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 Beatriz knows whether Leona's forehead is muddy. It is publicly announced that Beatriz does not know whether or not everyone's forehead is muddy. It is publicly announced that Leona knows whether or not everyone's forehead is muddy. It is publicly announced that Leona knows whether Beatriz's forehead is muddy.

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

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

Hypothesis: Beatriz can now know that 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 Heather knows whether Heather'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) ] Agenta knows that (Agentb knows whether 1)

Hypothesis: Heather can now know that Gregory can know whether Heather's forehead is muddy.


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

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

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

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


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

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

Hypothesis: Jody can now know that Annette can know whether Annette 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 Keith knows whether Bonnie's forehead is muddy. It is publicly announced that Keith knows whether Bonnie'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 2) ] [ ! (Agenta knows whether 2) ] Agentc knows that (Agentd 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) ] [ ! (Agenta knows whether 2) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Ruthie can now know that Ethyl can know that nobody'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 Donna knows whether Donna'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: Donna can now know that Kristy 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 Jorge does not know whether Jorge's forehead is muddy. It is publicly announced that Collette knows whether or not everyone's forehead is muddy. It is publicly announced that Jorge'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 whether (1&2&3&4)) ] [ ! 4 ] (Agenta 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) ] [ ! ~(Agentd knows whether 4) ] [ ! (Agenta knows whether (1&2&3&4)) ] [ ! 4 ] 0

Hypothesis: Collette can now know whether Collette'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 Jason knows whether or not nobody's forehead is muddy.

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jason 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)) ] Agentd knows that (Agentc knows that 3)

Hypothesis: Jason can now know that Joe can know that Joe'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 Rosie 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) ] [ ! ~(Agentb knows whether (1&2&3)) ] (Agentc knows that (1&2&3))

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

Hypothesis: Bonnie can now know that everyone 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 Mark does not know whether Mark's forehead is muddy. It is publicly announced that Kevin knows whether Kevin's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Kira can now know that Kira 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 Martha does not know whether Patricia is thirsty. It is publicly announced that Kari does not know whether Martha is thirsty. It is publicly announced that Martha does not know whether or not everyone 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) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! (1|2|3|4) ] (Agentd knows whether 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 3) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! (1|2|3|4) ] 0

Hypothesis: Jessica can now know whether Kari is thirsty.


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

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

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

Hypothesis: Dona can now know whether Leona can 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 not everyone is thirsty. It is publicly announced that Katherine 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) ] [ ! (~(1&2&3&4)) ] [ ! ~(Agentb knows whether 3) ] Agentc knows that (Agenta knows whether 2)

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

Hypothesis: Mary can now know that Reba can know whether Katherine is thirsty.


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

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

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

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


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Anne'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) ] Agenta knows whether (Agentd knows whether (1|2|3|4))

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Joyce does not know whether 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 3) ] (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) ] [ ! ~(Agentd knows whether 3) ] 0

Hypothesis: Joyce can now know that Lisa 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 not everyone's forehead is muddy.

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

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

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


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

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

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


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

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

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

Hypothesis: Ricardo can now know that everyone is thirsty.


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

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

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

Hypothesis: Olga can now know that Michael can 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). Karen's card is shown to William. Karen's card is shown to Karen. Sergio's card is shown to Karen. Karen's card is shown to Sergio. William's card is shown to Antonio. Karen's card is shown to Antonio. Antonio's card is shown to Antonio. It is publicly announced that someone picked a red card. It is publicly announced that Antonio picked a red card.

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

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

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


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

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

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

Hypothesis: Thomas can now know that Lewis 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 Christopher 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 that 2)

Hypothesis: Katharine can now know that Grace'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 Estelle does not know whether Alex is thirsty. It is publicly announced that Jonathan 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) ] [ ! 3 ] Agenta knows whether (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 1) ] [ ! 3 ] 0

Hypothesis: Alex can now know whether Jonathan can know that someone 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 Brooke 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)) ] 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: Brooke can now know whether Joseph can know whether someone's forehead is muddy.


