GamesNovatory/Performance/For Adults/Hat Magic

A very special performance

This is probably the most dramatic and effective performance children can give to their parents.

With 2 players

Black hat in front
Choose 2 children and place them in front of the class,
          in file, one behind the other, sideways to the class, the back child on your left.
In front of the file, show everyone, participants and class, the 3 hats,
     and have everyone notice, and remember, that there are 2 white, and only 1 black.
Place the black hat on the head of the front child,
     and bring the 2 remaining white hats behind the back player,
          holding them well in view of the class.
Ask the back child -
     "Do you see the color of the hat on the front child ?"
     "Even if you cannot see them, do you know the color of the hats I am holding ?"
          Ask help from the class if required.
     "Do you already know the color of the hat that I would put on your head ?"
          Again, ask help from the class if required.
Ask the front child -
     "Now that the back child has said that he already knows the color of his hat,
          do you know the color of your hat and of his hat ?"
          Again, ask help from the class if required.

Recapitulate the process -
     1. Everyone knows that there are 2 white hats and only 1 black hat.
     2. If the back player sees the black hat, he knows he has a white hat
          because there are only white hats left over,
               and we have a w-b Hat Disposition (from back to front).
     3. Once the front player hears the back player answer "Yes",
          he knows that he must have the black hat which the back player sees,
               and we have a "Yes", "Yes" Answer Combination,
                    with the front player knowing the color of both hats.

White hat in front
Choose 2 different children and place them in front of the class.
This time, place a white hat on the head of the front child,
     and bring the 2 remaining hats behind the back player,
          holding them well in view of the class.
Ask the back child -
     "Do you see the color of the hat on the front child ?"
     "Even if you cannot see them, do you know the color of the hats I am holding ?"
          Ask help from the class if required.
     "Do you already know the color of the hat that I would put on your head ?"
          Again, ask help from the class if required.
Ask the front child -
     "Now that the front child has said that he does not know the color of his hat,
          do you know the color of your own hat ?"
          Again, ask help from the class if required.

Recapitulate the process -
     1. Everyone knows that there are 2 white hats and only 1 black hat.
     2. If the back player sees a white hat, he cannot know the color of his hat
          because there is both a white hat and a black hat left over,
               and we could have a b-w or a w-w Hat Disposition.
     3. Once the front player hears the back player answer "No",
          he knows that he must have one of the white hats which the back player sees,
               and we have a "No", "Yes" Answer Combination,
                    with only the front player knowing the color of his own hat.

     NOTE that, in both cases,
               wearing either a black hat or a white hat,
          the front player knows the color of his own hat.

We're ready for a performance !
1. Choose 2 new players and place them in front of the class.
2. Ask a third child to carefully place hats on their heads,
          so that the performers do not see their own hats,
     and hide the remaining hat behind the back player.
3. A fourth child could replace you and ask the questions -
     To the back child, "Do you know the color of your hat ?"
          the answer will be simply "Yes" or "No".
     To the front child, "Do you know the color of your hat ?"
          the answer will always be "Yes".
     To the front child, "What is the color of your hat ?"
          The class applauds the answer.
     Still to the front child, "Do you know the color of the hat behind you ?"
          the answer should be, "Only if the back child knows himself".
     Still to the front child (only if the back knew), "What is the color of his hat ?"
          The answer is evidently "White", but no applause.
     To the back child, "Is that the correct answer",
          which is evidently confirmed and then the class applauds.
4. Repeat at will with new performers.

With 3 players

Black hats in front
Choose 3 children and place them in front of the class
In front of the file, show everyone, participants and class, the 5 hats,
     and have everyone notice, and remember, that there are 3 white, and only 2 black.
Place the 2 black hats on the heads of the 2 front children,
     a white hat on the head of the back child,
          and bring the 2 remaining white hats behind the back player,
               holding them well in view of the class.
Ask the back child -
     "Do you see the color of the hats on the 2 front children ?"
     "Even if you cannot see them, do you know the color of the hats left over ?"
          Ask help from the class if required.
     "So you know the color of the hat on your head ?"
          Again, ask help from the class if required.
Ask the 2 front children -
     "Now that the back child has said that he knows the color of his hat,
          do you know the color of your hats and of his hat ?"
          Again, ask help from the class if required.

Recapitulate the process -
     1. Everyone knows that there are 2 white hats and only 1 black hat.
     2. If the back player sees the 2 black hats, he knows he has a white hat
          because there are only white hats left,
               and we have a w-b-b Hat Disposition.
     3. Once the front players hear the back player answer "Yes",
          they know that they must have the 2 black hats which the back player sees,
               and we have a "Yes", "Yes", "Yes" Answer Combination,
                    with all the players knowing the colors of all the hats.

White hat in the middle
Choose 3 different children and place them in front of the class.
This time, place the white hat on the head of the middle child,
     the 2 black hats on the heads of the other children,
          and bring the 2 remaining white hats behind the back player,
               holding them well in view of the class.
Ask the back child -
     "Do you see the color of the hats on the 2 front children ?"
     "Even if you cannot see them, do you know the color of the hats left over ?"
          Ask help from the class if required.
     "Do you know the color of the hat on your head ?"
          Ask help from the class if required.
Ask the middle child -
     "Now that the back child has said that he does not know the color of his hat,
          and that you see the color of the hat on the front child,
               what does that tell you ?"
          Again, ask help from the class if required.
     "Do you know the color of the hat on your head ?"
          Ask help from the class if required.
Ask the front child -
     "Now that the middle child has said that he knows the color of his hat,
               do you know the color of your own hat and of his ?"
          Again, ask help from the class if required.

