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Omni-MATH / Let n>1 be a positive integer. Ana and Bob play a game with other n people. The group of n people form a circle, and Bob will put either a black hat or a white one on each person's…

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problem

Let n>1n>1 be a positive integer. Ana and Bob play a game with other nn people. The group of nn people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually. Before Bob distribute their hats, Ana gives nn people a strategy which is the same for everyone. For example, it could be "guessing the color just on your left" or "if you see an odd number of black hats, then guess black; otherwise, guess white". Ana wants to maximize the number of people who guesses the right color, and Bob is on the contrary. Now, suppose Ana and Bob are clever enough, and everyone forms a strategy strictly. How many right guesses can Ana guarantee? [i]
Plain-text mathematical notation (without MathML)
Let n>1 be a positive integer. Ana and Bob play a game with other n people. The group of n people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually.

Before Bob distribute their hats, Ana gives n people a strategy which is the same for everyone. For example, it could be "guessing the color just on your left" or "if you see an odd number of black hats, then guess black; otherwise, guess white".

Ana wants to maximize the number of people who guesses the right color, and Bob is on the contrary.

Now, suppose Ana and Bob are clever enough, and everyone forms a strategy strictly. How many right guesses can Ana guarantee?

[i]
Original LaTeX notation
Let $n>1$ be a positive integer. Ana and Bob play a game with other $n$ people. The group of $n$ people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually.

Before Bob distribute their hats, Ana gives $n$ people a strategy which is the same for everyone. For example, it could be "guessing the color just on your left" or "if you see an odd number of black hats, then guess black; otherwise, guess white".

Ana wants to maximize the number of people who guesses the right color, and Bob is on the contrary.

Now, suppose Ana and Bob are clever enough, and everyone forms a strategy strictly. How many right guesses can Ana guarantee?

[i]

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Source and history

task_id
dd7b1701-d5d3-5330-9615-9f820ee3ba8c
task_key
test--dd7b1701-d5d3-5330-9615-9f820ee3ba8c
task_revision_id
1
config / split
/ test

Official source

initial import