Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Tennie's card is shown to Harold. Harold's card is shown to Tennie. Tennie's card is shown to Earl. It is publicly announced that someone picked a red card. It is publicly announced that Earl does not know whether or not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:1 Agentc:2 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether (1&2&3)) ] Agenta knows that (Agentb knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2 Agentb:1 Agentc:2 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether (1&2&3)) ] 0

Hypothesis: Harold can now know that Tennie can know that everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentc knows whether (Agentd knows that 4)

Hypothesis: Herman can now know whether Andrea can know that Andrea's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ralph's card is shown to Missy. Sean's card is shown to Missy. Sean's card is shown to Ralph. William's card is shown to Sean. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:3,4 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] Agentb knows that (Agentc knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:3,4 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: William can now know that Ralph can know whether Ralph picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jane knows whether Robert's forehead is muddy. It is publicly announced that Robert knows whether Jane's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] [ ! (Agentb knows whether 1) ] (Agentc knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Maria can now know that nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Bridget's card is shown to Carol. Carol's card is shown to Bridget. It is publicly announced that someone picked a red card. It is publicly announced that Carol does not know whether or not everyone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] Agentb knows that (Agenta knows whether 1)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] 0

Hypothesis: Bridget can now know that Carol can know whether Carol picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Danny's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! 2 ] (Agentc knows whether (~(1|2|3)))

Hypothesis: John can now know whether or not nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] Agentc knows whether (Agenta knows whether 2)

Hypothesis: Irene can now know whether Frank can know whether Gregory's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that someone is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (1|2) ] Agentb knows whether (Agenta knows whether 2)

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (1|2) ] 0

Hypothesis: Justin can now know whether Ida can know whether Justin is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] Agenta knows that (Agentb knows that 2)

Hypothesis: Brad can now know that Helen can know that Helen's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Dion's card is shown to Ethel. Jennifer's card is shown to Ethel. Jennifer's card is shown to Dion. Ethel's card is shown to Jennifer. Dion's card is shown to Jennifer. Ethel's card is shown to Maynard. It is publicly announced that someone picked a red card. It is publicly announced that Maynard knows whether someone picked a red card. It is publicly announced that Jennifer does not know whether Jennifer picked a red card. It is publicly announced that Jennifer knows whether or not everyone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3 Agentb:3 Agentc:1,2 Agentd:1 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! ~(Agentc knows whether 3) ] [ ! (Agentc knows whether (1&2&3&4)) ] Agentc knows that (Agentb knows whether (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3 Agentb:3 Agentc:1,2 Agentd:1 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! ~(Agentc knows whether 3) ] [ ! (Agentc knows whether (1&2&3&4)) ] 0

Hypothesis: Jennifer can now know that Dion can know whether or not everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that John's forehead is not muddy. It is publicly announced that Della knows whether Della's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(3) ] [ ! (Agentb knows whether 2) ] Agenta knows that (Agentb knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(3) ] [ ! (Agentb knows whether 2) ] 0

Hypothesis: Chelsea can now know that Della can know whether Chelsea's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] (Agentb knows that 2)

Hypothesis: Frank can now know that Frank's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Susan's card is shown to Audrey. Susan's card is shown to Dorothy. Audrey's card is shown to Dorothy. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:1,2 VALID? [ ! (1|2|3) ] Agentb knows whether (Agenta knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:1,2 VALID? [ ! (1|2|3) ] 0

Hypothesis: Audrey can now know whether Susan can know whether Audrey picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Emilie does not know whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] Agentc knows whether (Agentb knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Dorothy can now know whether Lyman can know that Dorothy is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Lois does not know whether Sherry is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 3) ] (Agentc knows whether (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: Sherry can now know whether or not nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jennifer's card is shown to Darlene. Darlene's card is shown to Jennifer. Clorinda's card is shown to Jennifer. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:1,3 VALID? [ ! (1|2|3) ] (Agentb knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2 Agentb:1,3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Jennifer can now know whether or not everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Robert knows whether Angel's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (1&2&3) ] [ ! (Agenta knows whether 3) ] Agentc knows whether (Agenta knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (1&2&3) ] [ ! (Agenta knows whether 3) ] 0

Hypothesis: Angel can now know whether Robert can know whether Robert's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Leopoldo does not know whether or not everyone's forehead is muddy. It is publicly announced that Linda knows whether someone's forehead is muddy. It is publicly announced that Leopoldo does not know whether or not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] [ ! (Agentb knows whether (1|2)) ] [ ! ~(Agenta knows whether (1&2)) ] [ ! (~(1&2)) ] Agentb knows that (Agenta knows whether 1)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] [ ! (Agentb knows whether (1|2)) ] [ ! ~(Agenta knows whether (1&2)) ] [ ! (~(1&2)) ] 0

Hypothesis: Linda can now know that Leopoldo can know whether Leopoldo's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Victoria knows whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] Agenta knows whether (Agentb knows whether (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] 0

