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Find the Strategy (Posted on 2008-07-02) Difficulty: 2 of 5
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.

See The Solution Submitted by Praneeth    
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Question A question regarding the 'official solution' | Comment 14 of 15 |

I am a loss in the 'official solutions' explanation as to how a team member can identify the quantity of each teams red and blue caps. 

From what I understand from the explanation (please tell me if I misunderstood), if the quantity of red is four or one, a member with a red cap goes to the central room; if the quantity of blue is four or one, a member with a blue cap goes to the central room. What color cap does the team member representing two red caps and two blue caps wear?

There are four possible combinations where "Yes" would indicate Red > Blue, "No" would indicate Blue > Red, and "OK" would indicate Red = Blue, such that each "Yes", "No" and "OK" were shouted at least once:

4 Red, 0 Blue -- "Yes"
2 Red, 2 Blue -- "OK"
2 Red, 2 Blue -- "OK"
0 Red, 4 Blue -- "No"

3 Red, 1 Blue -- "Yes"
2 Red, 2 Blue -- "OK"
2 Red, 2 Blue -- "OK"
1 Red, 3 Blue -- "No" 

3 Red, 1 Blue -- "Yes"
3 Red, 1 Blue -- "Yes"
2 Red, 2 Blue -- "OK"
0 Red, 4 Blue -- "No"

4 Red, 0 Blue -- "Yes" 
2 Red, 2 Blue -- "OK" 
1 Red, 3 Blue -- "No"
1 Red, 3 Blue -- "No"

Your explanation did not seem to address how each member meeting in the final room would be able to identify the quantity of caps of each individual team. 

Edited on August 25, 2008, 12:58 pm
  Posted by Dej Mar on 2008-08-25 06:52:08

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