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

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

Hypothesis: Justin can now know that Donald can know that Donald 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 Jessica does not know whether Jessica's forehead is muddy. It is publicly announced that Dessie does not know whether or not everyone's forehead is muddy.

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

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

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

Hypothesis: Gary can now know whether Jonathan'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 Kristen does not know whether Tony is thirsty. It is publicly announced that Kristen knows whether Kristen is thirsty. It is publicly announced that Tony does not know whether Kristen is thirsty. It is publicly announced that Tina does not know whether Kristen is thirsty.

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

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

Hypothesis: Tony can now know that Kristen can know that Kristen is thirsty.


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. Robert's card is shown to Daniel. James's card is shown to James. Nicole's card is shown to James. Nicole's card is shown to Nicole. It is publicly announced that someone picked a red card. It is publicly announced that James knows whether someone picked a red card. It is publicly announced that James does not know whether Robert picked a red card. It is publicly announced that Daniel does not know whether James picked a red card.

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

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

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


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

Hypothesis: Sarah can now know that Jeffrey 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). Stephanie's card is shown to Mary. Mary's card is shown to Stephanie. Stephanie's card is shown to Stephanie. Elisha's card is shown to Stephanie. Edna's card is shown to Stephanie. Mary's card is shown to Edna. Edna's card is shown to Edna. It is publicly announced that someone picked a red card. It is publicly announced that Elisha does not know whether or not everyone picked a red card. It is publicly announced that Elisha does not know whether Elisha picked a red card. It is publicly announced that Stephanie knows whether Stephanie picked a red card.

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

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

Hypothesis: Mary can now know that Stephanie can know that nobody picked a red card.


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

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

Hypothesis: Refugio can now know that Duane can know whether Duane's forehead is muddy.


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

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

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

Hypothesis: Donna can now know whether Ralph can know whether Charles 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 John does not know whether Heather is thirsty. It is publicly announced that Heather does not know whether Benjamin 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 4) ] Agentd knows whether (Agentb knows whether 3)

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

Hypothesis: Benjamin can now know whether John can know whether Douglas is thirsty.


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

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

Hypothesis: Richard can now know whether Crystal 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 Natalie's forehead is not muddy. It is publicly announced that Amy knows whether Natalie's forehead is muddy.

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

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

Hypothesis: Rita can now know whether Natalie can know that Natalie'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 Jennifer knows whether or not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Herbert's forehead is muddy.

Hypothesis: Jennifer can now know whether Lewis'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 Suzanne is not thirsty.

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

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

Hypothesis: Arthur can now know whether Suzanne can know whether Josephine 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 (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: Roger can now 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). Tiffany's card is shown to Berta. Tiffany's card is shown to Tiffany. Derick's card is shown to Tiffany. Tiffany's card is shown to Derick. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Tiffany can now know that Berta 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 Curtis does not know whether John is thirsty.

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

Hypothesis: John can now know whether 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 Tara does not know whether Earl is thirsty. It is publicly announced that Tricia does not know whether Earl is thirsty.

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

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

Hypothesis: Taylor 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 Sharon is thirsty. It is publicly announced that Irene does not know whether or not everyone is thirsty. It is publicly announced that Sharon does not know whether Brandon is thirsty. It is publicly announced that not everyone is thirsty.

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

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

Hypothesis: Irene can now know that Sharon is thirsty.


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

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

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

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


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

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

Hypothesis: Sandra can now know whether Diane 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 Devin does not know whether or not everyone's forehead is muddy.

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

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


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

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

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


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

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

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

Hypothesis: Gabriel can now know whether someone 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 Donald knows whether Kathleen'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) ] Agenta knows whether (Agentb knows that 3)

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

Hypothesis: Kathleen can now know whether Manuel can know that Donald'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 Joshua does not know whether or not everyone's forehead is muddy. It is publicly announced that Cynthia knows whether Cynthia's forehead is muddy. It is publicly announced that Joshua knows whether Cynthia's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Ida can now know whether Lawrence can know whether 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 Lloyd knows whether or not nobody's forehead is muddy. It is publicly announced that Anne knows whether someone's forehead is muddy.

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

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

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


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

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

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

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


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

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

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


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

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

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

Hypothesis: Brenda can now know that Kevin can know that nobody is thirsty.