Hypothesis: Victoria can now know whether Truman can know whether or not nobody is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Maryjane's card is shown to Charles. Charles's card is shown to Maryjane. It is publicly announced that someone picked a red card. It is publicly announced that Charles knows whether someone picked a red card. It is publicly announced that Charles picked a black card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (1|2)) ] [ ! ~(1) ] (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (1|2)) ] [ ! ~(1) ] 0

Hypothesis: Maryjane can now know that Maryjane picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Catherine is not thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(3) ] Agenta knows whether (Agentc knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(3) ] 0

Hypothesis: Roger can now know whether Catherine can know that nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that someone is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Shalonda knows whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (~(1&2&3)) ] [ ! (Agenta knows whether (1&2&3)) ] (Agentb knows whether 3)

Hypothesis: Katina can now know whether Werner is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Peter's card is shown to Karen. Karen's card is shown to Peter. Peter's card is shown to Maryann. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:1 Agentc:2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] (Agentc knows that (1|2|3))

Hypothesis: Maryann can now know that someone picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Marjorie's forehead is muddy. It is publicly announced that Holly does not know whether Holly's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] [ ! ~(Agenta knows whether 1) ] Agentb knows that (Agenta knows whether (1&2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Marjorie can now know that Holly can know whether or not everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Sarah's card is shown to Lilly. Sarah's card is shown to Donna. It is publicly announced that someone picked a red card. It is publicly announced that Lilly does not know whether Lilly picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 1) ] Agenta knows that (Agentb knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Lilly can now know that Sarah can know that nobody picked a red card.


VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] Agenta knows whether (Agentc knows that 3)

Hypothesis: Michelle can now know whether Manuel can know that Manuel's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Roseanne's card is shown to Wm. Wm's card is shown to Bradley. Wm's card is shown to Roseanne. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] 0

Hypothesis: Wm can now know that Wm picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Linda knows whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] [ ! (Agentb knows whether (1&2&3&4)) ] Agentc knows that (Agenta knows whether (1|2|3|4))

Hypothesis: Chelsea can now know that Darlene can know whether someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Sonya does not know whether or not everyone is thirsty. It is publicly announced that Erica does not know whether Terri is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agentc knows whether 1) ] (Agentb knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agentc knows whether 1) ] 0

Hypothesis: Sonya can now know whether Sonya is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] Agenta knows that (Agentc knows whether (1&2&3&4))

Hypothesis: Jessica can now know that Joshua can know whether or not everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] (Agentc knows whether 3)

Hypothesis: Darryl can now know whether Darryl's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Chrystal knows whether Mary's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] Agentc knows that (Agenta knows that 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Jerome can now know that Mary can know that Jerome's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Dennis knows whether someone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether (1|2)) ] (Agentb knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether (1|2)) ] 0

Hypothesis: Dennis can now know whether or not nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Dawn does not know whether or not everyone is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] (Agentb knows whether (1|2))

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] 0

Hypothesis: Alan can now know whether someone is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Susan's card is shown to Candace. Candace's card is shown to Susan. It is publicly announced that someone picked a red card. It is publicly announced that Candace picked a black card. It is publicly announced that not everyone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(1) ] [ ! (~(1&2)) ] Agentb knows whether (Agenta knows whether 1)

Hypothesis: Susan can now know whether Candace can know whether Candace picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Dewey's card is shown to Luz. Gwendolyn's card is shown to Luz. Luz's card is shown to Gwendolyn. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,4 Agentb:1 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,4 Agentb:1 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Luz can now know whether Dewey can know whether Calvin picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Natalia does not know whether or not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether (1&2&3)) ] (Agentc knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether (1&2&3)) ] 0

Hypothesis: Donnie can now know that Natalia's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Phillip's card is shown to Kellie. Danny's card is shown to Phillip. Kellie's card is shown to Phillip. It is publicly announced that someone picked a red card. It is publicly announced that Kellie does not know whether Phillip picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] (Agenta knows whether 3)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] 0

Hypothesis: Danny can now know whether Phillip picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Thomas knows whether Thomas's forehead is muddy. It is publicly announced that Dennis knows whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 2) ] [ ! (Agenta knows whether (1&2)) ] Agentb knows that (Agenta knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 2) ] [ ! (Agenta knows whether (1&2)) ] 0

Hypothesis: Thomas can now know that Dennis can know whether or not nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Steve does not know whether Steve's forehead is muddy. It is publicly announced that William's forehead is not muddy. It is publicly announced that Steve's forehead is not muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 2) ] [ ! ~(4) ] [ ! ~(2) ] Agentc knows whether (Agentd knows that 4)

