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 James knows whether 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 whether (1|2|3|4)) ] Agenta knows whether (Agentd knows that 2)

Hypothesis: Billy can now know whether James can know that Lois'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 David knows whether Kimberly'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 that (Agenta 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) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Kimberly can now know that David can know that Kimberly'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. It is publicly announced that Bruce knows whether Bruce's forehead is muddy. It is publicly announced that Janet knows whether Bruce'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 2) ] [ ! (Agenta knows whether 2) ] (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) ] [ ! (~(1&2&3)) ] [ ! (Agentb knows whether 2) ] [ ! (Agenta knows whether 2) ] 0

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


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

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

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

Hypothesis: Micheal can now know that Mary can know that everyone 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 someone's forehead is muddy.

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

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

Hypothesis: Patrick can now 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 someone's forehead is muddy.

Hypothesis: Tyler can now know that nobody'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 Hannah does not know whether or not everyone is thirsty. It is publicly announced that Jonathan knows whether someone 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)) ] [ ! (Agenta 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) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! (Agenta knows whether (1|2|3)) ] 0

Hypothesis: Desirae 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 Peter does not know whether David is thirsty. It is publicly announced that Elizabeth does not know whether David 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 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) ] [ ! ~(Agentc knows whether 4) ] 0

Hypothesis: Elizabeth 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.

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

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

Hypothesis: Thomas can now know whether Michelle can know whether Andrew 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 Tammy does not know whether Tammy's forehead is muddy. It is publicly announced that Earl knows whether someone's forehead is muddy.

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

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

Hypothesis: Tammy can now know whether Earl'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 Angelique knows whether someone's forehead is muddy. It is publicly announced that June's forehead is not muddy. It is publicly announced that someone's forehead is muddy.

Hypothesis: Angelique can now 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 Pamela'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) ] [ ! ~(3) ] Agentd knows that (Agenta knows whether (1|2|3|4))

Hypothesis: Nancy can now know that Josephine can know whether 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 Cynthia's forehead is not muddy.

Hypothesis: Cynthia 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's forehead is muddy. It is publicly announced that Taylor's forehead is not muddy.

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

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

Hypothesis: Taylor can now know that Louise can know that Louise'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 Freddie knows whether Freddie'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 2) ] (Agenta knows whether 1)

Hypothesis: Julie can now know whether Julie'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 Dennis's forehead is muddy. It is publicly announced that 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) ] [ ! 2 ] [ ! (1&2&3) ] Agentb knows that (Agenta knows whether 2)

Hypothesis: Dennis can now know that Margret can know whether Dennis'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 Matthew knows whether or not everyone's forehead is muddy. It is publicly announced that Jonathan 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) ] [ ! (Agentb knows whether (1&2&3&4)) ] [ ! (Agentd knows whether (1&2&3&4)) ] Agenta knows that (Agentd knows that (~(1|2|3|4)))

Hypothesis: Julie can now know that Jonathan 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 Stephanie knows whether Marvin'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) ] Agentc knows that (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) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Adam can now know that Marvin can know that Adam'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 Georgia knows whether someone'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)

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

Hypothesis: Georgia can now know that Georgia'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 Brooke does not know whether Rebecca 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) ] (Agentb knows that 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 1) ] 0

Hypothesis: Brooke can now know that Brooke 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) ] [ ! (1|2|3|4) ] Agentd knows whether (Agentc knows whether 4)

Hypothesis: Addie can now know whether Etta can know whether Addie'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 Jorge knows 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) ] [ ! (Agentc knows whether (1&2&3)) ] Agentb knows that (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) ] [ ! (Agentc knows whether (1&2&3)) ] 0

Hypothesis: Teresa can now know that Jorge can know that Doris'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 Sue does not know whether Sue'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 4) ] Agenta knows that (Agentd knows whether (~(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) ] [ ! ~(Agentd knows whether 4) ] 0

Hypothesis: Benita can now know that Sue can 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. It is publicly announced that Jackie knows whether Jackie's forehead is muddy. It is publicly announced that Peter knows whether Jerome'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 2) ] [ ! (Agentd knows whether 1) ] (Agentd knows that 2)

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) ] [ ! (Agentb knows whether 2) ] [ ! (Agentd knows whether 1) ] 0

Hypothesis: Peter can now know that Jackie'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 Ronald does not know whether Nilda is thirsty. It is publicly announced that Charise does not know whether Ronald is thirsty. It is publicly announced that Nilda does not know whether Charise is thirsty.