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

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (1&2&3) ] (Agentc knows whether (~(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: Georgia can now 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). Lisa's card is shown to Lisa. Gregory's card is shown to Gregory. 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:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (~(1&2)) ] (Agenta knows whether (1|2))

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

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


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

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

Hypothesis: Margaret can now know that Nicholas can know that James is thirsty.


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

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

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

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

Hypothesis: Tomas 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). Lorenzo's card is shown to Arnold. Arnold's card is shown to Arnold. It is publicly announced that someone picked a red card. It is publicly announced that Arnold picked a black card.

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

Hypothesis: Lorenzo can now know whether Arnold 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 Amy does not know whether Lottie is thirsty.

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

Hypothesis: Amy can now know whether Lottie can know that Howard is thirsty.


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

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

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


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

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

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

Hypothesis: Maria can now know whether Suzanne can know that someone's forehead is muddy.


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

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

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


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

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

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

Hypothesis: James can now know whether Victor can 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 Charles does not know whether Carmen 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 that (Agenta knows whether 2)

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

Hypothesis: Carmen can now know that Cleta can know whether Freddie 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 Sarah knows whether Kimberly'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 2) ] Agenta knows whether (Agentb knows whether 3)

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

Hypothesis: Patrick can now know whether Kimberly can know whether Sarah'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 Tabitha does not know whether Adele is thirsty. It is publicly announced that Adele does not know whether Tabitha is thirsty. It is publicly announced that Tabitha does not know whether Adele 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) ] Agentd knows whether (Agentb 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) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether 1) ] 0

Hypothesis: Janie can now know whether Tabitha can know that Janie is thirsty.


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

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

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

Hypothesis: Takisha can now know that Jennifer can know that Jennifer 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 everyone's forehead is muddy. It is publicly announced that Kim knows whether Clarice's forehead is muddy.

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

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


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

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

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

Hypothesis: John can now know that Joyce 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 Lester does not know whether Jennifer is thirsty. It is publicly announced that Desiree does not know whether Margaret is thirsty. It is publicly announced that Margaret does not know whether Lester is thirsty. It is publicly announced that Lester knows whether someone is thirsty.

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

Hypothesis: Jennifer can now know that everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Albert'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) ] [ ! 1 ] Agentb knows that (Agenta knows whether 1)

Hypothesis: Geneva can now know that Albert can know whether Albert's forehead is muddy.


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

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

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

Hypothesis: Robert can now know that Robert 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 Travis 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)) ] (Agentb knows that 3)

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

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


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

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

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

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


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

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

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

Hypothesis: Robert can now know whether Felton 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 Tom knows whether Tom's forehead is muddy.

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

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

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

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agenta knows whether (1&2)) ] [ ! ~(Agenta knows whether 1) ] Agenta knows whether (Agentb knows whether (~(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: David can now know whether Jacqueline can know whether or not nobody's forehead is muddy.


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

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

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

Hypothesis: Spencer can now know that Spencer 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 Derek knows whether someone is thirsty. It is publicly announced that Derek knows whether Derek is thirsty.

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

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

Hypothesis: Darrell can now know that Robert can know that Derek is thirsty.


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

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

Hypothesis: Steven can now know whether Theresa can know that Theresa 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 Robert knows whether Gilbert's forehead is muddy. It is publicly announced that Robert does not know whether Robert's forehead is muddy.

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

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

Hypothesis: Judith 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). Lenora's card is shown to Jennifer. Jennifer's card is shown to Lenora. It is publicly announced that someone picked a red card. It is publicly announced that Jennifer knows whether someone picked a red card. It is publicly announced that Lenora picked a red card.

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Jeffrey does not know whether Priscilla is thirsty. It is publicly announced that Priscilla is not thirsty. It is publicly announced that Jeffrey 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) ] Agentb knows that (Agentd knows whether 1)

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

Hypothesis: Jeffrey can now know that Annmarie can know whether Priscilla is thirsty.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Gerald knows whether Charles's forehead is muddy. It is publicly announced that Charles 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) ] [ ! ~(Agenta knows whether (1&2)) ] (Agenta knows whether 1)

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

Hypothesis: Charles can now know whether Charles'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 Steven is thirsty.

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

Hypothesis: Joshua can now know whether Joshua 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 Lauren does not know whether or not everyone is thirsty. It is publicly announced that Lauren does not know whether or not everyone is thirsty.

