Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Josefina does not know whether Cindy is thirsty. It is publicly announced that Brenda does not know whether Cindy 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) ] [ ! ~(Agentb knows whether 3) ] (Agentc knows whether (1|2|3|4))

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

Hypothesis: Cindy can now know whether someone is thirsty.


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

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

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

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


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

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

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

Hypothesis: Michael can now know whether Marcia'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 Celeste does not know whether Celeste's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

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


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

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

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

Hypothesis: Traci can now know that Betty 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 Buddy is thirsty. It is publicly announced that Woodrow is not thirsty.

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

Hypothesis: Woodrow can now know that Buddy can know whether Jon 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 Joshua knows whether Joshua'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) ] (Agentc knows whether 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: Joshua can now know whether Erma'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 Ted knows whether someone is thirsty. It is publicly announced that Rachel does not know whether Ted is thirsty. It is publicly announced that Rachel is thirsty. It is publicly announced that James does not know whether Ted 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)) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] [ ! ~(Agentd knows whether 3) ] (Agenta knows that (~(1|2|3|4)))

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

Hypothesis: Jesse can now know that nobody is thirsty.


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

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

Hypothesis: David can now know that Dean'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 Rose's forehead is not muddy.

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

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

Hypothesis: Rose can now know whether Rose'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 Robert knows whether Robert's forehead is muddy. It is publicly announced that Shannon knows whether Mario's forehead is muddy.

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Diane 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))) ] Agentc 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) ] [ ! (Agenta knows whether (~(1|2|3|4))) ] 0

Hypothesis: Fred can now know whether Yon can know that nobody is thirsty.


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

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


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

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

Hypothesis: William can now know whether Thomas is thirsty.


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

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

Hypothesis: Jamaal can now know whether Deborah 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 Margarite does not know whether Margarite's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Tom can now know that Tom'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 Paul does not know whether or not everyone is thirsty. It is publicly announced that Paul does not know whether Clifton is thirsty.

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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Blanche does not know whether Andrea 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 (Agentc knows that 2)

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

Hypothesis: Cora can now know whether Andrea can know that Cora 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 Holli'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 ] (Agenta knows that (1&2&3))

Hypothesis: Adam 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 is thirsty. It is publicly announced that Vicente knows whether or not nobody is thirsty.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Helen can now know whether Adam 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 Patricia knows whether Patricia'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 2)

Hypothesis: Patricia can now know that Lou'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 Rey does not know whether Rey's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Rey 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) ] [ ! ~(Agenta knows whether 1) ] [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] Agenta knows whether (Agentb knows that (~(1|2|3)))

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

Hypothesis: Rey can now know whether Miguel 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). Anthony's card is shown to Anthony. Jennifer's card is shown to Carmen. Marguerite's card is shown to Marguerite. Jennifer's card is shown to Jennifer. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Jennifer can now know whether Carmen 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 Doris's forehead is not muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Doris knows whether or not everyone's forehead is muddy.

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


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lisa does not know whether or not everyone's forehead is muddy. It is publicly announced that Lisa 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 that (~(1|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: Lisa can now know that nobody's forehead is muddy.


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

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

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

Hypothesis: Peggy can now know that Bonnie 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 Diane knows whether or not everyone's forehead is muddy. It is publicly announced that Leslie does not know whether Leslie's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: James can now know that Vicky can know whether Vicky 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 Linda does not know whether or not everyone's forehead is muddy. It is publicly announced that Linda does not know whether Linda'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 that (1&2))

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

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


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

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

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

Hypothesis: James can now know that Clay 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). Natasha's card is shown to Monica. Teri's card is shown to Natasha. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Monica can now know whether Teri can know that Kenneth 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 Francis knows whether Francis's forehead is muddy.