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

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

Hypothesis: Ronald can now know that Ronald is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Clarence's card is shown to Ruth. Clarence's card is shown to Clarence. It is publicly announced that someone picked a red card. It is publicly announced that Clarence does not know whether or not everyone picked a red card. It is publicly announced that Ruth knows whether Clarence picked a red card. It is publicly announced that Clarence does not know whether Ruth picked a red card. It is publicly announced that Clarence knows whether or not nobody picked a red card.

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

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

Hypothesis: Clarence can now know whether 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 Carolyn does not know whether Carolyn's forehead is muddy. It is publicly announced that David knows whether or not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Edwin does not know whether Edwin'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)) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentb knows whether 2) ] (Agentc knows whether 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) ] [ ! ~(Agenta knows whether 1) ] [ ! (Agentc knows whether (1&2&3&4)) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentb knows whether 2) ] 0

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


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

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

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

Hypothesis: Joseph can now know that Karen 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 Christopher knows whether or not nobody'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))) ] (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 (~(1|2|3))) ] 0

Hypothesis: Christopher 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 Peter knows whether or not nobody's forehead is muddy. It is publicly announced that Abigail's forehead is not 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 (~(1|2|3))) ] [ ! ~(2) ] (Agentc knows that (1&2&3))

Hypothesis: Peter 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). Alice's card is shown to Alice. Nicholas's card is shown to Alice. Michael's card is shown to Nicholas. Alice's card is shown to Michael. Nicholas's card is shown to Michael. Michael's card is shown to Michael. It is publicly announced that someone picked a red card. It is publicly announced that Alice does not know whether Robbie picked a red card. It is publicly announced that Nicholas knows whether or not everyone picked a red card.

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

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

Hypothesis: Robbie can now know that Nicholas can know whether Michael 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 Raymond does not know whether Patricia is thirsty. It is publicly announced that Raymond 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 3) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agentb knows whether 3)

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

Hypothesis: Miguel can now know whether Patricia 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 Karla's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

Hypothesis: Dominic can now know that Armando 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 Alfredo does not know whether John is thirsty. It is publicly announced that someone 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) ] [ ! (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) ] [ ! ~(Agentd knows whether 2) ] [ ! (1|2|3|4) ] 0

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


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

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

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

Hypothesis: Kenneth can now know whether Kenneth 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 Barbara does not know whether Alfred 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) ] Agentd knows whether (Agenta knows that (1&2&3&4))

Hypothesis: Richard can now know whether Alfred can know that everyone 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 Angela's forehead is not muddy. It is publicly announced that Sherry's forehead is not 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) ] [ ! ~(3) ] [ ! ~(2) ] Agenta knows whether (Agentd knows that 4)

Hypothesis: Maynard can now know whether Genoveva can know that Genoveva's forehead is muddy.


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

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

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

Hypothesis: Cathleen 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 Karen knows whether or not everyone is thirsty. It is publicly announced that Olive 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) ] [ ! (Agentd knows whether (1&2&3&4)) ] [ ! (Agentc knows whether (~(1|2|3|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) ] [ ! (Agentd knows whether (1&2&3&4)) ] [ ! (Agentc knows whether (~(1|2|3|4))) ] 0

Hypothesis: Olive can now know that everyone is thirsty.


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

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

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

Hypothesis: Stacey can now know whether Mary picked a red card.


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

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

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

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


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

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

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

Hypothesis: Shannon can now know that George 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 Beverly is thirsty. It is publicly announced that Alicia does not know whether David is thirsty. It is publicly announced that Felicia knows whether or not nobody is thirsty. It is publicly announced that Felicia is not thirsty.

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

Hypothesis: David can now know that Felicia can know that Felicia is thirsty.


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

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

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

Hypothesis: Patricia can now know that Myra can know that nobody 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 Lucille knows whether or not nobody's forehead is muddy. It is publicly announced that Joyce knows whether or not nobody'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))) ] (Agentb knows whether 2)

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

Hypothesis: Joyce can now know whether Joyce'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 Elizabeth is not thirsty. It is publicly announced that Paul is thirsty.