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

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

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

Hypothesis: Sally can now know that Dwight can know whether Dwight 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 Mary's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Mary knows whether Robbie'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 ] [ ! (~(1&2&3)) ] [ ! (Agenta knows whether 2) ] Agenta knows whether (Agentb knows whether 1)

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

Hypothesis: Mary can now know whether Robbie can know whether Mary'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 Angela does not know whether Joseph 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 (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 1) ] 0

Hypothesis: Angela can now know that Mary can know that everyone is thirsty.


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

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

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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Robert does not know whether Robert'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) ] Agentc knows that (Agentb knows whether 3)

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

Hypothesis: Catherine can now know that Richard can know whether Catherine'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 Cynthia knows whether Irene'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) ] Agenta 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) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Cynthia can now know that Irene can 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 Nathan's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Benito knows whether Susan'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) ] [ ! (Agentc knows whether 1) ] Agentb knows that (Agentc knows whether (1&2&3))

Hypothesis: Nathan can now know that Benito 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's forehead is muddy. It is publicly announced that Christine's forehead is not muddy. It is publicly announced that Lee does not know whether Lee's forehead is muddy. It is publicly announced that Lee 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) ] [ ! ~(1) ] [ ! ~(Agentb knows whether 2) ] [ ! (Agentb knows whether (1&2&3)) ] Agentb knows whether (Agentc knows that 2)

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

Hypothesis: Lee can now know whether John can know that Lee'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 Jessica's forehead is muddy.

Hypothesis: Theodore can now know that Lonnie'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 does not know whether Cynthia's forehead is muddy. It is publicly announced that James knows whether James'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) ] [ ! (Agentc knows whether 3) ] Agenta knows whether (Agentb 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) ] [ ! ~(Agentd knows whether 4) ] [ ! (Agentc knows whether 3) ] 0

Hypothesis: Eric can now know whether Dwayne can know that Dwayne'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 Richard knows whether Richard'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 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 3) ] 0

Hypothesis: Jason can now know that Richard can know that Maria's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Suzanne's card is shown to Suzanne. Gary's card is shown to Gary. Gabriel's card is shown to Gary. Gary's card is shown to Gabriel. 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 Gabriel does not know whether Gabriel picked a red card.

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

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

Hypothesis: Suzanne can now know that 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.

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 that 1)

Hypothesis: Edmund can now know whether John can know that Ivan'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 Mary knows whether Leroy's forehead is muddy. It is publicly announced that Mary does not know whether or not everyone's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Ira can now know whether Hazel 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 Debbie does not know whether Louise is thirsty. It is publicly announced that Debbie does not know whether Louise 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 whether 1)

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

Hypothesis: Louise can now know whether Louise is thirsty.


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

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

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

Hypothesis: Richard can now know that Shirley can know whether Leah 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 Rogelio's forehead is not muddy. It is publicly announced that not everyone's forehead is muddy.

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

Hypothesis: Rogelio 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 Kimberly knows whether Kimberly's forehead is muddy. It is publicly announced that Linda knows whether or not everyone's forehead is muddy. It is publicly announced that Lindsey knows whether Kimberly'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 2) ] [ ! (Agenta knows whether (1&2&3&4)) ] [ ! (Agentc knows whether 2) ] Agentb knows whether (Agentc knows whether 3)

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

Hypothesis: Kimberly can now know whether Lindsey can know whether Lindsey'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 Lois does not know whether Shirley is thirsty. It is publicly announced that Shirley does not know whether Ethel is thirsty.

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

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

Hypothesis: Shirley can now know whether Michael 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 Maria knows whether or not nobody is thirsty. It is publicly announced that Annie 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 (1&2&3&4)) ] (Agentd 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|2|3|4))) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] 0

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

Hypothesis: Johnny can now know whether Tom 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 Grace knows whether or not nobody'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))) ] (Agentc knows that (1&2&3&4))

Hypothesis: Grace can now know that 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 Mattie 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 that 1)

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

Hypothesis: Clarence can now know that Mattie'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 Samantha's forehead is muddy.

