That's the other part of the puzzle
. Plasmid was very close, the answer to that part just needs to be refined to fit the specified rules laid down by his oh-so-munificent majesty. ;P
Well, here's are some strategies
Spoiler for general strategy
Suppose that whenever a participant raises his hand, the other 4 participant would be aware of this fact. Here's a general strategy that would allow all 5 participants to win.
I) After the first gong sounding, participant 1, who is the closest one to the king, is to raise his hand if he sees 2 black hats. Everyone should be able to figure their hat colors if this happens.
II) If the second gong sounds and participant 1 didn't raise his hand, then there are four classes of hat patterns, which are listed in the drawing below. (Note that the classes of patterns are listed as A, B, C, and D, each of which may include mirror images of one another along the vertical axis. For instance, in class A, the hat colors of participants 2a and 2b can be reversed without consequence on the following strategy).
III) After the second gong sounding, participants 2a and 2b are each supposed to raise his hand if he or she sees 2 hats of the same colors (2 whites or 2 blacks). Let's say that 2a raises his hand after the second gong sounds, then the hat patterns fall into class A or B. Participant 2b would then be able to figure out which pattern it is, and he needs to convey this information to participants 1, 3a, and 3b.
III-a) If the pattern is A, then participant 2b should raise his hand after participant 2a, but before the third gong sounding.
III-b) If the pattern is B, then participant 2b should raise his hand after the third gong sounding. Everyone else should be able to figure out his hat color. Participants 3a and 3b may need to observe each other's hat before being able to calculate their own color.
IV) If the third gong sounds and no one has raised his hand yet, then the hat pattern is either C or D. At this point, participants 2a and 2b would know which is the true pattern and they can convey it to the rest with the following:
IV-a) If the pattern is C, then participant 2a should raise his hand after the third gong.
IV-b) If the pattern is D, then participant 2b should raise his hand after the third gong. Everyone else should be able to figure out his color.
Suppose that whenever a participant raises his hand, the other 4 participant would be aware of this fact. Here's a general strategy that would allow all 5 participants to win.
I) After the first gong sounding, participant 1, who is the closest one to the king, is to raise his hand if he sees 2 black hats. Everyone should be able to figure their hat colors if this happens.
II) If the second gong sounds and participant 1 didn't raise his hand, then there are four classes of hat patterns, which are listed in the drawing below. (Note that the classes of patterns are listed as A, B, C, and D, each of which may include mirror images of one another along the vertical axis. For instance, in class A, the hat colors of participants 2a and 2b can be reversed without consequence on the following strategy).
III) After the second gong sounding, participants 2a and 2b are each supposed to raise his hand if he or she sees 2 hats of the same colors (2 whites or 2 blacks). Let's say that 2a raises his hand after the second gong sounds, then the hat patterns fall into class A or B. Participant 2b would then be able to figure out which pattern it is, and he needs to convey this information to participants 1, 3a, and 3b.
III-a) If the pattern is A, then participant 2b should raise his hand after participant 2a, but before the third gong sounding.
III-b) If the pattern is B, then participant 2b should raise his hand after the third gong sounding. Everyone else should be able to figure out his hat color. Participants 3a and 3b may need to observe each other's hat before being able to calculate their own color.
IV) If the third gong sounds and no one has raised his hand yet, then the hat pattern is either C or D. At this point, participants 2a and 2b would know which is the true pattern and they can convey it to the rest with the following:
IV-a) If the pattern is C, then participant 2a should raise his hand after the third gong.
IV-b) If the pattern is D, then participant 2b should raise his hand after the third gong. Everyone else should be able to figure out his color.
Minor details about the fish lines
Spoiler for details
In the general strategy, we assumed that whenever a participant raises his hand, the other 4 participant would be aware of this fact. The information in the OP doesn't specify whether this assumption is true or not.
1) If the assumption is true, things are all well and good.
2) If the assumption is not true, then simply use the fishlines to attach each participant to the other four. For instance, we could attach four lines from participant 1 to the thumbs of each of the other four, then attach four lines from participant 2a to the index finger of each of the other 4, and so on. Whenever a participant raises his hand, simply hold on to the fishlines as he or she raises the hand, and the other 4 would know who raised the hand.
In the general strategy, we assumed that whenever a participant raises his hand, the other 4 participant would be aware of this fact. The information in the OP doesn't specify whether this assumption is true or not.
1) If the assumption is true, things are all well and good.
2) If the assumption is not true, then simply use the fishlines to attach each participant to the other four. For instance, we could attach four lines from participant 1 to the thumbs of each of the other four, then attach four lines from participant 2a to the index finger of each of the other 4, and so on. Whenever a participant raises his hand, simply hold on to the fishlines as he or she raises the hand, and the other 4 would know who raised the hand.
Edited by bushindo, 13 May 2012 - 06:14 AM.
