follows.

The house will commit two coins on the table, oriented heads or tails each, and keep them covered. The player calls what the faces of the each of the coins are, either HH, HT, TH, or TT. The casino reveals the coins and if the player is correct, they win \$1, and otherwise they lose \$1.

Problem 1.
Prove that there is no strategy that can beat the casino.

After opening the customers stop coming by to play this boring game, so to boost attendance the casino modifies the game as follows.

The house will commit two coins on the table, oriented heads or tails each, and keep them covered. The player calls what the faces of each of the two coins are, either HH, HT, TH, or TT. The casino reveals one coin, of the players choice. After seeing revealed coin, the player can elect to back out of the game and neither win nor lose, or keep going, and see the second coin. If the player’s call is correct, they win \$1, and otherwise they lose \$1.

Problem 2.
Prove that there is no strategy that can beat the casino.

Even with the new, more fair, game, attendance at the casino starts dropping off again. The casino decides to offer a couples game.

The house will commit two coins on two tables, oriented heads or tails each, and keep them covered. The couple, together, calls what the the faces of each of the two coins are, either HH, HT, TH, or TT. Then, each player in the couple gets to see one coin each. Collectively they get to decide whether they are going to back out of the game or not by the following method. After seeing their revealed coin, each player will raise either a black flag or a red flag. If both players raise the different colour flags, the game ends and no one wins or loses. If both players raise the same colour flag, the game keeps going. If the couples original call was right, they win \$1, and otherwise, they lose \$1. To ensure that the couple cannot cheat, the two tables are places far enough apart such that each player’s decision on which flag to raise is space-like separated. Specifically the tables are placed 179 875 475 km apart and each player has 1 minute to decide which flag to raise otherwise a black flag will be raised on their behalf (or, more realistically, the tables are placed 400 m apart and each player has 100 nanoseconds to decide which flag to raise).

Problem 3.
Prove that there is no strategy for the couple that can beat the casino.
Problem 4.
Devise a physical procedure that a couple can follow to beat the casino on average at this last game without cheating.

The casino cannot figure out how they keep losing money on this game and, soon, Bell’s Casino goes bankrupt.

## Responses

Russell O’Connor: contact me