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


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

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

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

Hypothesis: Jon can now know that Jennifer 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 or not everyone is thirsty. It is publicly announced that Toby 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) ] [ ! ~(Agentc knows whether (1&2&3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agenta knows that 3)

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

Hypothesis: Toby can now know that William is thirsty.


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

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

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

Hypothesis: Joseph can now know that Delores can know whether Janice 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 Bob 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) ] [ ! (Agentc knows whether (1&2&3&4)) ] Agenta knows whether (Agentb knows that 3)

Hypothesis: Rebecca can now know whether Mark can know that Bob's forehead is muddy.


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

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

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

Hypothesis: Jeff can now know that Julie 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 Elaine knows whether Chris's forehead is muddy. It is publicly announced that Chris knows whether Evelyn's forehead is muddy. It is publicly announced that Elaine'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 2) ] [ ! (Agentb knows whether 1) ] [ ! ~(3) ] Agenta knows that (Agentc knows that (~(1|2|3)))

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

Hypothesis: Evelyn can now know that Elaine 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). Ronald's card is shown to Richard. Richard's card is shown to Richard. Harold's card is shown to Richard. Ronald's card is shown to Harold. Alisa's card is shown to Harold. Alisa's card is shown to Alisa. It is publicly announced that someone picked a red card. It is publicly announced that Alisa picked a red card.

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

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


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

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

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

Hypothesis: Debra can now know whether Richard 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 Willie does not know whether or not everyone is thirsty.

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

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

Hypothesis: Rosetta can now know that Rosetta is thirsty.


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

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

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

Hypothesis: Richard can now know that Claretha can know whether Claretha picked a red card.


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

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

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

Hypothesis: Claudia can now know that Janice 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 Daniel knows whether Daniel 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 that 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: Daniel can now know that Aaron can know that Aaron is thirsty.


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

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

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

Hypothesis: John 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's forehead is muddy. It is publicly announced that Shelton does not know whether or not everyone's forehead is muddy. It is publicly announced that Shelton does not know whether Shelton's forehead is muddy. It is publicly announced that Shelton does not know whether or not everyone's forehead is muddy.

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

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

Hypothesis: Shelton can now know whether John can know whether John'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 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 whether 3)

Hypothesis: Frances can now know that Lisa can know whether Glenn'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 Edna's forehead is not muddy. It is publicly announced that Howard's forehead is muddy. It is publicly announced that Edna knows whether Edna'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 ] [ ! (Agenta knows whether 1) ] Agentb knows that (Agentc knows that (1&2&3))

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

Hypothesis: Howard can now know that Ralph can 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 Kimberly knows whether someone's forehead is muddy. It is publicly announced that Curtis knows whether Kimberly's forehead is muddy. It is publicly announced that Irma does not know whether Irma's forehead is muddy.

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

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

Hypothesis: Kimberly can now know that Irma can know whether Kimberly'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 Constance knows whether or not everyone is thirsty.

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

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

Hypothesis: Robert can now know that Constance can know that nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Cynthia's card is shown to Cynthia. Jamie's card is shown to Cynthia. Charles's card is shown to Cynthia. Jamie's card is shown to Jamie. Charles's card is shown to Jamie. Cynthia's card is shown to Charles. Annette's card is shown to Annette. It is publicly announced that someone picked a red card. It is publicly announced that Jamie does not know whether Annette 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) ] (Agentc knows whether 3)

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: Charles can now know whether Charles 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 Charles does not know whether Brenda is thirsty. It is publicly announced that Charles knows whether or not everyone is thirsty.

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

Hypothesis: Ronald can now know that Brenda can know whether Brenda is thirsty.


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) ] Agenta knows whether (Agentb knows whether 2)

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

Hypothesis: Charles can now know whether Ruth can know whether Ruth 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 Michael's forehead is muddy.

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

Hypothesis: Steven can now know that Catherine'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 Elaine does not know whether Elaine's forehead is muddy.

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

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

Hypothesis: Jennifer can now know that Elaine can know whether Jennifer'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 Carina'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) ] [ ! 3 ] Agentb knows that (Agentc knows that 2)

Hypothesis: Jesus can now know that Carina can know that Jesus'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.