Recapitulate the process -
     1. Everyone knows that there are 3 white hats and only 2 black hats.
     2. If the back player sees a white hat, he cannot know the color of his hat
          because there is at least 1 black hat left with the white hats,
               and we could have a b-w-b or a w-w-b Hat Disposition.
     3. Once the middle player hears the back player answer "No",
          he knows that there is a white hat in the 2 front hats,
          and since he sees a black hat on the front child,
               he knows that he has the white hat.
     4. Once the front child hears the middle child answer "Yes",
          he knows that he has a black hat,
          that the middle child has a white hat.
               and we have a "No", "Yes", "Yes" Answer Combination.

White hat in front
Choose 3 different children and place them in front of the class.
This time, place the white hat on the head of the front child,
     the 2 black hats on the heads of the back children,
          and bring the 2 remaining white hats behind the back player,
               holding them well in view of the class.
Ask the back child -
     "Do you see the color of the hats on the 2 front children ?"
     "Even if you cannot see them, do you know the color of the hats left over ?"
          Ask help from the class if required.
     "Do you know the color of the hat on your head ?"
          Again, ask help from the class if required.
Ask the middle child -
     "Do you see the color of the hat on the front child ?"
     "The back child has said that he does not know the color of his hat,
          what does that tell you ?"
          Again, ask help from the class if required.
     "Do you know the color of the hat on your head ?"
          Again, ask help from the class if required.
Ask the front child -
     "Now both players behind you said they do not know the color of their hats,
               do you know the color of your own hat ?"
          Again, ask help from the class if required.

Recapitulate the process -
     1. Everyone knows that there are 3 white hats and only 2 black hats.
     2. If the 2 back players see a white hat, they cannot know the color of their hats
          because there will be at least 1 black hat left with the white hats.
     3. Once the front child hears the other 2 answer "No", "No",
          he knows that he has the white hat
               which made the others say "No",
                    and we have a "No", "No", "Yes" Answer Combination.
          He only knows the color of his own hat.
     4. This is the specific Answer Combination involved in the original Enigma.

     NOTE that, in the three cases,
               with the "Yes", "Yes", "Yes", the "No", "Yes", "Yes",
                    or the "No", "No", "Yes" Answer Combinations,
          the front player always knows the color of his own hat,
               whether he is wearing either a black hat or a white hat.

We're ready for a performance !
1. Choose 3 players and place them in front of the class.
2. Ask a fourth child to carefully place hats on their heads,
          so that the players do not see their own hats,
     and hide the remaining hats behind the back player.
3. A fifth child could replace you and ask the questions -
     To the back player, "Do you know the color of your hat ?"
          the answer will be simply "Yes" or "No".
     To the middle player, "Do you know the color of your hat ?"
          the answer will be simply "Yes" or "No".
     To the front player, "Do you know the color of your hat ?"
          the answer will always be "Yes".
     To the front child, "What is the color of your hat ?"
          The class applauds the answer.
     Still to the front child, "Do you know the color of the hats behind you ?"
          the answer should be, "Only if the player knows".
     Still to the front child (only if the player behind knew), "What is the color of his hat ?"
          Hold the applause.
     To the player behind, "Is that the correct answer",
          which is evidently confirmed and the class applauds.
     Repeat for the other player if required.
4. Repeat the whole performance at will with new players.

We're ready for an audience !
Even though the 4- and 5-player game is more impressive
          and more difficult to decypher,
     the 3-player game already makes a very good show,
               especially when performed by 5- and 6-year-olds,
          and will leave the parents quite amazed and perplexed.

With 4 players

Indications can now be far more condensed -
     1. Show the 4 w hats and the 3 b hats to the 4 players.
     2. Use the 3 bs and 1 w for all the exercises.
     3. When the w is in the back,
          Answers will be "YYYY", and Colors will be wbbb.
     4. When the w is in a central position,
          Answers will be "NYYY", and colors will be ?wbb, or
          Answers will be "NNYY", and colors will be ??wb.
     5. When the w is in the front,
          Answers will be "NNNY", and colors will be ???w.
               (? evidently represents either of the 2 colors.)

Recapitulate the process -
1. The first player to see only bs knows he has a w, and
     his "Y" tells all those in front that they have bs.
2. The front player must remember which player was the first to say "Y" and have a w,
     all the others ahead having bs.

The performance is evidently as it was for 3 players.

With 5 players

Another condensed set of indications -
     1. Show the 5 w hats and the 4 b hats to the 5 players.
     2. Use the 4 bs and 1 w for all the exercises.
     3. When the w is in the back,
          Answers will be "YYYYY", and Colors will be wbbbb.
     4. When the w is in one of the central positions,
          Answers will be "NYYYY", and colors will be ?wbbb,
          Answers will be "NNYYY", and colors will be ??wbb, or
          Answers will be "NNNYY", and colors will be ???wb.
     5. When the w is in the front,
          Answers will be "NNNNY", and colors will be ????w.

The recapitulation is the same -
1. The first player to see only bs knows he has a w, and
     his "Y" tells all those in front that they have bs.
2. The front player must remember which player was the first to say "Y" and have a w,
     all the others ahead having bs.

The performance is evidently as it was for 3 players.