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 Latasha knows whether someone's forehead is muddy. It is publicly announced that Lawrence knows whether or not nobody's forehead is muddy. It is publicly announced that Lawrence knows whether Latasha's forehead is muddy. It is publicly announced that Christopher knows whether Christopher's forehead is muddy.

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Nancy does not know whether Theresa is thirsty. It is publicly announced that Nancy does not know whether William 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) ] (Agentd knows whether 3)

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

Hypothesis: William can now know whether Nancy 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 Len does not know whether or not everyone's forehead is muddy. It is publicly announced that everyone'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) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! (1&2&3) ] [ ! (1|2|3) ] (Agenta knows whether 1)

Hypothesis: Len can now know whether Len'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 Virginia'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) ] Agentc knows that (Agentd knows that (~(1|2|3|4)))

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


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

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

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

Hypothesis: Irma can now know that Annie can know whether Irma'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 Wayne does not know whether or not everyone's forehead is muddy.

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

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


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

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

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

Hypothesis: Derrick can now know that Keith can know whether someone is thirsty.


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

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

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

Hypothesis: Amy can now know that Billy can know whether Billy picked a red card.


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

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

Hypothesis: Jacqueline can now know that David can know that Jacqueline 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 Kam knows whether Joyce'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 3) ] Agentd knows that (Agentb knows whether 1)

Hypothesis: Blanca can now know that Kam can know whether David'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 Brett does not know whether Frances is thirsty. It is publicly announced that Laura does not know whether Frances is thirsty.

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

Hypothesis: Laura can now know whether Zane can know that someone 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 Leo knows whether someone's forehead is muddy. It is publicly announced that Leo knows whether or not nobody's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

Hypothesis: Patrick 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). Roxanne's card is shown to Roxanne. Roxanne's card is shown to Mark. It is publicly announced that someone picked a red card.

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

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

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


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

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

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


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

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

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

Hypothesis: Howard can now know whether Brian can know that Howard 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 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) ] Agentc knows whether (Agentb knows that (1&2&3&4))

Hypothesis: Charles can now know whether Christopher 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). Jim's card is shown to Jerilyn. Michael's card is shown to Michael. Jim's card is shown to Michael. Jerilyn's card is shown to Jim. It is publicly announced that someone picked a red card. It is publicly announced that Michael picked a red card.

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

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

Hypothesis: Jim can now know that Jerilyn 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 Babette does not know whether or not everyone is thirsty. It is publicly announced that Babette does not know whether Sam is thirsty.

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

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

Hypothesis: Babette can now know whether Sam can know that Sam 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 Janet is thirsty. It is publicly announced that Janet does not know whether James is thirsty. It is publicly announced that John does not know whether James is thirsty.

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

Hypothesis: George can now know that George 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 Sarah knows whether or not nobody's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Bill can now know that Clarence 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 Denise knows whether or not everyone's forehead is muddy. It is publicly announced that Connie does not know whether Connie's forehead is muddy.

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

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

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


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

Hypothesis: Naomi can now know that Christopher can know that Naomi'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 Laticia knows whether or not nobody's forehead is muddy.

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

Hypothesis: Chung can now know whether Carol can know that Carol's forehead is muddy.


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

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

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

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


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Barbara knows whether Eric's forehead is muddy. It is publicly announced that Eric knows whether Eric'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 2) ] [ ! (Agentb knows whether 2) ] Agentc knows that (Agentb knows that 2)

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

Hypothesis: Barbara can now know that Eric can know that Eric'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 Harry knows whether Harry is thirsty.

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

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

Hypothesis: Harry can now know whether William can know that someone is thirsty.


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

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

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

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


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

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

Hypothesis: Sonya can now know that Nicholas'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 Sylvia does not know whether Cynthia is thirsty. It is publicly announced that Sandra does not know whether Sylvia is thirsty. It is publicly announced that Sandra does not know whether Sylvia is thirsty. It is publicly announced that Tanya knows whether or not nobody is thirsty.

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

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

Hypothesis: Sylvia can now know whether Tanya can know whether Sylvia 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 not everyone is thirsty. It is publicly announced that Marguerite knows whether Marguerite is thirsty. It is publicly announced that Marguerite 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)) ] (Agentb knows that 3)

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

Hypothesis: Rhoda can now know that Marguerite 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 Donald does not know whether or not everyone is thirsty.

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

Hypothesis: Elisa can now know that Elisa 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 Goldie is thirsty. It is publicly announced that David 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) ] [ ! 2 ] [ ! ~(Agentd knows whether (1&2&3&4)) ] (Agentd knows whether (1|2|3|4))

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

Hypothesis: David can now know whether someone is thirsty.


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

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

Hypothesis: Burl can now know whether Leticia 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 Alexander is not thirsty.

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

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

Hypothesis: Alexander can now know that someone is thirsty.


