The Game of Death (Part I)

Discrete Mathematics Level 5

Five undercover agents from an elite special force unit are caught by a criminal organization. The gangster leader decides to execute them by playing a deadly game. Anyone who can survive from this game will be released without a single wound. These are the details of the game:

  1. All five agents will be lined up in random order.
  2. Each agent can see the agents lined up in front of him, but cannot see any of the agents lined up behind him.
  3. After they line up, the gangster members will spray airbrush paint on each agent's back with blue or red color. No agent can see the color on their back.
  4. Then one of the gangster members will ask each agent for the color of paint on his back, starting from the most behind one.
  5. The agent who is asked can only answer one word: "BLUE" or "RED", otherwise he will be killed.
  6. If the agent answers incorrectly, the gangster will kill him.
  7. All agents must be silent during the game. If not, they will all be killed.

The agents are given an opportunity to gather for a day before the game is started. After long discussion, one of the agents, the smartest one, has the best strategy to survive from the game. The strategy will yield the greatest number of agents that can be saved. Assume that, no matter what happens, all the agents will follow the strategy and since they are from the elite special force unit, it is reasonable to assume that they are all physically and mentally healthy so no one will ruin the plan. If the expected rate of survival using the best strategy can be expressed as \(\dfrac{p}{q}\), then what is \(p+q\)?

Next step : The Game of Death (Part II)


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