At a competition, a group of 16 members is divided into 4 teams of 4 members each. The procedure of the competition is as follows:

Each team will be seated in a different room and after which every one will be given a cap either red or blue, i.e., total of 16 caps out of which 8 are red and 8 are blue and one from each room has to shout one of these words: Yes/No/OK and only once. They have to shout loud enough so that all other teams of their group can hear.

Then, one from each group(all 4 together) has to go to another room where they should tell the number of red and blue caps each of the other teams wore.

Before the competition started, all 16 of them are asked to devise a strategy together for every team to accomplish the task successfully. They are also told that they should not take the caps off their heads and should not talk to members of other teams until the competition is over.

Can you find out the strategy they used? It is known that all the 3 words have been shouted at least once and one of the guys said,"In this case, anyone one of four of us could have told you the distribution of hats even if we came here individually"?

Assume all their voices are indistinguishable.

(In reply to re(2): Possible solution by Praneeth)

Befor the competiton, each member is assigned a sequential number such that each individual knows his effective position compared to the others. The purpose of this is to identify which member of the groups is to shout; which member will meet the three others in the final declaration room; and an order for the chosen members to make their declarations.

When first divided and relocated in a room with with three other members, the member with the lowest assigned number looks at his team's caps. If he sees three red caps or three blue caps, he shouts "YES". If he sees two blue caps and one red cap, he shouts "NO". If he sees two red caps and one blue cap, he shouts "OK".

The member with the highest assigned number in the room, who shall meet in the final room, observes the shouting member's cap and notes the color.

If his team member shouted "YES" with a red cap , and the shouter is wearing a red cap, and he sees the other two members wearing red caps, he knows his team has four red caps.

If his team member  shouted "YES" with a red cap, and he sees the other two members wearing blue caps, he knows his team has three blue caps and one red cap.

If his team member shouted "YES" with a blue cap, and he sees the other two members wearing red caps, he knows his team has three red caps and one blue cap.

If his team member shouted "YES" with a blue cap, and he sees the other two members wearing blue caps, he knows his team has four blue caps.

If his team member shouted "NO" with a red cap, he knows his team has two blue caps and two red caps.

If his team member shouted "NO" with a blue cap, he knows his team has three blue caps and one red cap.

If his team member shouted "OK" with a red cap, he knows his team has three red caps and one blue cap.

If his team member shouted "OK" with a blue cap, he knows his team has two blue and two red caps.

In the final room -- as their is no rule or restriction given as to sequence of declaration -- each member, in assigned sequential order, declares first their own team's cap composition. Then, in same sequential order, each, then having heard the composition of each of the other teams, may declare the cap composition of  the other teams.

Posted by Dej Mar on 2008-07-04 11:53:40