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

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

Hypothesis: Jack can now know whether Lauren can 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 Chris knows whether Chris's forehead is muddy.

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

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

Hypothesis: Thomas 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 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) ] [ ! (~(1&2&3)) ] Agenta knows that (Agentb knows that 1)

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


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

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

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

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


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

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

Hypothesis: Steven can now know that Thomas'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 Kathryn knows whether someone's forehead is muddy. It is publicly announced that Regina knows whether someone's forehead is muddy. It is publicly announced that Debbie knows whether Regina'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)) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! (Agenta knows whether 4) ] (Agentd knows that 4)

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

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


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

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

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

Hypothesis: Gladys can now know whether Gerald can know that Gerald's forehead is muddy.


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

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

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

Hypothesis: Paula can now know that Sharon 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 Anthony knows whether someone's forehead is muddy.

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

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

Hypothesis: Marie 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). Roger's card is shown to Roger. Roger's card is shown to David. David's card is shown to David. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card.

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

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

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


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

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


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

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

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

Hypothesis: Andrea can now 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). Crystal's card is shown to Crystal. Michael's card is shown to Nicole. Eddie's card is shown to Eddie. Nicole's card is shown to Michael. 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:2 VALID? [ ! (1|2|3|4) ] Agentb knows that (Agentc knows that 2)

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

Hypothesis: Nicole can now know that Eddie can know that Nicole 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 Timothy does not know whether Ruth is thirsty. It is publicly announced that Timothy does not know whether Daisy 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) ] [ ! ~(Agentc knows whether 4) ] Agentc knows whether (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) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agentc knows whether 4) ] 0

Hypothesis: Timothy can now know whether Ruth can 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 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 (~(1|2|3)))

Hypothesis: Leslie 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 Jason knows whether someone's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Cory 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 Darrell knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Darrell can now know that Reginald 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 Tempie's forehead is not muddy.

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

Hypothesis: Tempie can now know whether Robert can know that Tempie'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 Clayton does not know whether Robert is thirsty. It is publicly announced that Evelynn does not know whether Clayton is thirsty. It is publicly announced that Robert does not know whether Willie is thirsty. It is publicly announced that Willie does not know whether Evelynn 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 2) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agentc knows whether 1) ] (Agenta knows whether 2)

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

Hypothesis: Evelynn can now know whether Clayton 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 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) ] [ ! 3 ] Agentd knows that (Agentc knows that 4)

Hypothesis: Marcos can now know that John can know that Marcos'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 Mark knows whether or not everyone's forehead is muddy. It is publicly announced that Mark knows whether Bonnie'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) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! (Agentb knows whether 1) ] [ ! (1|2|3) ] Agenta knows that (Agentb knows whether (~(1|2|3)))

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

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


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

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

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

Hypothesis: Donald can now know that David 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 Leland knows whether Curtis's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Eugene knows whether Curtis's forehead is muddy.

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

Hypothesis: Eugene can now know that Dennis can know whether Leland'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 Lisa 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))) ] Agentc knows whether (Agenta knows whether 4)

Hypothesis: Lisa can now know whether Dustin can know whether Antonio'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 Travis's forehead is not muddy. It is publicly announced that Barbara'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 ] Agentb knows that (Agenta knows whether (1&2&3))

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


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

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

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

Hypothesis: William can now know that Eugene is thirsty.


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

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

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

Hypothesis: Jill can now know that Barbara can know that 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 Rebecca does not know whether Elizabeth is thirsty. It is publicly announced that Rebecca does not know whether Elizabeth is thirsty.

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

Hypothesis: Elizabeth can now know that Rebecca can 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.

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

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


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Arthur knows whether Kevin's forehead is muddy. It is publicly announced that Kevin'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 ] Agentc knows whether (Agenta knows that 2)

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

Hypothesis: Kevin can now know whether Darin can know that Arthur's forehead is muddy.


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

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

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

Hypothesis: John can now know whether Donald 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). Randall's card is shown to Brian. Frances's card is shown to Brian. Brian's card is shown to Frances. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Randall can now know that Brian picked a red card.


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

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

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

Hypothesis: Jeffrey can now know whether Manuel 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). Brad's card is shown to Brad. John's card is shown to Brad. John's card is shown to John. Mildred's card is shown to John. Duane's card is shown to Mildred. It is publicly announced that someone picked a red card.

