Maths puzzle

 

Assessment started: undefined.

Item 1

There are N$N$ people in a room, each wear a birthday hat of a specific colour.

They can see other people's hats but not their own.

One of them shouted, "If you can see at least 6$\mathbf{6}$ yellow hats and at least 6$\mathbf{6}$ white hats, raise your glass!"

Exactly 12$\mathbf{12}$ people raised their glasses.

What is the possible value of N$N$that fits this case?

Solution

We know that there are 6$6$ people with yellow hats and 6$6$ people with white hats.

But they cannot see their hats, so there must be more people.

If there are 7+$7+$ people with a yellow hat and 7+$7+$ people with a white hat, then there are 14+$14+$ people who would raise theirglasses-- more than the actual 12$12$.

Therefore there must be 6$6$ people with a hat of one colour (say, yellow), and more than 6$6$ with a hat of the other colour (white). 

The people with a yellow hat can see only 5$5$ other yellow hats, so they don't raise their glasses.

But all people with a white hat will raise their glasses.

Therefore there are 12$12$ people with white hats.

Add the 6$6$ with a yellow hat, and we get a total of 18$18$.