Strategic Hats

First of all, we know there must be at least 5 people with red hats and at least 5 people with white hats (or else no one would be raising their hands).

Hence, denoting $R$ as the number of red hats and $W$ as the number of white hats, there are only 3 cases:

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Case 1: If $R>5$ and $W>5$ , then everyone will be able to see at least 5 red hats and at least 5 white hats and $n\geqslant12$ , but then that would lead to at least 12 people raising their hands, which is a contradiction.

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Case 2: One of $R$ and $W$ equals 5 while the other greater than 5 , without loss of generality, let's suppose that $R=5$ and $W>5$ , then everyone except those wearing red hats will raise their hands. Since there are 10 people raising hands and none of them are the 5 people wearing red hats, this makes $n=10+5=15$ .

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Case 3: If $R=W=5$ , then only those who are not wearing red or white hats will raise their hands, making $n=20$ .

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Thus, the minimum value of $n$ is $15$ .

We know that there are five people with red hats and five people with white hats. But they cannot see their own hats, so there must be more people.

If there are $6+$ people with a red hat and $6+$ people with a white hat, then there are $12+$ people who would raise their hands-- more than the actual 10.

Therefore there must be five people with a hat of one color (say, red), and more than five with a hat of the other color (white). The people with a red hat can see only four other red hats, so they don't raise their hands. But all people with a white hat will raise their hands. Therefore there are 10 people with white hats. Add the five with a red hat, and we get a total of $\boxed{15}$ .

The analysis must start at 11 since there must be at least five red and white hats plus one person (person 11) who can see them all.

If person 11 wears a neutral color (not red nor white) hat that person will be the only hand raised since the others will still only be able to see 4 hats matching their own while seeing 5 of the other color.

If person 11 wears either red or white (the majority color) 6 hands will be raised as all 6 wearing the majority color hat can mutually see 5 hats of the majority hat color and the five of the minority hat color. This, maximizing the number of raised hands, must be the choice so far to achieve a minimum.

If a person 12 is added with the minority color hat the result will 12 raised hands, exceeding the given value of 10. Therefore, no additional hats of the minority color can be added under the given condition of 10 hands raised.

Additional persons wearing either the majority color hat or any neutral color hat will add one hand to the total as each can see the original 5 red and 5 white hats.

Starting from 11 people with 6 hands, 4 more will be required to get the count to 10.

11 + 4 = 15 is the minimum number of people required to have exactly 10 hands raised.

Lets take 5 people with red hat and 10 with white. As people with red hat are will see 4 red hats - : they will not raise their hands. And as people with white hats are seeing 5 red and at least 5 white, they all will raise hands.