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

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

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


Hypothesis: Kecia can now know that Linda'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 Shirley 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) ] [ ! (Agenta knows whether (~(1|2|3|4))) ] (Agenta 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) ] [ ! (Agenta knows whether (~(1|2|3|4))) ] 0

Hypothesis: Shirley can now know that James is thirsty.


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

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

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

Hypothesis: Leslie 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 Jeffrey does not know whether Donald is thirsty. It is publicly announced that Donald does not know whether Jeffrey is thirsty. It is publicly announced that Jeffrey is thirsty. It is publicly announced that Charity does not know whether Donald 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) ] [ ! ~(Agentb knows whether 1) ] [ ! 1 ] [ ! ~(Agentc knows whether 2) ] (Agenta knows whether (1&2&3&4))

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

Hypothesis: Jeffrey can now know whether or not everyone is thirsty.


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

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

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

Hypothesis: Melinda can now know whether Karen can know whether or not everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Robert knows whether or not nobody'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 2)

Hypothesis: Joan can now know whether Robert'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 Frank knows whether Tanika'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) ] Agentd knows whether (Agentb knows whether 1)

Hypothesis: Bill can now know whether Rosemary can know whether Tanika'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 everyone is thirsty. It is publicly announced that Hazel knows whether someone is thirsty.

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

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

Hypothesis: Hazel can now know whether or not everyone is thirsty.


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

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

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

Hypothesis: Michael can now 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). Robbie's card is shown to Robbie. Tammy's card is shown to Robbie. Aaron's card is shown to Tammy. Aaron's card is shown to Aaron. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Robbie can now know that Tammy can 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). Joan's card is shown to Joan. Joan's card is shown to Leslie. Leslie's card is shown to Leslie. Leslie's card is shown to Ruth. Joan's card is shown to Emma. It is publicly announced that someone picked a red card. It is publicly announced that Joan does not know whether Emma picked a red card.

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

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

Hypothesis: Emma can now know whether Joan 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 Wendell does not know whether Wendell'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) ] Agentb 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) ] [ ! ~(Agentd knows whether 4) ] 0

Hypothesis: Omar can now know whether Wendell can know that 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. It is publicly announced that Jessie knows whether or not everyone's forehead is muddy.

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

Hypothesis: Crystal can now know that Jessie can know that Irene'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 Jacqueline knows whether Joseph'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 2) ] (Agenta knows whether 3)

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

Hypothesis: Jacqueline can now know whether Louie'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 Monika is thirsty. It is publicly announced that Rose 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) ] [ ! 3 ] [ ! ~(Agentb knows whether (1&2&3)) ] Agentc 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 ] [ ! ~(Agentb knows whether (1&2&3)) ] 0

Hypothesis: Monika can now know whether Elizabeth can know whether Monika 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 Elizabeth's forehead is not muddy.

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

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

Hypothesis: Elizabeth can now know that Mary'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 Barbara's forehead is not muddy.

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

Hypothesis: Barbara can now know that Michael can know that Michael'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 Jeremy knows whether someone's forehead is muddy. It is publicly announced that Angela knows whether Angela's forehead is muddy.

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

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

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


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

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

Hypothesis: Jerry can now know that Nichole 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 Kenny does not know whether Robert is thirsty. It is publicly announced that Michael 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) ] [ ! ~(Agenta knows whether 4) ] [ ! (Agentb knows whether (1|2|3|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 4) ] [ ! (Agentb knows whether (1|2|3|4)) ] 0

Hypothesis: Kenny can now know that Anthony is thirsty.


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

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

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

Hypothesis: Stephen can now know that 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 Paul does not know whether Beverly is thirsty. It is publicly announced that Ben does not know whether Lois is thirsty. It is publicly announced that Lois does not know whether Paul is thirsty. It is publicly announced that Paul 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) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether 4) ] [ ! ~(Agentd knows whether 3) ] [ ! (Agentc knows whether (1|2|3|4)) ] (Agentb knows that 1)

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

Hypothesis: Beverly can now know that Ben 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 Darius knows whether someone's forehead is muddy.