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

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


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

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

Hypothesis: Brian can now know whether William can 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). Daniel's card is shown to Daniel. Daniel's card is shown to Junior. Junior's card is shown to Junior. Wilma's card is shown to Junior. Junior's card is shown to Wilma. Junior's card is shown to Bruce. Bruce's card is shown to Bruce. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Daniel can now know that Bruce can know whether Wilma 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 Richard knows whether or not nobody's forehead is muddy. It is publicly announced that Daniel knows whether or not nobody's forehead is muddy. It is publicly announced that Daniel does not know whether or not everyone's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Raul can now know whether someone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joanne knows whether Joanne'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) ] (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) ] [ ! (Agentc knows whether 3) ] 0

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


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

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

Hypothesis: Billy can now know whether Lena can know that Billy 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 Jamie'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) ] (Agentb knows whether 2)

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


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

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

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

Hypothesis: Ericka can now know that Eddie can know that Jane's forehead is muddy.


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

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

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

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

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

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

Hypothesis: Kimberly can now know whether Lisa is thirsty.


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

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

Hypothesis: Albert can now know that Gregg can know that Gregg'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 Jeanette knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Jeanette can now know that Ivan can know whether someone's forehead is muddy.


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

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

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


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

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

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

Hypothesis: Helen can now know whether Beverly can know whether Helen 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 Victor knows whether John's forehead is muddy.

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

Hypothesis: Treena can now know that John'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 Joan knows whether Joan is thirsty. It is publicly announced that Joan is thirsty.

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

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

Hypothesis: Joan can now know that nobody is thirsty.


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

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

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


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

Hypothesis: Rosemarie can now know that Donna 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 Anthony does not know whether Sheila is thirsty.

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

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

Hypothesis: Anthony can now know that Sheila can know whether or not nobody is thirsty.


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

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

Hypothesis: Debbie can now know that Carlo can know that Debbie'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 Brandon's forehead is not muddy. It is publicly announced that Soila knows whether or not everyone's forehead is muddy.

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

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


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

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

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

Hypothesis: Richard can now know whether someone is thirsty.


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

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

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

Hypothesis: Robert can now know that Timothy can know whether or not 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). Mary's card is shown to Mary. Mary's card is shown to Lucas. Lucas's card is shown to Lucas. Rachel's card is shown to Rachel. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Lucas can now know that Mary can know whether Lorraine picked a red card.


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

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

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

Hypothesis: James can now know that Angelic can know that nobody picked a red card.


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

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

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


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

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

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

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


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

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

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

Hypothesis: Thelma can now know whether Marguerite can know that Thelma 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 Margaret does not know whether Kathryn is thirsty.

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

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

Hypothesis: Heather can now know that Margaret can know that Heather 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 Gary's forehead is muddy. It is publicly announced that Edward knows whether someone's forehead is muddy.

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

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

Hypothesis: Edward can now know that Gary can know that Edward'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 Clara knows whether someone's forehead is muddy.

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

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


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

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

Hypothesis: Dianne can now know that Ernie 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 Jocelyn knows whether Sharon is thirsty. It is publicly announced that Jocelyn is thirsty.

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

Hypothesis: Rose can now know whether Beth can know whether Jocelyn is thirsty.


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 Kimberly. Kyle's card is shown to Kimberly. Kimberly's card is shown to Laura. Laura's card is shown to Laura. Kyle's card is shown to Laura. Kimberly's card is shown to Kyle. It is publicly announced that someone picked a red card. It is publicly announced that Laura knows whether or not nobody picked a red card.

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

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

Hypothesis: Laura can now know whether Kyle can 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. It is publicly announced that Lawrence knows whether Connie'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 (~(1|2|3|4)))

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


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Donald 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) ] [ ! (Agentc knows whether (1|2|3)) ] (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 (1|2|3)) ] 0

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Thomas does not know whether Thomas'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) ] (Agenta knows that (~(1|2|3)))

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Daniel is thirsty. It is publicly announced that Billy 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) ] [ ! 4 ] [ ! ~(Agenta knows whether 3) ] 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) ] [ ! 4 ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: Maurice can now know that Joseph can know that nobody 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 Danielle does not know whether or not everyone is thirsty. It is publicly announced that Danielle does not know whether or not everyone is thirsty.