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

Hypothesis: Paul can now know that Elizabeth 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 David 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 (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 (1&2&3&4)) ] 0

Hypothesis: Gloria can now know that everyone 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 Odette'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) ] [ ! 2 ] Agentc knows that (Agentd knows whether 2)

Hypothesis: Kathy can now know that Cynthia can know whether Odette'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 William knows whether someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Betty knows whether John'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)) ] [ ! (~(1&2&3&4)) ] [ ! (Agenta knows whether 4) ] Agentb knows whether (Agentc knows whether (1|2|3|4))

Hypothesis: Hollis can now know whether William 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 Gina knows whether Gina is thirsty.

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

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

Hypothesis: Gina can now know that Linda can know whether Linda is thirsty.


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 David knows whether James's forehead is muddy.

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

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

Hypothesis: James can now know whether David can 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 is thirsty. It is publicly announced that Enrique does not know whether Beverly is thirsty. It is publicly announced that Enrique knows whether or not everyone is thirsty.

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

Hypothesis: Enrique can now know that Tia can know that Enrique 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 Howard 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)) ] (Agentc 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: Daniel can now know that Daniel'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 Kim does not know whether Kim's forehead is muddy. It is publicly announced that Deb does not know whether or not everyone's forehead is muddy. It is publicly announced that Christen knows whether Deb'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) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! (Agentd knows whether 2) ] (Agenta knows that 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) ] [ ! ~(Agenta knows whether 1) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! (Agentd knows whether 2) ] 0

Hypothesis: Kim can now know that Deb'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 Bertha 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) ] [ ! (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: Bertha can now know that 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 Doris 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) ] [ ! (Agentc knows whether (1&2&3)) ] Agenta knows whether (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) ] [ ! (Agentc knows whether (1&2&3)) ] 0

Hypothesis: Herbert can now know whether Doris can know that Dorothy's forehead is muddy.


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) ] (Agentb 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: Lucy can now know that Lucy'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 Yvonne does not know whether Juan is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that June does not know whether Juan 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) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentd knows whether 1) ] (Agentc 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) ] [ ! ~(Agentb knows whether 1) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentd knows whether 1) ] 0

Hypothesis: Gloria can now know that Juan 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 Billy 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) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] Agenta knows whether (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&2&3&4)) ] 0

Hypothesis: Robert can now know whether Margaret can know whether 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 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) ] [ ! (~(1&2&3&4)) ] Agenta knows that (Agentc knows whether (1&2&3&4))

Hypothesis: Kay can now know that Iona can know whether or not everyone'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 everyone's forehead is muddy. It is publicly announced that Luis knows whether or not nobody's forehead is muddy. It is publicly announced that Luis knows whether Kevin's forehead is muddy.

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

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

Hypothesis: Kevin can now know whether Luis can know that someone'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 William knows whether William'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 1) ] Agentc knows that (Agentd knows that 4)

Hypothesis: Aaron can now know that Mary can know that Mary'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 someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

Hypothesis: Kimberly 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.

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

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

Hypothesis: Hazel can now know that Hazel 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 Patricia knows whether 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 whether (1|2|3|4)) ] Agentb knows whether (Agentd knows whether (1&2&3&4))

Hypothesis: Patricia can now know whether Wallace 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. It is publicly announced that Myra knows whether someone's forehead is muddy. It is publicly announced that Myra knows whether Myra's forehead is muddy. It is publicly announced that Paul knows whether 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) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether 1) ] [ ! (Agentc knows whether (1|2|3)) ] [ ! (1|2|3) ] (Agentb 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)) ] [ ! (Agenta knows whether 1) ] [ ! (Agentc knows whether (1|2|3)) ] [ ! (1|2|3) ] 0

Hypothesis: Seth can now know that Paul'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 everyone's forehead is muddy. It is publicly announced that Andrea's forehead is muddy.

Hypothesis: William can now know that Tim'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 Lisa knows whether Mark'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 3) ] Agentb knows that (Agenta knows that 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) ] [ ! (Agenta knows whether 3) ] 0

Hypothesis: John can now know that Lisa can know that Lisa'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 Thomas does not know whether or not everyone is thirsty. It is publicly announced that Thomas does not know whether Candace is thirsty. It is publicly announced that Thomas knows whether Thomas is thirsty.