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

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

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


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

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

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

Hypothesis: Yvonne can now know that Shawnta'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 Debra knows whether or not everyone is thirsty.

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

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

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


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

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

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

Hypothesis: Nelson can now know that Marcus can know that Marcus 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 Scott does not know whether Scott's forehead is muddy. It is publicly announced that Fannie knows whether or not everyone's forehead is muddy. It is publicly announced that Fannie knows whether someone's forehead is muddy.

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

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

Hypothesis: Scott 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). Sara's card is shown to Sara. Jody's card is shown to Jody. It is publicly announced that someone picked a red card. It is publicly announced that Jody knows whether or not nobody picked a red card.

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

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

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

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

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

Hypothesis: Mark can now know whether Mark'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 Joshua 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)) ] Agentb knows that (Agentc knows that 1)

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

Hypothesis: Carmen can now know that Nancy can know that Joshua'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 Thomas does not know whether Mary is thirsty.

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

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

Hypothesis: Mary can now know that everyone is thirsty.


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

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

Hypothesis: Doris can now know that Brenda'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 Erin does not know whether James is thirsty. It is publicly announced that James does not know whether Erin is thirsty.

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

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

Hypothesis: Erin can now know that James 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). Judith's card is shown to Judith. Robert's card is shown to Judith. Judith'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 everyone picked a red card.

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

Hypothesis: Judith can now know whether Robert 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 Laura knows whether or not everyone is thirsty.

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

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

Hypothesis: Doris can now know whether Bennie 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 Susan knows whether Cole's forehead is muddy.

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Shirley does not know whether Tommy is thirsty. It is publicly announced that Raymond does not know whether Shirley 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 1) ] Agenta 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) ] [ ! ~(Agenta knows whether 3) ] [ ! ~(Agentb knows whether 1) ] 0

Hypothesis: Shirley can now know that Connie can 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). George's card is shown to Robert. Joseph's card is shown to Joseph. George's card is shown to Joseph. George's card is shown to George. It is publicly announced that someone picked a red card.

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

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

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


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

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

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


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

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

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

Hypothesis: Juan can now know that Cody 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). Karen's card is shown to Jack. Gilbert's card is shown to Jack. Jack's card is shown to Karen. Jack's card is shown to Gilbert. It is publicly announced that someone picked a red card. It is publicly announced that Jack picked a red card. It is publicly announced that Karen knows whether or not nobody picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that Jack knows whether Gilbert picked a red card.

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

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

Hypothesis: Jack can now know whether Karen can know that nobody picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Nicholas does not know whether Nicholas's forehead is muddy. It is publicly announced that Deanne knows whether or not nobody's forehead is muddy. It is publicly announced that Deanne knows whether Deanne's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

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

Hypothesis: Nicholas 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 Phillip knows whether or not everyone's forehead is muddy. It is publicly announced that Kenneth does not know whether Kenneth's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Mark can now know whether Peter can know whether Mark picked a red card.


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

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

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

Hypothesis: Richard can now know that Clair can know whether Clair 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 Marjorie'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 ] Agentc knows that (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) ] [ ! 1 ] 0

Hypothesis: Olive can now know that Eileen can know that Olive'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 Robert does not know whether or not everyone is thirsty. It is publicly announced that Robert does not know whether or not everyone 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 that (Agenta knows that 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)) ] 0

Hypothesis: Robert can now know that James can know that James 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 Donna is not thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Donna does not know whether Oma 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) ] [ ! ~(2) ] [ ! (~(1&2&3)) ] [ ! ~(Agentb knows whether 3) ] [ ! (1|2|3) ] (Agenta knows whether (1|2|3))

Hypothesis: Barbara 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 not everyone's forehead is muddy. It is publicly announced that David knows whether David's forehead is muddy. It is publicly announced that Katharine knows whether David's forehead is muddy.

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

Hypothesis: Katharine 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). Tammy's card is shown to Joseph. Paul's card is shown to Joseph. Joseph's card is shown to Paul. Paul's card is shown to Paul. It is publicly announced that someone picked a red card. It is publicly announced that Joseph does not know whether or not everyone picked a red card.