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

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

Hypothesis: Eric can now know that Mary can know whether or not nobody is thirsty.


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

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

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

Hypothesis: Mary can now know whether or not 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). Josh's card is shown to Josh. Joshua's card is shown to Josh. Joshua's card is shown to Joshua. Vonda's card is shown to Joshua. Josh's card is shown to James. Joshua's card is shown to James. Josh's card is shown to Vonda. Vonda's card is shown to Vonda. It is publicly announced that someone picked a red card. It is publicly announced that Josh knows whether Josh picked a red card.

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

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

Hypothesis: James can now know whether Vonda can know that someone 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 Ben does not know whether or not everyone is thirsty. It is publicly announced that Alfonso 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 whether (1&2)) ] (Agentb knows that 2)

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

Hypothesis: Alfonso can now know that Alfonso 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 James knows whether Michelle'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) ] (Agentc knows that 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) ] 0

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


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

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

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

Hypothesis: Betty can now know whether Elisabeth 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 is thirsty. It is publicly announced that Helen does not know whether Marion is thirsty. It is publicly announced that Helen does not know whether or not everyone is thirsty.

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

Hypothesis: Johanna can now know that everyone is thirsty.


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

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

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

Hypothesis: Marshall can now know that everyone 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 someone is thirsty. It is publicly announced that Jerry does not know whether Theresa is thirsty.

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

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

Hypothesis: Theresa can now know whether Jerry can know that someone is thirsty.


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

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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Marilyn does not know whether Anthony is thirsty. It is publicly announced that Nick knows whether someone is thirsty. It is publicly announced that Marilyn does not know whether or not everyone is thirsty. It is publicly announced that Marlene 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 3) ] [ ! (Agenta knows whether (1|2|3|4)) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] Agenta knows that (Agentc knows that (1|2|3|4))

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

Hypothesis: Nick can now know that Anthony can know that someone is thirsty.


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

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

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

Hypothesis: Sandra can now know whether Cathy can know whether Dwight 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 Julia's forehead is muddy.

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

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


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

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

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

Hypothesis: Frank can now know that Shelby can know that Frank'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 Denis does not know whether Ellen is thirsty. It is publicly announced that Ellen does not know whether Denis is thirsty. It is publicly announced that Denis does not know whether Ellen 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) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether 1) ] Agentc knows that (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) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether 1) ] 0

Hypothesis: Edna can now know that Denis can know whether Michael 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 Charles knows whether or not nobody's forehead is muddy. It is publicly announced that Patricia does not know whether Patricia's forehead is muddy. It is publicly announced that Patricia does not know whether Patricia's forehead is muddy. It is publicly announced that Patricia does not know whether or not everyone's forehead is muddy.

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

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

Hypothesis: Justin can now know whether Charles'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 Tomas knows whether Miguel's forehead is muddy. It is publicly announced that Tomas knows whether or not everyone's forehead is muddy. It is publicly announced that Miguel knows whether Tomas'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) ] Agenta knows that (Agentb 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) ] 0

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


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

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

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

Hypothesis: Trina can now know whether Steven can know whether Steven'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 Jody knows whether Jody is thirsty. It is publicly announced that Nancy 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 2) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] (Agentc knows that 3)

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

Hypothesis: John can now know that John 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 Rick does not know whether Rick's forehead is muddy.

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

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

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


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

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

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


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

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

Hypothesis: Mary can now know that Martha'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 Bernice's forehead is not muddy.

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

Hypothesis: Cora can now know that Bernice 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 Dallas knows whether Dallas's forehead is muddy.

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

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

Hypothesis: Tami can now know that Dallas can know that Tami'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 Modesto does not know whether Clarence is thirsty.

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

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

Hypothesis: Mckinley can now know whether Susan can know whether Susan 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 Andy'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) ] Agentd knows whether (Agentb knows that (1|2|3|4))

Hypothesis: George can now know whether Jacob can know that someone's forehead is muddy.