Hypothesis: Sharell can now know whether William can know that William's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Carla's card is shown to Josefina. Carla's card is shown to Debbie. Josefina's card is shown to Carla. Josefina's card is shown to Douglass. Debbie's card is shown to Douglass. It is publicly announced that someone picked a red card. It is publicly announced that Douglass does not know whether Carla picked a red card. It is publicly announced that Carla does not know whether Carla picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:3 Agentb:3 Agentc:1 Agentd:1,2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agentc knows whether 3) ] (Agentc knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:3 Agentb:3 Agentc:1 Agentd:1,2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Carla can now know that everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Humberto does not know whether Elisabeth is thirsty. It is publicly announced that Robert does not know whether Elisabeth is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agenta knows whether 2) ] Agentb knows whether (Agenta knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agenta knows whether 2) ] 0

Hypothesis: Elisabeth can now know whether Robert can know whether Emmanuel is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Donald's card is shown to Robert. Maria's card is shown to Robert. Robert's card is shown to Donald. It is publicly announced that someone picked a red card. It is publicly announced that Donald does not know whether or not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether (1&2&3)) ] (Agentb knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether (1&2&3)) ] 0

Hypothesis: Donald can now know whether Donald picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Paul's forehead is muddy. It is publicly announced that James does not know whether James's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] [ ! ~(Agenta knows whether 1) ] Agentb knows that (Agenta knows that 1)

Hypothesis: Paul can now know that James can know that James's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jared's card is shown to Briana. Briana's card is shown to Samuel. Briana's card is shown to Jared. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] (Agentb knows whether 2)

Hypothesis: Samuel can now know whether Samuel picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Teresa does not know whether Andrew is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] (Agentc knows that (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] 0

Hypothesis: Candice can now know that nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Michael's card is shown to Gerald. Vicki's card is shown to Gerald. Gerald's card is shown to Roger. Michael's card is shown to Roger. Gerald's card is shown to Vicki. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,4 Agentb:1,2 Agentc:1 VALID? [ ! (1|2|3|4) ] (Agentc knows that 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,4 Agentb:1,2 Agentc:1 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Roger can now know that Vicki picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] Agenta knows that (Agentc knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Larry can now know that Julian can know whether or not everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Marshall knows whether Duane's forehead is muddy. It is publicly announced that Duane does not know whether or not everyone's forehead is muddy. It is publicly announced that Duane does not know whether Duane's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether 3) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] [ ! ~(Agentc knows whether 3) ] [ ! (~(1&2&3&4)) ] Agentd knows whether (Agenta knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether 3) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] [ ! ~(Agentc knows whether 3) ] [ ! (~(1&2&3&4)) ] 0

Hypothesis: Marshall can now know whether Joseph can know that Joseph's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Clara knows whether or not nobody's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (~(1|2|3|4))) ] Agenta knows whether (Agentd knows that 2)

Hypothesis: Erich can now know whether Clara can know that Delbert's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] Agenta knows that (Agentd knows that (~(1|2|3|4)))

Hypothesis: Rudolph can now know that Brian can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Mark knows whether Nichole's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] Agentd knows that (Agentc knows that (1|2|3|4))

Hypothesis: Melissa can now know that Kim can know that someone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Bertha does not know whether Bertha's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] (Agenta knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Bertha can now know that everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Clayton's card is shown to Earnestine. Nathan's card is shown to Earnestine. Earnestine's card is shown to Clayton. Nathan's card is shown to Lorenza. Earnestine's card is shown to Nathan. Lorenza's card is shown to Nathan. It is publicly announced that someone picked a red card. It is publicly announced that Earnestine knows whether or not nobody picked a red card. It is publicly announced that everyone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,4 Agentb:1 Agentc:4 Agentd:1,3 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (~(1|2|3|4))) ] [ ! (1&2&3&4) ] Agentb knows whether (Agentc knows whether (~(1|2|3|4)))

Hypothesis: Clayton can now know whether Lorenza can know whether or not nobody picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Rita knows whether someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (1|2|3|4)) ] Agentd knows whether (Agentb knows that (~(1|2|3|4)))

Hypothesis: Maxine can now know whether Rita can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentd knows that (Agentb knows that 2)

Hypothesis: Terry can now know that James can know that James's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jose's card is shown to Michelle. Jose's card is shown to Karen. Jane's card is shown to Karen. Jane's card is shown to Jose. Michelle's card is shown to Jane. Jose's card is shown to Jane. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card. It is publicly announced that Jane does not know whether Karen picked a red card. It is publicly announced that Michelle does not know whether Karen picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:3 Agentb:3,4 Agentc:4 Agentd:1,3 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agenta knows whether 2) ] Agenta knows that (Agentb knows that 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:3 Agentb:3,4 Agentc:4 Agentd:1,3 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agenta knows whether 2) ] 0

Hypothesis: Michelle can now know that Karen can know that Jose picked a red card.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Roger knows whether or not nobody's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (~(1|2))) ] (Agentb knows that (1&2))

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (~(1|2))) ] 0

Hypothesis: Ronald can now know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Scott knows whether or not nobody's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (~(1|2|3))) ] (Agentc knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (~(1|2|3))) ] 0

