the riddle isn't available any more, come to see answer

[HomieDolphine]

The prize isn't available anymore. DP of the tokens will be at 15:00 today (server time)
the answer: The final answer is (2N)!, which is equal to 1*2*3*...*(2N-1)*2N. the explanation: Firstly decide which guests get the white hats. we have 2N guests and N white hats, so there are 2NcN ways to do this, which is ((2N)!/(N!)^2). Now in order to define the circles, we will define two functions which are onto and one to one between the group of people with white hat and the group of people with black hat (make sure you understand why defining those functions gives all the possible circles arrangements and every arrangement is given exactly one time). The first function will be for every person with white hat to decide excactly which person with black hat will stand to his right, there are N! ways to define this function. The second function will be for every person with black hat the decide exactly which person with white hat will stand to this right, there are N! ways to define this function. so we have ((2N)!/(N!)^2)*(N!)^2=(2N)!. Final answer: there are (2N)! ways.

 

[HomieDolphine]

lets just say N is 2 it means there are 4 guests and maximum of 2 circles then there can be 2 arrangements. then if N is 3 then there can be a maximum of 3 circles then there can be 3 arrangements

so the answer is N?

i know im completely wrong but i tried for fun

This isn't correct. In every circle, you can rearrange the people inside it (but how many arrangements are there for a circle with N people? Try to find it yourself). Also, you can rearrange the peoppe between the circles. For example if our people are a,b,c,d so 2 different arrangements are (a, b) + (c,d) and (a, c) + (b, d). Ofc their hats are black and white in same circle.
So 2 things not taken into account:
1) inside circle arrangements
2) alternative people between circles

 

[HomieDolphine]

i mean, there is an n number of guests so you're saying that there are the same number of circles as guests so i dont think the ans is correct, but then again i am not the greatest at math.

If you wanna take an hint: the answer is way more then N.

 

[Koshi]

just gonna assure you that you're not gonna find an someone that can solve imo problems playing on a cracked minecraft server

this is probably why your grammar is like this

 

[HomieDolphine]

Another major hint was added.
Good luck!

 

[AncestorGamingYT]

whoever answers this riddle get 6m prison tokens:

John is having a birthday, and exactly 2N guests have arrived to John's birthday. John has N white hats and N black hats. He want his guests to dance in a number of circles (one circle or more), such that every circle contains at least 2 people, and people with the same hat color will not be nearby each other (but they can be in the same circle). Assume that all of the guests can be distinguished, and all of the hats with the same color are exactly the same, therefore, rearranging all of the white hats will not count as another arrangement and rearranging the people WILL count as another arrangement . In how many ways can John arrange this? Show your steps.

reminder: a correct answer without the steps will NOT count as solving the riddle.
gl to everybody.
note 1: I assure you that you can't find the answer on the internet. the riddle is made of another riddle that was at a certain mathematics olympiad, and has been slightly changed.
note 2: the prize was 2m tokens, now its 6m tokens.
note 3: now the prize is 15m tokens
Hint 1: the final answer is going to involve N. for example N! (1*2*3*...*N) or N^2 could be valid answers (and one of those might be correct, who knows)
Hint 2: An important part of the solution would be considering all of the combinations to assign N guests to the N white hats, and the rest to the black hats. There are exactly 2nCn ways to do it, or (2N)!/(N!)², for example if N=3 then (2N)!=6!=720, 3!=6, so there are 720/6²=20 ways to assign the guests to white hats.

tell me the time limit for it

 

[HomieDolphine]

tell me the time limit for it

The prize will be available until reset

 

[AncestorGamingYT]

The prize will be available until reset

aight thanks imma try doing it XD

 

[Citan]

420N

420-1=419
419x1=419
419+1=420
420/1=420
this is too ez

 

[htsdfisy]

This isn't correct. In every circle, you can rearrange the people inside it (but how many arrangements are there for a circle with N people? Try to find it yourself). Also, you can rearrange the peoppe between the circles. For example if our people are a,b,c,d so 2 different arrangements are (a, b) + (c,d) and (a, c) + (b, d). Ofc their hats are black and white in same circle.
So 2 things not taken into account:
1) inside circle arrangements
2) alternative people between circles

so its N^2 then? because if 4 people was in the party then there would be 16 arrangements right?

 

[Mixu_]

i didn't read this but... john. thats my answer.