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

Hypothesis: Flossie can now know that Norma can know that Norma'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 Mary does not know whether Chester is thirsty. It is publicly announced that Chester does not know whether Savannah 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) ] [ ! ~(Agentc knows whether 1) ] (Agentc knows whether 3)

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

Hypothesis: Chester can now know whether Chester 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 Edith 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) ] [ ! (Agentc knows whether (~(1|2|3|4))) ] Agentb knows that (Agenta knows that (~(1|2|3|4)))

Hypothesis: Robert can now know that Charles 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 Risa's forehead is muddy. It is publicly announced that Risa's forehead is muddy. It is publicly announced that Diane's forehead is not muddy.

Hypothesis: Risa can now know that John'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 Deborah knows whether Lettie's forehead is muddy.

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

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

Hypothesis: Deborah can now 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 is thirsty. It is publicly announced that Ryan does not know whether Mark is thirsty. It is publicly announced that Joyce does not know whether Walter 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) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether 4) ] [ ! (1|2|3|4) ] Agentc knows that (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 2) ] [ ! ~(Agenta knows whether 4) ] [ ! (1|2|3|4) ] 0

Hypothesis: Ryan can now know that Walter can know that everyone 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 Tina's forehead is not muddy. It is publicly announced that Clinton knows whether someone's forehead is muddy.

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

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

Hypothesis: Tina can now know that Keith'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 Alejandrina does not know whether Lela is thirsty. It is publicly announced that Paul does not know whether Alejandrina is thirsty. It is publicly announced that Paul knows whether or not nobody is thirsty. It is publicly announced that Lela does not know whether James is thirsty.

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

Hypothesis: Paul can now know whether Alejandrina 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 Melissa knows whether Florentina's forehead is muddy. It is publicly announced that Melissa knows whether or not nobody's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Larry can now know that nobody is thirsty.


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

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

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

Hypothesis: Cathy can now know that Kevin can know whether or not nobody is thirsty.


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

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

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

Hypothesis: Sarah can now know that Jackie 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 Reynaldo is not thirsty.

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

Hypothesis: Joyce can now know whether Randal can know that Randal is thirsty.


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

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

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

Hypothesis: Velma can now know that Velma 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 Edna does not know whether Brian is thirsty. It is publicly announced that Mary 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 4) ] [ ! 1 ] (Agentc 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) ] [ ! ~(Agentc knows whether 4) ] [ ! 1 ] 0

Hypothesis: Edna can now know that Agnes is thirsty.


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

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

Hypothesis: Maire can now know that James can 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 Maria does not know whether Wayne is thirsty. It is publicly announced that Maria 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 2) ] [ ! ~(Agenta knows whether (1&2&3)) ] Agenta knows that (Agentb knows that 1)

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

Hypothesis: Maria can now know that Wayne can know that Maria is thirsty.


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

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

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


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Krista does not know whether Kristy 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) ] (Agentc knows whether 2)

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

Hypothesis: Kristy can now know whether Krista is thirsty.


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

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

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

Hypothesis: David can now know that Madie can know that someone 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 Karen knows whether Karen'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 whether (Agentb knows whether (1&2))

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

Hypothesis: Karen can now know whether Shirley 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 is thirsty. It is publicly announced that Amanda 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)) ] Agentb knows that (Agenta knows whether (1&2&3&4))

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

Hypothesis: Amanda can now know that Katrina can 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). Shanda's card is shown to Dale. Shanda's card is shown to Shanda. It is publicly announced that someone picked a red card. It is publicly announced that Dale knows whether Shanda picked a red card.