Hypothesis: David can now know that Scott's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that James knows whether someone's forehead is muddy. It is publicly announced that Carrie knows whether Steven's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! (Agenta knows whether 2) ] (Agentc knows whether (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: James can now know whether or not nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jonna's card is shown to Ana. Jonna's card is shown to Misty. Ana's card is shown to Jonna. Misty's card is shown to Jonna. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:3 Agentc:1,2 VALID? [ ! (1|2|3) ] (Agenta knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:3 Agentc:1,2 VALID? [ ! (1|2|3) ] 0

Hypothesis: Ana can now know whether Ana picked a red card.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Eva knows whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (1&2)) ] (Agenta knows that (1|2))

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (1&2)) ] 0

Hypothesis: Eva can now know that someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that William knows whether Steven's forehead is muddy. It is publicly announced that William does not know whether William's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agenta knows whether 1) ] (Agentc knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Nancy can now know that Steven's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Gretchen's card is shown to Jill. Gretchen's card is shown to Dinah. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:3 VALID? [ ! (1|2|3) ] Agenta knows whether (Agentb knows that (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Jill can now know whether Dinah can know that someone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Mary knows whether Dwayne's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] [ ! (Agentc knows whether 1) ] Agentc knows that (Agenta knows whether 4)

Hypothesis: Mary can now know that Dwayne can know whether Joshua's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] (Agentc knows that (1&2&3))

Hypothesis: Shirley can now know that everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Eric does not know whether Natalie is thirsty. It is publicly announced that George knows whether someone is thirsty. It is publicly announced that Milton does not know whether or not everyone is thirsty. It is publicly announced that Milton does not know whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] Agentc knows that (Agenta knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] 0

Hypothesis: Eric can now know that Milton can know whether Eric is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jeanne knows whether Jeanne's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether 4) ] (Agentd knows whether (~(1|2|3|4)))

Hypothesis: Jeanne can now know whether or not nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jonathan does not know whether or not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agentb knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Lloyd can now know whether or not everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Vincent's card is shown to Cheryl. Cheryl's card is shown to Vincent. Cheryl's card is shown to Mary. It is publicly announced that someone picked a red card. It is publicly announced that Mary knows whether someone picked a red card. It is publicly announced that Vincent picked a black card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] [ ! ~(2) ] (Agentc knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] [ ! ~(2) ] 0

Hypothesis: Mary can now know that Vincent picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Brenda does not know whether Ronald is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 2) ] (Agentc knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 2) ] 0

Hypothesis: Jessica can now know that everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Helen does not know whether John is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] Agentc knows that (Agentb knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Helen can now know that John can know whether or not everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] Agenta knows whether (Agentc knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] 0

Hypothesis: Paul can now know whether Bernadette can know that Emma's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Martha knows whether Martha's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] Agenta knows whether (Agentb knows that 2)

Hypothesis: Mary can now know whether Lajuana can know that Lajuana's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Daniel is not thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(2) ] Agentd knows that (Agenta knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(2) ] 0

Hypothesis: Ashley can now know that Lawrence can know that Lawrence is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). John's card is shown to Derek. Derek's card is shown to John. It is publicly announced that someone picked a red card. It is publicly announced that Derek picked a red card. It is publicly announced that John does not know whether or not everyone picked a red card. It is publicly announced that John does not know whether or not everyone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 1 ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 1 ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] 0

Hypothesis: John can now know that John picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Anna knows whether someone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (1|2)) ] Agenta knows that (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (1|2)) ] 0

Hypothesis: Anna can now know that Teresa can know that Teresa's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] (Agentc knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Lloyd can now know that nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Chris does not know whether Jo is thirsty. It is publicly announced that everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 1) ] [ ! (1&2&3) ] (Agenta knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 1) ] [ ! (1&2&3) ] 0

Hypothesis: Jo can now know that Paula is thirsty.


VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] Agentb knows whether (Agentc knows whether 3)

Hypothesis: Mae can now know whether Maria can know whether Maria's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ernesto's card is shown to Kate. Carrie's card is shown to Jamel. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:4 Agentb:3 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows that 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:4 Agentb:3 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Kate can now know whether Jamel can know that Ernesto picked a red card.


VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] Agentb knows whether (Agentc knows that (1|2|3))

Hypothesis: Nicole can now know whether Antione can know that someone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lori does not know whether Lori's forehead is muddy. It is publicly announced that Nicole knows whether someone's forehead is muddy. It is publicly announced that Thomas does not know whether or not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] [ ! (~(1&2&3&4)) ] (Agentc knows that 4)

Hypothesis: Nicole can now know that Thomas's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Jennifer does not know whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] Agenta knows whether (Agentc knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] 0

Hypothesis: Katherine can now know whether Jennifer can know whether Jennifer is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lillian knows whether or not nobody's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (~(1|2|3|4))) ] Agentd knows whether (Agentb knows whether (1&2&3&4))

Hypothesis: Lillian can now know whether Ruby can know whether or not everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Helen's card is shown to Lisa. Brian's card is shown to Lisa. Helen's card is shown to Brian. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1,3 Agentb:1 VALID? [ ! (1|2|3) ] (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1,3 Agentb:1 VALID? [ ! (1|2|3) ] 0

Hypothesis: Helen can now know that Helen picked a red card.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Racheal's card is shown to Vera. Vera's card is shown to Racheal. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card.