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

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

Hypothesis: Darius can now know that Jacqueline can know whether Mary'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 4)

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


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

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

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

Hypothesis: Joshua can now know that George can know that Joshua is thirsty.


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

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

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

Hypothesis: Carolyn can now know whether Stephanie can know whether or not 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 Rachelle's forehead is not muddy.

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

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

Hypothesis: Rachelle can now know that Anita 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). Richard's card is shown to Richard. Malena's card is shown to Richard. Malena's card is shown to Malena. Leon's card is shown to Malena. Malena's card is shown to Lisa. Lisa's card is shown to Lisa. Leon's card is shown to Leon. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Richard can now know whether Leon can know whether someone 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 Joseph does not know whether Christopher is thirsty. It is publicly announced that Richard 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) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agentb knows whether (1&2&3))

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

Hypothesis: Christopher can now know whether or not everyone is thirsty.


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

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

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

Hypothesis: Moses can now know that Colleen can know that someone 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 Myrtle knows whether Mary's forehead is muddy.

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

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

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


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

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

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

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


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that 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)) ] Agentc knows that (Agenta knows that 3)

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

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


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

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

Hypothesis: Melissa can now know whether Wallace can know whether Jessica picked a red card.


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

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

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

Hypothesis: Bruce can now know whether Lisa can know that Lisa 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 not everyone is thirsty. It is publicly announced that Gary does not know whether Joan is thirsty. It is publicly announced that Joan knows whether or not nobody is thirsty.

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

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

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


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

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

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

Hypothesis: Donald 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 John does not know whether Michael is thirsty. It is publicly announced that Christine does not know whether Allyson 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 3) ] Agentb knows whether (Agenta 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) ] [ ! ~(Agentd knows whether 3) ] 0

Hypothesis: John can now know whether Michael can know that John 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 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) ] [ ! (~(1&2&3&4)) ] Agentb knows that (Agentc knows whether 4)

Hypothesis: April can now know that Terrence can know whether Carter 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 Dolores does not know whether or not everyone's forehead is muddy.

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

Hypothesis: Steven can now know that Dolores can know whether or not 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 Gregory 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) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] (Agentd knows whether 1)

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


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

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

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

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


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

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

Hypothesis: Anthony can now know that Anthony'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 Scott knows whether or not nobody's forehead is muddy. It is publicly announced that Scott does not know whether Scott's forehead is muddy.

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

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: Lillian can now know whether Lillian's forehead is muddy.


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

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

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

Hypothesis: Willie can now know that Millicent 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). Laura's card is shown to Laura. Geoffrey's card is shown to Laura. Laura's card is shown to Geoffrey. Alan's card is shown to Geoffrey. Laura's card is shown to Alan. It is publicly announced that someone picked a red card.

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

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

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


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Rachell's forehead is muddy. It is publicly announced that Rachell knows whether Rachell'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) ] [ ! 3 ] [ ! (Agentc knows whether 3) ] Agentb knows whether (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) ] [ ! 3 ] [ ! (Agentc knows whether 3) ] 0

Hypothesis: Melissa can now know whether Charles can know that nobody's forehead is muddy.


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

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

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

Hypothesis: Brian can now know whether Brian'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 Stanley knows whether Shameka's forehead is muddy. It is publicly announced that Shameka does not know whether or not everyone's forehead is muddy. It is publicly announced that Stanley knows whether Dennis's forehead is muddy. It is publicly announced that Stanley knows whether Esmeralda'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 (1&2&3&4)) ] [ ! (Agenta knows whether 2) ] [ ! (Agenta knows whether 3) ] (Agenta 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 4) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] [ ! (Agenta knows whether 2) ] [ ! (Agenta knows whether 3) ] 0

Hypothesis: Stanley can now know whether Shameka'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 Shari knows whether Shari's forehead is muddy. It is publicly announced that Tim knows whether Violet'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) ] [ ! (Agentc knows whether 1) ] (Agentd knows whether 3)

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) ] [ ! (Agentc knows whether 1) ] 0

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


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

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

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

Hypothesis: Maria can now know that Debra can know whether someone 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 John does not know whether or not everyone is thirsty. It is publicly announced that John does not know whether or not everyone is thirsty.