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

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

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

Hypothesis: Michael can now know that Michael 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 John's forehead is not muddy.

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

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


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

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

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

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


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

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

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

Hypothesis: Kevin can now know whether Kevin is thirsty.


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

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

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

Hypothesis: Leroy can now know whether Joseph picked a red card.


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

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

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

Hypothesis: Nathaniel can now know whether Charles picked a red card.


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

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

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


Premise: There are three persons. Everyone is visible to others. 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 Antonio knows whether or not nobody's forehead is muddy.

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

Hypothesis: Lauren can now know that Antonio can know whether Antonio'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 Todd's forehead is muddy. It is publicly announced that Philip knows whether or not everyone's forehead is muddy.

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

Hypothesis: Todd can now know whether Todd'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 Maude knows whether Derrick'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 (Agenta knows that (1&2&3&4))

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


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

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

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

Hypothesis: Herbert can now know whether Sharon can 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 Antonia's forehead is muddy.

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

Hypothesis: Marsha can now know that Antonia can know that Frederick'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 Clarence knows whether or not nobody is thirsty.

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

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

Hypothesis: Clarence can now know that Michael 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 Rolland knows whether Michael'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) ] [ ! (Agentc knows whether 2) ] [ ! (~(1&2&3)) ] Agentb knows whether (Agentc knows that (~(1|2|3)))

Hypothesis: Michael can now know whether Rolland can know that 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 Sharon does not know whether Sharon's forehead is muddy. It is publicly announced that Ashley does not know whether or not everyone's forehead is muddy.

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

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

Hypothesis: Ashley can now know whether Sharon can know that Ashley'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 Stephen does not know whether or not everyone is thirsty. It is publicly announced that Stephen does not know whether Vicky 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 2) ] (Agentb knows that 1)

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

Hypothesis: Vicky can now know that Kimberly is thirsty.


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

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

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

Hypothesis: Amelia can now know whether Isabel can know whether Isabel 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 John knows whether or not everyone's forehead is muddy. It is publicly announced that John's forehead is muddy.

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

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

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

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

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

Hypothesis: Keith can now know that Katie can know whether Keith's forehead is muddy.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: James can now know that Kimberly can know that Kimberly is thirsty.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Kenneth can now know that Sharon'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 Xavier knows whether Thomas's forehead is muddy.

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

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

Hypothesis: Xavier can now know that Thomas can know whether Kevin'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 Guadalupe does not know whether or not everyone's forehead is muddy.

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

Hypothesis: Joseph can now know whether Darrin can know whether Catherine's forehead is muddy.


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

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

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

Hypothesis: Bobby can now know whether Ruby can know that someone's forehead is muddy.


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

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

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: Luis can now know whether Nancy can know that Nancy is thirsty.


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

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

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

Hypothesis: Michael can now know that Ruby 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 Christopher does not know whether Christopher's forehead is muddy. It is publicly announced that Clarence knows whether Christopher'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) ] (Agentc knows whether 3)

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

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


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

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

Hypothesis: Ruthie can now know whether Jon can know whether Ruthie'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 Milton knows whether someone's forehead is muddy. It is publicly announced that Robert does not know whether or not everyone's forehead is muddy.

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

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

Hypothesis: Marvin can now know whether Robert 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 Michael knows whether Michael's forehead is muddy. It is publicly announced that Janice knows whether Michael's forehead is muddy.

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

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

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

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

Hypothesis: Joel can now know whether Joel'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 Rachel 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) ] [ ! (Agentc knows whether (1|2|3)) ] (Agentb knows that 2)

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

Hypothesis: Sara can now know that Sara'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 Jason is thirsty.

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

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

Hypothesis: Christopher can now know whether Jason can 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 Kevin knows whether Gregory'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) ] [ ! (Agentc knows whether 2) ] [ ! (~(1&2&3)) ] Agenta knows that (Agentc knows that 2)

Hypothesis: Betty can now know that Kevin can know that Gregory's forehead is muddy.


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

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

Hypothesis: Lillian can now know that Rhonda picked a red card.


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

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

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

Hypothesis: Joann can now know that Eva can know whether Eva picked a red card.


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Gerald knows whether Gerald'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)) ] Agenta knows that (Agentc knows that (1|2|3|4))

Hypothesis: Richard can now know that Gerald 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 Michael knows whether Phyllis's forehead is muddy.

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

Hypothesis: Ashley can now know that Michael can know that Michael'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 Isaiah is thirsty. It is publicly announced that Evelyn knows whether Isaiah is thirsty.

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

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

Hypothesis: Isaiah can now know that Jo 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 Cynthia is not thirsty. It is publicly announced that Cynthia does not know whether Jonathan is thirsty. It is publicly announced that Cynthia knows whether Cynthia is thirsty.