Hypothesis: Vera can now know that someone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] (Agenta knows whether 4)

Hypothesis: Faye can now know whether Harold's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Sandra knows whether someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (1|2|3|4)) ] Agentc knows that (Agentd knows that (~(1|2|3|4)))

Hypothesis: Glen can now know that Marie can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Richard does not know whether Cleo is thirsty. It is publicly announced that Betty does not know whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] Agentb knows whether (Agenta knows whether (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] 0

Hypothesis: Angela can now know whether Richard can know whether or not nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Maria's card is shown to Julia. Carman's card is shown to Julia. Elizabeth's card is shown to Julia. Carman's card is shown to Maria. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:3 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows that 1)

Hypothesis: Julia can now know whether Maria can know that Julia picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Tara knows whether or not nobody's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (~(1|2|3))) ] (Agentb knows whether (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (~(1|2|3))) ] 0

Hypothesis: Tara can now know whether someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lisa knows whether or not everyone's forehead is muddy. It is publicly announced that Lisa does not know whether Lisa's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether 2) ] (Agentc knows that (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether 2) ] 0

Hypothesis: Helen can now know that someone's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Spencer knows whether or not nobody's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (~(1|2))) ] Agenta knows whether (Agentb knows that 1)

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (~(1|2))) ] 0

Hypothesis: Spencer can now know whether Charles can know that Spencer's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). George's card is shown to Rachel. Rachel's card is shown to George. It is publicly announced that someone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] Agenta knows whether (Agentb knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] 0

Hypothesis: Rachel can now know whether George can know whether or not nobody picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] Agentb knows that (Agenta knows that 3)

Hypothesis: Joshua can now know that Frank can know that Pamela's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Greta does not know whether or not everyone's forehead is muddy. It is publicly announced that Greta does not know whether or not everyone's forehead is muddy. It is publicly announced that Robin knows whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! (Agenta knows whether (1&2)) ] (Agentb knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! (Agenta knows whether (1&2)) ] 0

Hypothesis: Greta can now know whether or not nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Kurtis does not know whether or not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] Agentc knows whether (Agentb knows whether 3)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Billy can now know whether Felipe can know whether Billy's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Alvin knows whether someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1|2|3)) ] (Agenta knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1|2|3)) ] 0

Hypothesis: Alvin can now know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] (Agenta knows whether 3)

Hypothesis: Robbie can now know whether Eddy is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Paul does not know whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] Agentd knows whether (Agentb knows that 4)

Hypothesis: Donald can now know whether Paul can know that Donald's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Charles's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! 1 ] Agentb knows whether (Agenta knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! 1 ] 0

Hypothesis: Ella can now know whether Charles can know that Ella's forehead is muddy.


VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] Agenta knows that (Agentb knows whether 1)

Hypothesis: Carla can now know that William can know whether Carla's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentc knows that (Agentb knows that 2)

Hypothesis: Ronald can now know that Wesley can know that Wesley's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Joyce does not know whether Annette is thirsty. It is publicly announced that Lisa does not know whether Jeffrey is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 4) ] Agentb knows that (Agentc knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 4) ] 0

Hypothesis: Joyce can now know that Annette can know that everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Darlene does not know whether Kelly is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] (Agentc knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] 0

Hypothesis: Terry can now know whether Darlene is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Brian does not know whether Elizabeth is thirsty. It is publicly announced that Brian is not thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(3) ] (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(3) ] 0

Hypothesis: Raymond can now know that Raymond is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jana's card is shown to Ronnie. Edgar's card is shown to Jana. Jana's card is shown to Edgar. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:3 Agentc:2 VALID? [ ! (1|2|3|4) ] (Agentc knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:3 Agentc:2 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Edgar can now know whether Jana picked a red card.


VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentb knows that (Agenta knows that 3)

Hypothesis: Maria can now know that James can know that Dean's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] (Agentb knows that 1)

Hypothesis: Matthew can now know that Melody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Tyler's card is shown to Cecil. Flo's card is shown to Cecil. Flo's card is shown to Tyler. Cecil's card is shown to Flo. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:3 Agentc:1 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] [ ! (1|2|3) ] (Agentc knows that 1)

Hypothesis: Flo can now know that Cecil picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ronnie's card is shown to Carmen. Carolyn's card is shown to Brandon. Ronnie's card is shown to Brandon. Brandon's card is shown to Carolyn. Ronnie's card is shown to Carolyn. Carmen's card is shown to Ronnie. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:4 Agentb:3,4 Agentc:2,4 Agentd:1 VALID? [ ! (1|2|3|4) ] (Agentb knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:4 Agentb:3,4 Agentc:2,4 Agentd:1 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Brandon can now know that Brandon picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Betty does not know whether Erin is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] (Agentc knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Betty can now know that Philip is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Sarah does not know whether Rickey is thirsty. It is publicly announced that Sarah knows whether or not everyone is thirsty. It is publicly announced that Jill knows whether or not nobody is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] [ ! (Agentc knows whether (1&2&3)) ] [ ! (Agentb knows whether (~(1|2|3))) ] (Agentc knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] [ ! (Agentc knows whether (1&2&3)) ] [ ! (Agentb knows whether (~(1|2|3))) ] 0

Hypothesis: Sarah can now know that nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Cindy's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! 2 ] (Agentc knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! 2 ] 0

Hypothesis: Lena can now know that everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Michael does not know whether Devon is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 3) ] Agenta knows whether (Agentd knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 3) ] 0

Hypothesis: David can now know whether Michael can know that someone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Patricia does not know whether Cathy is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 2) ] Agentd knows that (Agentc knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 2) ] 0

Hypothesis: William can now know that Crystal can know whether Crystal is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Danny's card is shown to Julia. Jonathon's card is shown to Julia. Jonathon's card is shown to Danny. Danny's card is shown to Jonathon. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:3 Agentc:2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] (Agentb knows that (1|2|3))

Hypothesis: Danny can now know that someone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Andrew knows whether Elaine's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] Agentd knows whether (Agentc knows whether 4)

Hypothesis: Freda can now know whether Sam can know whether Freda's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jerry's card is shown to Ingrid. Rebecca's card is shown to Jerry. Ingrid's card is shown to Jerry. It is publicly announced that someone picked a red card. It is publicly announced that Ingrid does not know whether Jerry picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] Agenta knows whether (Agentb knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] 0

Hypothesis: Rebecca can now know whether Ingrid can know that nobody picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Kim knows whether Charles's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 3) ] (Agentd knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 3) ] 0

Hypothesis: Sarah can now know whether Kim's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Sheryl does not know whether or not everyone's forehead is muddy. It is publicly announced that Carol knows whether Carol's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether (1&2&3)) ] [ ! (Agenta knows whether 1) ] [ ! (1&2&3) ] (Agentc knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether (1&2&3)) ] [ ! (Agenta knows whether 1) ] [ ! (1&2&3) ] 0

Hypothesis: Sheryl can now know that Floyd's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Dwayne's card is shown to Richard. Richard's card is shown to Dwayne. It is publicly announced that someone picked a red card. It is publicly announced that Dwayne knows whether Dwayne picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 2) ] (Agentb knows whether (1|2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 2) ] 0

Hypothesis: Dwayne can now know whether someone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that America knows whether or not everyone's forehead is muddy. It is publicly announced that America does not know whether America's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether 2) ] (Agentc knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether 2) ] 0

Hypothesis: Karen can now know that Cindy's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Vivian's card is shown to Cynthia. Vivian's card is shown to Jeffrey. Cynthia's card is shown to Vivian. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:3 Agentc:1 VALID? [ ! (1|2|3) ] Agentc knows that (Agentb knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:3 Agentc:1 VALID? [ ! (1|2|3) ] 0

Hypothesis: Vivian can now know that Jeffrey can know whether or not everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Clifton's forehead is muddy.

Hypothesis: Rudolph can now know that someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] (Agentc knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Kareem can now know that Barbara is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Henry's card is shown to Maria. Michael's card is shown to Henry. It is publicly announced that someone picked a red card. It is publicly announced that Michael knows whether or not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (1&2&3)) ] Agenta knows whether (Agentb knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (1&2&3)) ] 0

Hypothesis: Maria can now know whether Michael can know that Henry picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentb knows that (Agentd knows that (~(1|2|3|4)))

Hypothesis: Kevin can now know that Lisa can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Waneta knows whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1&2&3&4)) ] Agentc knows that (Agentb knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1&2&3&4)) ] 0

Hypothesis: Carla can now know that Christopher can know that Terry is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Clarissa's card is shown to Francis. Francis's card is shown to Felix. Francis's card is shown to Jarred. Francis's card is shown to Clarissa. Felix's card is shown to Clarissa. It is publicly announced that someone picked a red card. It is publicly announced that Jarred does not know whether Jarred picked a red card. It is publicly announced that Felix knows whether or not everyone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:4 Agentb:1 Agentc:1 Agentd:1,2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 3) ] [ ! (Agentb knows whether (1&2&3&4)) ] (Agenta knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:4 Agentb:1 Agentc:1 Agentd:1,2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 3) ] [ ! (Agentb knows whether (1&2&3&4)) ] 0