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

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

Hypothesis: Thomas can now know whether Candace can know whether Thomas is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Mary's card is shown to Mary. Mary's card is shown to Seth. Michele's card is shown to Michele. It is publicly announced that someone picked a red card. It is publicly announced that Mary knows whether or not nobody picked a red card. It is publicly announced that Mary does not know whether Michele picked a red card.

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

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

Hypothesis: Mary can now know that Seth can know whether Seth 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 someone is thirsty.

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

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

Hypothesis: Trevor can now know that Bessie can know that everyone is thirsty.


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 Mary. Ralph's card is shown to Mary. Ralph's card is shown to Ralph. Marilyn's card is shown to Ralph. Dana's card is shown to Marilyn. Ralph's card is shown to Dana. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Dana can now know whether Ralph can know that 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 Jana 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)) ] Agentc knows that (Agentb knows that 3)

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


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

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

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

Hypothesis: Patricia can now know that Juan can know that nobody picked a red card.


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

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

Hypothesis: Joanne can now know that Noemi can know whether or not everyone picked a red card.


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

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

Hypothesis: Cherish can now know that 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. It is publicly announced that Rosalia's forehead is not 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) ] (Agenta knows whether 1)

Hypothesis: Curtis can now know whether Curtis'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 Richard knows whether someone's forehead is muddy. It is publicly announced that Jacqueline knows whether Jacqueline's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Richard knows whether someone's forehead is muddy.

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

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

Hypothesis: Jacqueline can now know that Richard'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) ] Agentb knows whether (Agentd knows whether 2)

Hypothesis: Herbert can now know whether Marcene can know whether Herbert'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 George knows whether George's forehead is muddy.

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

Hypothesis: George can now know that Amber can 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. It is publicly announced that Marshall is not thirsty.

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

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

Hypothesis: Helen can now know whether Michael can know whether Marshall is thirsty.


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

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

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

Hypothesis: Laura can now know that Scott can know that 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.

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

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


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

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

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

Hypothesis: Kim can now know that Kim 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 Julia does not know whether Christopher is thirsty. It is publicly announced that not 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 whether (Agentc knows whether (~(1|2|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: Christopher can now know whether Jacob 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). Van's card is shown to Kimberly. Ruth's card is shown to Kimberly. Van's card is shown to Van. Van's card is shown to Roland. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Roland can now know whether or not nobody 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 Sharita does not know whether Brandi is thirsty. It is publicly announced that Sharita does not know whether Brandi is thirsty.

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

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

Hypothesis: Brandi can now know that Sharita 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). Dwayne's card is shown to Arturo. Arturo's card is shown to Linda. Linda's card is shown to Linda. Dwayne's card is shown to Linda. Maria's card is shown to Dwayne. Arturo's card is shown to Dwayne. Linda's card is shown to Dwayne. Dwayne's card is shown to Dwayne. It is publicly announced that someone picked a red card.

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

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


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

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

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

Hypothesis: Cynthia can now know that Rhonda 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 Edward 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) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] Agenta knows that (Agentd knows that (~(1|2|3|4)))

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


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

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

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

Hypothesis: Donald can now know whether Thelma can 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 Mike's forehead is muddy. It is publicly announced that Michael does not know whether or not everyone's forehead is muddy.

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

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

Hypothesis: Mike can now know that Michael'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 Mildred does not know whether Mildred'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) ] Agentb knows that (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) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Henry can now know that Mildred can know that Mildred'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 (Agenta knows whether (~(1|2|3)))

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


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

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

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

Hypothesis: Dale can now know that nobody 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 Frances does not know whether Leticia 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) ] (Agentb 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 1) ] 0

Hypothesis: Frances can now know whether William is thirsty.


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

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

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

Hypothesis: Madelyn can now know whether Eric 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 Wayne knows whether Juan'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) ] Agentc knows whether (Agentb knows that 4)

Hypothesis: Jane can now know whether Pedro can know that Juan'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 not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Anne's forehead is not muddy. It is publicly announced that Irma's forehead is muddy.

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

Hypothesis: Anne can now know that Irma can know whether Irma's forehead is muddy.