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

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

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

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

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


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

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

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

Hypothesis: Edgar can now know that Jean can 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). Rosemary's card is shown to Judith. Judith's card is shown to Rosemary. Rosemary's card is shown to Rosemary. It is publicly announced that someone picked a red card. It is publicly announced that Judith picked a red card.

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

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

Hypothesis: Rosemary can now 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. It is publicly announced that David's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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


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

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

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

Hypothesis: Elodia can now know that Carmelo can know that Carmelo is thirsty.


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

Hypothesis: Andre can now know whether Jacquelyn'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 Heath does not know whether Bruce is thirsty. It is publicly announced that Erica does not know whether or not everyone is thirsty.

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

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

Hypothesis: Erica can now know whether Heath 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 Brittney does not know whether Betty is thirsty.

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

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

Hypothesis: Robert can now know whether Brittney can know that Betty is thirsty.


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

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

Hypothesis: Sandy can now know that Sarah can know whether someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Corey does not know whether or not everyone 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) ] [ ! ~(Agentc knows whether (1&2&3)) ] [ ! (~(1&2&3)) ] Agentb knows whether (Agenta knows whether 3)

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

Hypothesis: Eugene can now know whether Richard can know whether Corey 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 Peter does not know whether Michael is thirsty. It is publicly announced that Amanda knows whether someone is thirsty.

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

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

Hypothesis: Peter can now know that Michael 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 Dale does not know whether Daniel 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 that (Agenta knows that (1&2&3))

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

Hypothesis: Daniel can now know that Kathy can know that everyone 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 George 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)) ] (Agenta knows whether (1&2))

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

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


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

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

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

Hypothesis: Tina can now know that Meta can know whether or not everyone picked a red card.


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

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

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

Hypothesis: Ralph can now know that Linda can 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 Linda does not know whether Alton is thirsty. It is publicly announced that Cindy does not know whether Kathryn is thirsty. It is publicly announced that Linda knows whether or not nobody is thirsty.

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

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

Hypothesis: Alton can now know that Cindy can know whether Linda 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 Linda does not know whether Glen is thirsty.

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

Hypothesis: Glen can now know that nobody is thirsty.


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

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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Marlene's card is shown to Marlene. Don's card is shown to Marlene. Marlene's card is shown to Don. It is publicly announced that someone picked a red card. It is publicly announced that Thomas knows whether or not everyone picked a red card. It is publicly announced that Marlene knows whether 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)) ] [ ! (Agentb knows whether (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) ] [ ! (Agenta knows whether (1&2&3)) ] [ ! (Agentb knows whether (1|2|3)) ] 0

Hypothesis: Thomas 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 someone's forehead is muddy. It is publicly announced that Hortencia 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) ] [ ! (1|2|3) ] [ ! (Agentc knows whether (1&2&3)) ] (Agenta knows that (1&2&3))

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


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

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

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

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


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

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

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

Hypothesis: Brian can now know whether Ray can know whether Brian'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 John 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) ] [ ! ~(Agentb knows whether 4) ] 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) ] [ ! ~(Agentb knows whether 4) ] 0

Hypothesis: Jim can now know whether David can know that Shirley 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 Retta does not know whether Glory 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 (Agentb knows whether (1&2&3))

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


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

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

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

Hypothesis: Michael can now know that Basil can know that Michael 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 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 (1&2&3&4)) ] Agenta knows that (Agentd knows whether 4)

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

Hypothesis: Michael can now know that Lloyd can know whether Lloyd is thirsty.


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

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

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

Hypothesis: Colleen can now know whether Ginger 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 Amanda does not know whether Alfred is thirsty. It is publicly announced that Amanda does not know whether David is thirsty.

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

Hypothesis: Alfred can now know that David can know that someone 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 Samuel's forehead is not muddy.

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

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


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

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

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

Hypothesis: Clarence can now know whether Thomas can know that Thomas 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 Neva does not know whether Rosalie is thirsty.

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

Hypothesis: Kate can now know whether Rosalie can know whether Neva 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 Linda does not know whether or not everyone's forehead is muddy.

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

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


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

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

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

Hypothesis: Kenneth can now know that Ricky can know whether or not nobody picked a red card.


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

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

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

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


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

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

Hypothesis: Michelle can now know that Thomas can know whether 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 not everyone's forehead is muddy.

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

Hypothesis: Samuel can now know that William can know that Ronald'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 Kaylee knows whether or not nobody's forehead is muddy.

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

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


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

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

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

Hypothesis: Wanda can now know whether Cecil can know that Cecil's forehead is muddy.


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Barbra does not know whether or not everyone is thirsty. It is publicly announced that Barbra knows whether Barbra 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)) ] [ ! (Agentb knows whether 2) ] 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) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! (Agentb knows whether 2) ] 0

Hypothesis: Barbra can now know whether Glenn 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 Kathleen knows whether Kathleen is thirsty. It is publicly announced that Deborah does not know whether Kathleen is thirsty.