Hypothesis: Francis can now know that Felix picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Mary knows whether Henrietta's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 3) ] (Agenta knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 3) ] 0

Hypothesis: Ramona can now know whether Ramona's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Stephen does not know whether Anna is thirsty. It is publicly announced that Robert does not know whether Jesus is thirsty. It is publicly announced that Jesus knows whether Jesus is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 4) ] [ ! (Agentd knows whether 4) ] (Agentc knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 4) ] [ ! (Agentd knows whether 4) ] 0

Hypothesis: Anna can now know whether Anna is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Glen's card is shown to Harvey. Harvey's card is shown to Joseph. Glen's card is shown to Joseph. Harvey's card is shown to Glen. Joseph's card is shown to Glen. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card. It is publicly announced that Glen does not know whether Glen picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] [ ! ~(Agentc knows whether 3) ] Agentb knows that (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Joseph can now know that Harvey can know that Harvey picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Jonas does not know whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agenta knows that (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Jonas can now know that someone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Kevin does not know whether Kate is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] (Agenta knows that (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] 0

Hypothesis: Mary can now know that nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Mark knows whether someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (1|2|3)) ] Agenta knows that (Agentb knows whether 3)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (1|2|3)) ] 0

Hypothesis: Sue can now know that Mark can know whether Christina's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joe knows whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1&2&3&4)) ] Agentc knows that (Agenta knows that 2)

Hypothesis: Rita can now know that Elizabeth can know that James's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Patricia's card is shown to Margaret. Margaret's card is shown to Patricia. It is publicly announced that someone picked a red card. It is publicly announced that Patricia picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] Agentb knows that (Agenta knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] 0

Hypothesis: Patricia can now know that Margaret can know that Patricia picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Ronald's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! 2 ] Agentc knows whether (Agenta knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! 2 ] 0

Hypothesis: Danny can now know whether Herman can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Michelle does not know whether Annabelle is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 1) ] Agentc knows that (Agentd knows whether (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 1) ] 0

Hypothesis: Monique can now know that Effie can know whether someone is thirsty.


VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentd knows that (Agentc knows whether 2)

Hypothesis: Debra can now know that Brittany can know whether Peter's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Tyler's card is shown to Pedro. Melissa's card is shown to Pedro. Melissa's card is shown to Tyler. Joyce's card is shown to Melissa. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:3,4 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] Agentc knows that (Agenta knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:3,4 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Tyler can now know that Pedro can know that Joyce picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Henrietta knows whether Eric's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 3) ] (Agentc knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 3) ] 0

Hypothesis: Eric can now know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Melissa does not know whether Mary is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 1) ] Agenta knows whether (Agentb knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 1) ] 0

Hypothesis: Mary can now know whether Melissa can know that Melissa is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Jesse's card is shown to Catherine. Irene's card is shown to William. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:3 VALID? [ ! (1|2|3|4) ] (Agenta knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:3 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Catherine can now know whether Catherine picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Mary's card is shown to Edwin. Jo's card is shown to Edwin. Edwin's card is shown to Mary. Angela's card is shown to Mary. Edwin's card is shown to Jo. Mary's card is shown to Jo. Edwin's card is shown to Angela. Mary's card is shown to Angela. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that Angela picked a black card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3 Agentb:1,4 Agentc:1,2 Agentd:1,2 VALID? [ ! (1|2|3|4) ] [ ! (1|2|3|4) ] [ ! ~(4) ] Agentc knows that (Agentb knows whether (1&2&3&4))

Hypothesis: Jo can now know that Mary can know whether or not everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Cesar knows whether Cesar's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] Agenta knows that (Agentb knows that (~(1|2|3|4)))

Hypothesis: Carl can now know that Eleanor can know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Joseph's card is shown to Raymond. Raymond's card is shown to Rafael. Raymond's card is shown to Joseph. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] Agentb knows that (Agentc knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] 0

Hypothesis: Rafael can now know that Joseph can know that Joseph picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Shirley knows whether Ronald's forehead is muddy. It is publicly announced that Shirley's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 3) ] [ ! 2 ] Agenta knows whether (Agentb knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 3) ] [ ! 2 ] 0

Hypothesis: Lorrie can now know whether Shirley can know that everyone's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Graciela knows whether Jessica's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (1&2) ] [ ! (Agenta knows whether 2) ] [ ! (1&2) ] (Agenta knows whether 1)

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (1&2) ] [ ! (Agenta knows whether 2) ] [ ! (1&2) ] 0

Hypothesis: Graciela can now know whether Graciela's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Dena knows whether or not nobody is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (~(1|2|3|4))) ] (Agenta knows whether (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (~(1|2|3|4))) ] 0