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

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

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

Hypothesis: Sammy can now know whether Sammy 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 Lorie knows whether someone is thirsty. It is publicly announced that Patrick knows whether Lorie is thirsty.

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

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

Hypothesis: Patrick can now know that nobody 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 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) ] [ ! (1|2|3|4) ] Agentd knows that (Agentb knows that 4)

Hypothesis: John can now know that Elvis can know that John'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 Betty knows whether someone's forehead is muddy. It is publicly announced that Betty knows whether someone's forehead is muddy. It is publicly announced that Betty knows whether someone's forehead is muddy. It is publicly announced that Betty 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 whether (1|2|3)) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1&2&3)) ] (Agenta 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) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1&2&3)) ] 0

Hypothesis: Betty can now 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 Moises does not know whether or not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

Hypothesis: Florence 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. It is publicly announced that Juanita knows whether Juanita's forehead is muddy. It is publicly announced that Carolyn'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 1) ] [ ! 3 ] Agentc knows whether (Agenta knows whether (1|2|3|4))

Hypothesis: Carolyn can now know whether Juanita can know whether 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 Robert does not know whether or not everyone's forehead is muddy. 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 (1&2&3&4)) ] [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows that 2)

Hypothesis: Carl can now know whether Stanley can know that Stanley's forehead is muddy.


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

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

Hypothesis: Raymond can now 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 Maria knows whether Maria's forehead is muddy. It is publicly announced that Maria's forehead is not 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 1) ] [ ! ~(1) ] Agentd knows that (Agenta knows that 4)

Hypothesis: Paul can now know that Maria can know that Paul'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 Harvey'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) ] [ ! ~(2) ] Agentd knows that (Agentc knows that 4)

Hypothesis: Mildred can now know that Billy can know that Mildred'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 Melissa knows whether Melissa'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) ] (Agentd knows whether 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: Robert can now know whether Robert's forehead is muddy.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Clara can now know whether Gregg can know whether or not nobody 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 William does not know whether Samantha is thirsty. It is publicly announced that Samantha is not thirsty. It is publicly announced that Samantha does not know whether William is thirsty. It is publicly announced that William knows whether Samantha is thirsty.

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

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

Hypothesis: Samantha can now know that William can know that Joy 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 someone's forehead is muddy.

Hypothesis: Mary 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). Christopher's card is shown to Abigail. Abigail's card is shown to Abigail. Abigail's card is shown to Francis. It is publicly announced that someone picked a red card. It is publicly announced that Francis picked a red card.

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

Hypothesis: Christopher 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's forehead is muddy. It is publicly announced that Donna 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) ] [ ! (Agenta knows whether 4) ] Agentd knows whether (Agenta knows that 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 4) ] 0

Hypothesis: Mary can now know whether Donna can know that Mary'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 Mary knows whether Wanda'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 1) ] Agentc knows that (Agentb 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) ] [ ! (Agentc knows whether 1) ] 0

Hypothesis: Mary can now know that Paula 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 Robert 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) ] [ ! (Agentb knows whether (1&2&3&4)) ] (Agentc knows whether 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: Kevin can now know whether Patricia 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 Michael knows whether Michael'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) ] (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) ] 0

Hypothesis: Robert 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 Jacqueline knows 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)) ] Agentb knows whether (Agentc 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) ] [ ! (Agentb knows whether (1&2&3)) ] 0

Hypothesis: Jacqueline can now know whether Charles can know whether or not 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). Rosa's card is shown to Daniel. Rosa's card is shown to Raymond. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Daniel can now know whether Raymond picked a red card.


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

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

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

Hypothesis: Ann can now know that Ann 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 Patrick'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) ] [ ! 4 ] Agenta knows whether (Agentb knows whether 3)

Hypothesis: Salvatore can now know whether Janice can know whether Christine'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 not everyone's forehead is muddy. It is publicly announced that Mary knows whether or not nobody's forehead is muddy. It is publicly announced that Neoma knows whether Nicholas'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))) ] [ ! (Agentd knows whether 3) ] Agenta knows that (Agentd knows that 2)

Hypothesis: Craig can now know that Neoma can know that Mary'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 John's forehead is muddy. It is publicly announced that Elizabeth does not know whether Elizabeth's forehead is muddy.