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

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

Hypothesis: Joseph can now know whether Tammy can know that 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 David does not know whether David'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) ] Agenta knows that (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) ] 0

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


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Edna knows whether or not everyone's forehead is muddy. It is publicly announced that Edna knows whether Brian'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) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! (Agentb knows whether 1) ] [ ! (1|2|3) ] Agentc knows that (Agenta knows that 1)

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

Hypothesis: Joy can now know that Brian can know that Brian'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 Corey is thirsty. It is publicly announced that Heidi does not know whether John is thirsty. It is publicly announced that Corey knows whether Corey is thirsty. It is publicly announced that Heidi does not know whether or not everyone is thirsty.

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

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

Hypothesis: Corey can now know that someone is thirsty.


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

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

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

Hypothesis: Gretchen can now know that Gretchen'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 Linda is not thirsty. It is publicly announced that Lucy 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) ] (Agentd knows whether (~(1|2|3|4)))

Hypothesis: Willie can now 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 Callie does not know whether Callie'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) ] Agenta knows whether (Agentc knows that 1)

Hypothesis: Callie can now know whether Jamie can know that Callie'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 Elizabeth's forehead is muddy.

Hypothesis: Benjamin 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 Judith knows whether someone's forehead is muddy.

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

Hypothesis: Judith 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's forehead is muddy. It is publicly announced that Samuel knows whether someone's forehead is muddy. It is publicly announced that Laura 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|2|3|4)) ] [ ! (Agentb knows whether (1&2&3&4)) ] Agentb 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) ] [ ! (Agenta knows whether (1|2|3|4)) ] [ ! (Agentb knows whether (1&2&3&4)) ] 0

Hypothesis: Laura can now know that Chad can know whether Laura'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 Elsie 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)

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

Hypothesis: Elsie can now know that Frances can know that Elsie'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 Doris knows whether Doris is thirsty. It is publicly announced that Doris is not 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 whether (1|2|3))

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

Hypothesis: Doris can now know whether someone 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 Sue knows whether Sue'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 that 4)

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


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

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

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


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

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

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


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

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

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

Hypothesis: Ryan 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). Shakira's card is shown to Anna. Anna's card is shown to Dave. Dave's card is shown to Dave. Anna's card is shown to Shakira. Dave's card is shown to Rebecca. Rebecca's card is shown to Rebecca. It is publicly announced that someone picked a red card. It is publicly announced that Rebecca does not know whether Shakira picked a red card. It is publicly announced that Anna knows whether Shakira picked a red card.

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

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

Hypothesis: Rebecca can now know that Shakira can know that Anna picked a red card.


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

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


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

Hypothesis: Jazmin can now know that Sheila can know that William 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 Lester knows whether Roger's forehead is muddy.

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

Hypothesis: Lester can now know that Roger can know that Roger'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 Casey knows whether Adrian'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 4) ] Agentc knows that (Agenta knows that 3)

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

Hypothesis: Virginia can now know that Robert can know that Virginia'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 Carlos does not know whether or not everyone's forehead is muddy. It is publicly announced that Carlos does not know whether Carlos's forehead is muddy. It is publicly announced that Joan's forehead is muddy. It is publicly announced that Joan knows whether Carlos'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 ] [ ! (Agentb knows whether 1) ] (Agentb knows whether (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 ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Joan can now know whether or not everyone'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 Whitney knows whether Jo'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 2) ] (Agentb knows that (~(1|2|3|4)))

Hypothesis: Jo 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 James knows whether someone is thirsty.

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

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

Hypothesis: Penny can now know whether Harriet 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 everyone's forehead is muddy. It is publicly announced that Kathryn knows whether someone's forehead is muddy.

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

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

Hypothesis: Elsie can now know whether Kathryn'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 Jennifer's forehead is not muddy.

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

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

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


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

Hypothesis: Landon can now know that Rachael'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) ] (Agentb knows that (~(1|2)))

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

Hypothesis: Gerry can now 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). Robert's card is shown to Robert. Benjamin's card is shown to Robert. Robert's card is shown to Benjamin. Paul's card is shown to Benjamin. Benjamin's card is shown to Paul. Robert's card is shown to Devon. It is publicly announced that someone picked a red card. It is publicly announced that Benjamin knows whether someone picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that Benjamin picked a black card. It is publicly announced that Robert does not know whether Devon picked a red card.

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

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

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


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

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

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

Hypothesis: Randolph can now know that June 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 Linda's forehead is not muddy.

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

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


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

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