Hypothesis: Alan can now know whether someone is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Maryann's card is shown to Frank. Alexander's card is shown to Frank. Myra's card is shown to Frank. Alexander's card is shown to Maryann. Myra's card is shown to Maryann. Frank's card is shown to Alexander. Myra's card is shown to Alexander. Maryann's card is shown to Myra. It is publicly announced that someone picked a red card. It is publicly announced that Alexander does not know whether Maryann picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:3,4 Agentc:1,4 Agentd:2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] Agentb knows whether (Agentd knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:3,4 Agentc:1,4 Agentd:2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Maryann can now know whether Myra can know that Frank picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Angela knows whether someone's forehead is muddy. It is publicly announced that Rebecca knows whether Georgia's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! (Agenta knows whether 2) ] (Agentc knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Angela can now know whether Rebecca's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentd knows whether 3)

Hypothesis: Kathryn can now know whether Christian can know whether Kimberly's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Andrew's card is shown to James. Gwyn's card is shown to Linda. Linda's card is shown to Gwyn. Linda's card is shown to Andrew. It is publicly announced that someone picked a red card. It is publicly announced that Linda does not know whether Andrew picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:4 Agentb:3 Agentc:2 Agentd:2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] (Agentb knows that 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:4 Agentb:3 Agentc:2 Agentd:2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 4) ] 0

Hypothesis: Linda can now know that Andrew picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Maria knows whether or not nobody is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (~(1|2))) ] Agentb knows whether (Agenta knows that 1)

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether (~(1|2))) ] 0

Hypothesis: Emma can now know whether Maria can know that Maria is thirsty.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Carole does not know whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] 0

Hypothesis: Carole can now know that Carole's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Anisa does not know whether Thomas is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] (Agenta knows that 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] 0

Hypothesis: Jessica can now know that Thomas is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Eileen does not know whether Allison is thirsty. It is publicly announced that Heather is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] [ ! 2 ] (Agentb knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] [ ! 2 ] 0

Hypothesis: Heather can now know that everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Mona knows whether Andre's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (1&2&3) ] [ ! (Agenta knows whether 3) ] Agentb knows whether (Agentc knows whether (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (1&2&3) ] [ ! (Agenta knows whether 3) ] 0

Hypothesis: David can now know whether Andre can know whether or not nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Michael does not know whether Carl is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether 2) ] Agenta knows that (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether 2) ] 0

Hypothesis: Michael can now know that Carl can know that Carl is thirsty.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Latasha does not know whether or not everyone's forehead is muddy. It is publicly announced that Latasha does not know whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] Agenta knows that (Agentb knows whether (1|2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] 0

Hypothesis: Dennis can now know that Latasha can know whether someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that James knows whether or not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] (Agenta knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] 0

Hypothesis: James can now know that Joey's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ramon's card is shown to Kathleen. Kathleen's card is shown to Mary. Ramon's card is shown to Mary. Kathleen's card is shown to Ramon. Mary's card is shown to Ramon. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card. It is publicly announced that Ramon does not know whether Ramon picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] [ ! ~(Agentc knows whether 3) ] Agentc knows that (Agenta knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Ramon can now know that Kathleen can know whether Mary picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Charles knows whether Betty's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 3) ] (Agenta knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 3) ] 0

Hypothesis: Charles can now know that someone's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Becky's forehead is not muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(1) ] Agenta knows whether (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(1) ] 0

Hypothesis: Becky can now know whether Gabrielle can know that Gabrielle's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Gerald does not know whether Gerald's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Gerald does not know whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether 2) ] [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] Agentb knows that (Agenta knows that (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether 2) ] [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] 0

Hypothesis: Gerald can now know that Ryan can know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Norma's forehead is muddy. It is publicly announced that Andrea knows whether someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! 2 ] [ ! (Agentc knows whether (1|2|3)) ] Agenta knows that (Agentc knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! 2 ] [ ! (Agentc knows whether (1|2|3)) ] 0

Hypothesis: Michael can now know that Andrea can know whether Norma's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Robert's card is shown to Alberto. Gabriel's card is shown to Mildred. Mildred's card is shown to Gabriel. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:3 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] (Agentc knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:3 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Robert can now know that Alberto picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Terry does not know whether Daryl is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 1) ] (Agentc knows that (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 1) ] 0

Hypothesis: Terry can now know that nobody is thirsty.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Richard's forehead is muddy. It is publicly announced that Rebecca does not know whether Rebecca's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] [ ! ~(Agenta knows whether 1) ] Agenta knows that (Agentb knows whether (1|2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! 2 ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Rebecca can now know that Richard can know whether someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Wesley's card is shown to Barbara. Barbara's card is shown to Cindi. Wesley's card is shown to Cindi. Barbara's card is shown to Wesley. Cindi's card is shown to Wesley. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] (Agentb knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] 0

Hypothesis: Cindi can now know whether Cindi picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joseph knows whether Frank's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] (Agentc knows that 4)

Hypothesis: Mercedes can now know that Veronica's forehead is muddy.