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

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

Hypothesis: Elizabeth can now know whether John can know that John'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 Henry does not know whether Darnell is thirsty. It is publicly announced that Henry does not know whether Charles is thirsty. It is publicly announced that Darnell does not know whether Velma is thirsty. It is publicly announced that Velma 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) ] [ ! ~(Agenta knows whether 3) ] [ ! ~(Agentb knows whether 4) ] [ ! 4 ] (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) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agenta knows whether 3) ] [ ! ~(Agentb knows whether 4) ] [ ! 4 ] 0

Hypothesis: Henry can now know that Charles is thirsty.


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 someone's forehead is muddy.

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

Hypothesis: Allan can now know that Arthur can know whether or not 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 Carol does not know whether Virginia is thirsty.

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

Hypothesis: Eric can now 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). Evelyn's card is shown to Evelyn. Johnie's card is shown to Evelyn. German's card is shown to Evelyn. Johnie's card is shown to Johnie. German's card is shown to Johnie. Evelyn's card is shown to German. Robert's card is shown to Robert. It is publicly announced that someone picked a red card. It is publicly announced that Johnie does not know whether Robert picked a red card.

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

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

Hypothesis: Robert can now know whether Johnie 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 Juan 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|2|3))) ] Agenta knows that (Agentc knows whether 1)

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

Hypothesis: Albertha can now know that Juan can know whether Albertha 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 Alan knows whether Fidela's forehead is muddy. It is publicly announced that Cecilia knows whether Cecilia's forehead is muddy. It is publicly announced that Cecilia'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 4) ] [ ! (Agenta knows whether 1) ] [ ! 1 ] Agentd knows that (Agenta knows that (~(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 4) ] [ ! (Agenta knows whether 1) ] [ ! 1 ] 0

Hypothesis: Fidela can now know that Cecilia 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). Chris's card is shown to Maritza. Maritza's card is shown to Elida. Elida's card is shown to Elida. Chris's card is shown to Elida. Maritza's card is shown to Chris. Elida's card is shown to Chris. Chris's card is shown to Chris. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Elida can now know whether Chris can know that Chris 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 Tina's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Leonard does not know whether Leonard's forehead is muddy.

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

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

Hypothesis: Tina can now 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 is thirsty. It is publicly announced that Malissa does not know whether or not everyone is thirsty. It is publicly announced that Malissa does not know whether Steven is thirsty. It is publicly announced that Jeffery does not know whether Steven 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)) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(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) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: Steven can now know whether 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 Brittany knows whether Harvey'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 1) ] Agentb knows that (Agentd 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) ] [ ! (Agentc knows whether 1) ] 0

Hypothesis: Leonard can now know that Ron can know that Harvey'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 Robert. Robert's card is shown to Katherine. Katherine's card is shown to Katherine. Gary's card is shown to Gary. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Lloyd can now know whether Robert can know that Katherine picked a red card.


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

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

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

Hypothesis: Enriqueta can now know whether James can know whether someone 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 Rita does not know whether Fred 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) ] (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) ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: Rita can now know that Rita is thirsty.


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

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

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

Hypothesis: Ernestina can now know whether Cynthia 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 Kali knows whether someone's forehead is muddy. It is publicly announced that Kali knows whether someone's forehead is muddy. It is publicly announced that Kali knows whether someone's forehead is muddy. It is publicly announced that Kali 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 whether (1|2|3)) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1&2&3)) ] (Agentb 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)) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! (Agenta knows whether (1&2&3)) ] 0

Hypothesis: John 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). George's card is shown to Norma. Dora's card is shown to Norma. Greg's card is shown to George. Norma's card is shown to Dora. Dora's card is shown to Dora. Dora's card is shown to Greg. Greg's card is shown to Greg. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Norma can now know whether Greg can know whether George 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 Raymond does not know whether Marlene is thirsty. It is publicly announced that everyone is thirsty. It is publicly announced that Lisa is thirsty. It is publicly announced that Lisa 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) ] [ ! (1&2&3&4) ] [ ! 1 ] [ ! 1 ] (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) ] [ ! ~(Agentd knows whether 2) ] [ ! (1&2&3&4) ] [ ! 1 ] [ ! 1 ] 0

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

Hypothesis: Norma can now know that Alice'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 Steven knows whether someone's forehead is muddy. It is publicly announced that Steven knows whether Marie's forehead is muddy. 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 whether (1|2|3|4)) ] [ ! (Agentd knows whether 3) ] [ ! (1|2|3|4) ] Agentc knows that (Agenta knows whether 3)

Hypothesis: Marie can now know that Mellisa can know whether Marie'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 Megan is not thirsty.