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

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

Hypothesis: Doris can now know that John can know whether Doris 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 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) ] Agenta knows whether (Agentd knows that (1|2|3|4))

Hypothesis: Marilyn can now know whether Harriet can know that someone's forehead is muddy.


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

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

Hypothesis: Brian can now know whether Heather'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 not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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


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

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

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

Hypothesis: Billy can now know that Billy 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 Sharon is not thirsty. It is publicly announced that Jeffery does not know whether Anthony is 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 3) ] (Agentd knows that 4)

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 3) ] 0

Hypothesis: Sharon can now know that Sharon is thirsty.


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

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

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

Hypothesis: Jose can now know that everyone 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 Alberta 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) ] [ ! ~(Agentc knows whether (1&2&3)) ] Agentb 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) ] [ ! ~(Agentc knows whether (1&2&3)) ] 0

Hypothesis: Daniel can now know whether Alberta can 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). James's card is shown to Brian. Reba's card is shown to James. James's card is shown to James. Brian's card is shown to Genevieve. James's card is shown to Genevieve. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Brian can now know that Reba 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 Tracy does not know whether Tracy'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 3) ] Agenta 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 3) ] 0

Hypothesis: Philip can now know that Nevada can know that Philip'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 Heather does not know whether John is thirsty. It is publicly announced that Deborah does not know whether Ashley 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) ] [ ! ~(Agentd knows whether 2) ] (Agentd knows whether 2)

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

Hypothesis: Deborah can now know whether Ashley is thirsty.


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

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

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

Hypothesis: Adelaide can now know whether Kevin 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). Robert's card is shown to Arianne. Robert's card is shown to Robert. Arianne's card is shown to Maxine. Maxine's card is shown to Maxine. It is publicly announced that someone picked a red card.

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

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

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

Hypothesis: William can now know that Robin can know that 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 Rex knows whether Rex'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) ] Agentc knows whether (Agenta knows whether (1&2&3&4))

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


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

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

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

Hypothesis: Una can now know that Rodney 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 not everyone 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 1)

Hypothesis: Kathleen can now know that Kathleen is thirsty.


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

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

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

Hypothesis: Robert can now know whether Judy can know whether Robert 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 Diana knows whether Edith'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 4) ] (Agenta knows that 4)

Hypothesis: Casey can now know that Edith'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 Christine does not know whether Patricia is thirsty. It is publicly announced that Christine does not know whether or not everyone is thirsty. It is publicly announced that Christine does not know whether Charles is thirsty. It is publicly announced that Patricia does not know whether Charles is thirsty.

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

Hypothesis: Patricia can now know whether Scott can know whether Christine is thirsty.


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

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

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

Hypothesis: Julie can now know that Gregory can know whether Gregory 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 James'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)

Hypothesis: James can now know that James'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 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: Cynthia can now know that Philip can know whether or not 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 Lavonne knows whether someone's forehead is muddy.

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

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

Hypothesis: Raymond can now know that Svetlana can know that Lavonne'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 Amy does not know whether Sandra is thirsty. It is publicly announced that Lynn does not know whether Amy is thirsty. It is publicly announced that Sandra does not know whether Lynn 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) ] (Agentc knows whether 2)

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: Sandra can now know whether Lynn is thirsty.


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

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

Hypothesis: Cristina can now know whether Beverly can know whether Janet'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 Linda's forehead is muddy.

Hypothesis: Clayton can now know that Clayton'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.

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

Hypothesis: Lyndsay can now know that Tony can know that Lyndsay'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 Kylie does not know whether Diane is thirsty. It is publicly announced that James does not know whether John is thirsty.

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

Hypothesis: James can now know whether Diane 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 Kurt does not know whether or not everyone is thirsty. It is publicly announced that Richard knows whether Kurt is thirsty.

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

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

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

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

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

Hypothesis: Mary can now know whether Richard can know that Richard'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 Kimberly 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) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] Agentc knows whether (Agentd knows that (1|2|3|4))

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