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

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

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


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

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

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

Hypothesis: Leonard can now know that Katherine can know whether someone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Johnnie does not know whether Johnnie'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) ] (Agentc knows whether 2)

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

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


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

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

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

Hypothesis: Tony can now know that Deborah can know whether Aaron 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 Shelly's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

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

Hypothesis: Katherine can now know whether Shelly can know that Katherine'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 Julian does not know whether or not everyone is thirsty. It is publicly announced that Jose knows whether someone is thirsty. It is publicly announced that Jose 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)) ] [ ! (Agentb knows whether (1|2|3|4)) ] [ ! 2 ] (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) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] [ ! (Agentb knows whether (1|2|3|4)) ] [ ! 2 ] 0

Hypothesis: Esther can now know whether Ruby 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 Donna's forehead is muddy.

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


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

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

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


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

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

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

Hypothesis: Dylan can now know that Dylan 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 Joseph'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) ] [ ! 2 ] [ ! (1|2) ] Agentb knows that (Agenta knows that (1&2))

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


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

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

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

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


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Albert does not know whether Barbara is thirsty. It is publicly announced that Douglas does not know whether Albert is thirsty. It is publicly announced that Barbara does not know whether Douglas 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) ] [ ! ~(Agentb knows whether 4) ] [ ! ~(Agentc knows whether 2) ] (Agentc knows whether 3)

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

Hypothesis: Barbara can now know whether Barbara 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 Jeremiah does not know whether Jeremiah's forehead is muddy.

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

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

Hypothesis: Richard can now know that Christopher can know whether Jacque's forehead is muddy.


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

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

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

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


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Christine's card is shown to Christine. Christine's card is shown to Timothy. 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 that (Agenta knows that (~(1|2)))

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

Hypothesis: Timothy can now know that Christine can know that nobody picked a red card.


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

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

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

Hypothesis: Eldon can now know whether Paul 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 Jane's forehead is not muddy.

Hypothesis: Alfred can now know that Donald'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 Douglas knows whether Doretha's forehead is muddy. It is publicly announced that Doretha's forehead is muddy.

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

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


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

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

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

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


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

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

Hypothesis: Henry can now know that Martin 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 Arlene knows whether Arlene's forehead is muddy. It is publicly announced that Arlene knows whether or not everyone's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Carroll can now know whether Blanche can know whether Blanche 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 Lester's forehead is not muddy. It is publicly announced that Hae's forehead is muddy. It is publicly announced that Hae's forehead is muddy.

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

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

Hypothesis: James can now know whether Cynthia can know that Cynthia'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 David knows whether or not everyone's forehead is muddy. It is publicly announced that David knows whether someone's forehead is muddy. It is publicly announced that David knows whether David's forehead is muddy.

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

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

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


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

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

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

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

Hypothesis: Robert can now know that Shaina'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 Danny 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 whether (Agentc knows that (1|2|3))

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

Hypothesis: Danny can now know whether Sharon can know that 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 Lawrence does not know whether Hazel 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 4) ] (Agentb knows that 2)

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

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

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

Hypothesis: Dannie can now know whether Patrick'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 Carl knows whether Carl's forehead is muddy.

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

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

Hypothesis: Casey can now know that Casey'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) ] Agentd knows that (Agentc knows that (1&2&3&4))

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


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

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

Hypothesis: Mandy can now know that 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). Paul's card is shown to Paul. David's card is shown to Paul. Paul's card is shown to David. Raymond's card is shown to David. David's card is shown to David. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Raymond can now know that Paul can know whether Paul 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 Lenore knows whether or not nobody's forehead is muddy. It is publicly announced that Quintin's forehead is not muddy. It is publicly announced that Doris does not know whether Doris'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))) ] [ ! ~(2) ] [ ! ~(Agenta knows whether 1) ] Agentb knows whether (Agenta knows that (1|2|3))

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

Hypothesis: Quintin can now know whether Doris can know that someone'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 Jean does not know whether James is thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: James can now know that Jean can know whether Jean is thirsty.


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

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

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

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


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

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

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

Hypothesis: Ruth can now know whether Nikki 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 Frank does not know whether Alfred is thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: Frank can now know that Delmer can know that nobody is thirsty.