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

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

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


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

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

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


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

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


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

Hypothesis: Brent can now know that Carl can know that Brent'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 Jonathan does not know whether Jonathan's forehead is muddy. It is publicly announced that Paul knows whether or not nobody's forehead is muddy.

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

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

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

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

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

Hypothesis: Sidney can now know whether George can know that George'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 Kara knows whether or not everyone's forehead is muddy. It is publicly announced that Wanda's forehead is not muddy.

Hypothesis: Kenneth can now know that 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 Joan knows whether someone's forehead is muddy.

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


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

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

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

Hypothesis: Jack can now know that Lisa can know that Jack'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 someone is thirsty. It is publicly announced that Paul is not 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) ] [ ! ~(2) ] Agentb knows whether (Agenta knows that 2)

Hypothesis: Paul can now know whether Jennie can know that Paul is thirsty.


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

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

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

Hypothesis: Tiesha can now know whether Sharon can know whether Sharon 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 Shawn is not thirsty.

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

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

Hypothesis: Toya can now know that Maryann 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 Phyllis does not know whether James is thirsty. It is publicly announced that Phyllis does not know whether or not everyone is thirsty. It is publicly announced that Phyllis does not know whether Hugh is thirsty. It is publicly announced that James does not know whether Hugh is thirsty.

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

Hypothesis: James can now know that Gina can know that Gina 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 Wm does not know whether Wm's forehead is muddy.

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

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


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

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

Hypothesis: Donna can now know whether Deborah can know whether Donna'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 someone's forehead is muddy. It is publicly announced that Vernon knows whether Vernon's forehead is muddy. It is publicly announced that Wendi knows whether Vernon's forehead is muddy. It is publicly announced that Vernon knows whether Luther'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 3) ] [ ! (Agenta knows whether 3) ] [ ! (Agentc knows whether 2) ] Agentd knows that (Agenta knows that (1|2|3|4))

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

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


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

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

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


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

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) ] Agentb knows whether (Agenta knows whether 3)

Hypothesis: Rose can now know whether Patricia can know whether Robert'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 Viola's forehead is muddy. It is publicly announced that Maria knows whether or not everyone's forehead is muddy. It is publicly announced that Heather knows whether Viola's forehead is muddy.

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

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


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

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

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

Hypothesis: Melanie 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 George knows whether Rebecca's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Glenda knows whether Rebecca'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 (Agenta knows whether 1)

Hypothesis: Glenda can now know that Rebecca can know whether Rebecca'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 Charles does not know whether Pat is thirsty. It is publicly announced that Charles does not know whether Justin is thirsty.

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

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

Hypothesis: Charles can now know whether Justin can know whether Justin 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 Richard knows whether Raquel's forehead is muddy. It is publicly announced that Clay's forehead is not muddy.

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

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


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

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

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

Hypothesis: Mario can now know whether Theresa picked a red card.


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Christian'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) ] [ ! ~(4) ] Agentd knows that (Agentc knows that (1&2&3&4))

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


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

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

Hypothesis: Melissa can now know that Sarah can know that Melissa is thirsty.


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

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

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

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


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

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

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

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


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Kyle's card is shown to Kyle. Gustavo's card is shown to Kyle. Kyle's card is shown to Gustavo. It is publicly announced that someone picked a red card. It is publicly announced that Kyle knows whether or not everyone picked a red card. It is publicly announced that Gustavo knows whether or not nobody 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 whether (~(1|2))) ] (Agentb knows that 2)

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

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


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

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

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

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


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

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

Hypothesis: Craig can now know that Norman can know that Craig 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 Danette knows whether someone's forehead is muddy.

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

Hypothesis: Tyler can now know whether Catherine can know whether Catherine'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 Amy knows whether or not nobody's forehead is muddy. It is publicly announced that Brian's forehead is not muddy.

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

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

Hypothesis: Amy can now know whether Brian 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). Lonnie's card is shown to Anthony. Betty's card is shown to Lonnie. Anthony's card is shown to Betty. Betty's card is shown to Betty. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Anthony can now know that Lonnie 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 Chester does not know whether Travis is thirsty. It is publicly announced that John does not know whether or not everyone is thirsty. It is publicly announced that John does not know whether Travis is thirsty. It is publicly announced that Travis 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: Manuel can now know that John can know that everyone 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 Holly knows whether or not nobody's forehead is muddy. It is publicly announced that Arthur knows whether or not everyone'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 whether 2)

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

Hypothesis: Arthur can now know whether Holly can know whether Holly'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 Linda does not know whether Linda's forehead is muddy. It is publicly announced that Moises knows whether or not nobody's forehead is muddy.

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

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

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


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

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

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

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