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

Hypothesis: Megan can now know whether Michelle can know whether or not nobody 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 someone's forehead is muddy.

Hypothesis: Christine can now know that Brandy can know that nobody'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 Sandra is thirsty. It is publicly announced that Lamont knows whether Sandra is thirsty.

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

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

Hypothesis: Lamont can now know that Donald 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 Shawn does not know whether Deborah is thirsty. It is publicly announced that Deborah does not know whether Paul is thirsty. It is publicly announced that Deborah is thirsty. It is publicly announced that Deborah knows whether someone 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) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] [ ! (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) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] [ ! (Agentb knows whether (1|2|3|4)) ] 0

Hypothesis: Cynthia can now know whether someone 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 (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) ] 0

Hypothesis: Julia can now know whether Hazel can 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. It is publicly announced that 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)) ] Agentd knows that (Agentb knows that 3)

Hypothesis: Raymond can now know that Matt can know that Alberto'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 Martha is thirsty.

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

Hypothesis: Martha 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). Judy's card is shown to Judy. Larry's card is shown to Tyler. Judy's card is shown to Larry. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Judy can now know whether Martin can know that Larry 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 Elaine knows whether or not everyone's forehead is muddy. It is publicly announced that Elaine does not know whether Elaine's forehead is muddy. It is publicly announced that Curtis knows whether Curtis'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)) ] [ ! ~(Agenta knows whether 1) ] [ ! (Agentc knows whether 3) ] Agenta knows whether (Agentc 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)) ] [ ! ~(Agenta knows whether 1) ] [ ! (Agentc knows whether 3) ] 0

Hypothesis: Elaine can now know whether Curtis 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 is thirsty. It is publicly announced that Robert does not know whether Mary is thirsty. It is publicly announced that Tonya does not know whether Mary 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 whether 1) ] (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) ] [ ! ~(Agentc knows whether 1) ] 0

Hypothesis: Mary can now 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 Virginia does not know whether or not everyone's forehead is muddy. It is publicly announced that Monica knows whether someone's forehead is muddy. It is publicly announced that Virginia 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)) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] Agenta knows whether (Agentd 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 (1&2&3&4)) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] 0

Hypothesis: Theresa can now know whether Aline can know that everyone'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 Deborah knows whether or not nobody's forehead is muddy. It is publicly announced that Deborah's forehead is muddy. It is publicly announced that Juan'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 (~(1|2|3|4))) ] [ ! 2 ] [ ! ~(3) ] (Agentc knows that 2)

Hypothesis: Juan can now know that Deborah'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 Candice's forehead is not 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) ] (Agenta knows that (1&2&3))

Hypothesis: Candice 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's forehead is muddy. It is publicly announced that Cheryl 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)) ] (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) ] [ ! ~(Agentb knows whether (1&2&3)) ] 0

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


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

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

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

Hypothesis: Willie can now know whether Rebecca can know that Robert 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 someone's forehead is muddy. It is publicly announced that Cheryl knows whether Corey's forehead is muddy. It is publicly announced that Corey's forehead is muddy. It is publicly announced that Douglas does not know whether Douglas'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 4) ] [ ! 4 ] [ ! ~(Agenta knows whether 1) ] Agentd knows whether (Agentb knows whether 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) ] [ ! (1|2|3|4) ] [ ! (Agentc knows whether 4) ] [ ! 4 ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Corey can now know whether Kenneth can know whether Douglas'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 Karol's forehead is not muddy. It is publicly announced that June knows whether Karol'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 1) ] Agenta knows that (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) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Karol can now know that Christine can know that Karol'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 that (Agentb knows whether 3)

Hypothesis: Audrey can now know that Joan can know whether Ruth'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 Elsie's forehead is muddy. It is publicly announced that Christopher does not know whether Christopher'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 3) ] Agenta knows that (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) ] [ ! 2 ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Michael can now know that Christopher can know that Elsie's forehead is muddy.