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

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

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


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

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

Hypothesis: Nancy can now know that Willie can know that everyone is thirsty.


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

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

Hypothesis: Brian can now know that Daniel can know that Dusty'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.

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

Hypothesis: Orlando 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). Aaron's card is shown to Pearlie. Edith's card is shown to Pearlie. Pearlie's card is shown to Aaron. Edith's card is shown to Aaron. Benjamin's card is shown to Benjamin. Aaron's card is shown to Edith. It is publicly announced that someone picked a red card. It is publicly announced that Pearlie knows whether or not nobody picked a red card. It is publicly announced that Pearlie picked a red card.

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

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

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


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

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

Hypothesis: Charles can now know whether Lillie can know that Lillie'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 Gwendolyn knows whether someone is thirsty.

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

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

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


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

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

Hypothesis: Kathy can now know that Veronica can know that Barbara is thirsty.


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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that James does not know whether Marie is thirsty. It is publicly announced that Frank 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)) ] (Agentc knows that (1&2&3))

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

Hypothesis: Marie can now know that everyone 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 Albert knows whether someone is thirsty. It is publicly announced that Frances knows whether or not nobody is thirsty. It is publicly announced that Richard knows whether Richard is thirsty. It is publicly announced that Frances does not know whether Albert is thirsty.

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

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

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


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

Hypothesis: Eilene can now know whether Laurence can know whether Helen'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 Edith knows whether Matthew'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) ] (Agentc knows that 2)

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

Hypothesis: Edith can now know that Matthew'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 Brenda knows whether or not nobody's forehead is muddy. It is publicly announced that Ashley's forehead is not muddy. It is publicly announced that Brenda'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) ] [ ! (Agentd knows whether (~(1|2|3|4))) ] [ ! ~(1) ] [ ! ~(4) ] Agenta knows that (Agentb knows whether 1)

Hypothesis: Ashley can now know that Scott can know whether Ashley'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 Edwina knows whether or not everyone's forehead is muddy. It is publicly announced that Edwina does not know whether Edwina's forehead is muddy.

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

Hypothesis: Wilma can now know that Edwina'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 Eleanor does not know whether or not everyone's forehead is muddy. It is publicly announced that Margie knows whether Eleanor's forehead is muddy. It is publicly announced that Margie knows whether Eleanor's forehead is muddy.

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

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

Hypothesis: Margie can now know whether Thomas can know that someone's forehead is muddy.


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

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

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

Hypothesis: Vernell can now know whether Patrick can know whether Robert 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.

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

Hypothesis: Richard can now know that John can know that Thurman'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 William does not know whether or not everyone's forehead is muddy. It is publicly announced that William 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)) ] [ ! (Agentb knows whether (~(1|2))) ] (Agentb knows whether (~(1|2)))

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

Hypothesis: William can now 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 Lori'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) ] [ ! ~(3) ] Agentc knows that (Agentb knows whether (1&2&3))

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


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

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

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


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

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

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


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

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

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

Hypothesis: Nancy can now know that Ruth can know whether Nancy's forehead is muddy.


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

Hypothesis: James can now know that Delores can know whether or not nobody is thirsty.


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

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

Hypothesis: Jennette can now know whether Matthew 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 Brian 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 that (Agentb knows whether (~(1|2|3)))

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

Hypothesis: Andrew can now know that Brian can know whether or not 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 Rita is thirsty.

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

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

Hypothesis: Sophia can now know whether Rita is thirsty.


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

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

Hypothesis: Heidi can now know whether Michelle can know whether Rachel 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 Jesse knows whether someone is thirsty. It is publicly announced that Susan does not know whether Paul is thirsty. It is publicly announced that Karen is thirsty.

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

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

Hypothesis: Karen can now know whether Jesse is thirsty.


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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Robert does not know whether Jennifer is thirsty. It is publicly announced that Jennifer does not know whether Laura 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) ] [ ! ~(Agentd knows whether 3) ] Agentd knows whether (Agentc knows that 1)

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

Hypothesis: Jennifer can now know whether Laura can know that Thomas 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 Marion'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 that (~(1|2|3)))

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


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Jennifer knows whether Jennifer 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 3) ] (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) ] [ ! (Agentc knows whether 3) ] 0

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

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

Hypothesis: Mark can now know that Arla can know that Betty's forehead is muddy.


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

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

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

Hypothesis: Thomas can now know whether Amanda can know whether Margaret 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 Concetta knows whether or not everyone's forehead is muddy.

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


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

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

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

Hypothesis: Robert can now know whether Hazel can know that Hazel is thirsty.


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

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

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

Hypothesis: Steven can now know that Andrea can know whether Andrea 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 Lawrence does not know whether or not everyone is thirsty.

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

Hypothesis: Kimberly can now know that Kimberly is thirsty.


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

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

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

Hypothesis: Walter can now know whether Wayne 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 Jordan knows whether or not nobody's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

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

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


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

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

Hypothesis: Lizette can now know whether Mary can know that Valerie's forehead is muddy.


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

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

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

Hypothesis: Mildred can now know whether Joseph can know whether Mildred 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 Crystal does not know whether Elliot is thirsty. It is publicly announced that Rita does not know whether or not everyone is thirsty.

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

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

Hypothesis: Sylvia can now know that Rita can know that Rita is thirsty.


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

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

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

Hypothesis: Rose can now know that Rose 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 someone is thirsty. It is publicly announced that Dona does not know whether Margaret is thirsty.

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

Hypothesis: Karen can now know that Dona can know that nobody 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 Sharon knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Sharon can now know that David can know that Sharon'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 Betty knows whether Betty's forehead is muddy.

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

Hypothesis: George can now know that Betty can know that Melody'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 Eileen does not know whether or not everyone is thirsty. It is publicly announced that Juliet does not know whether Asuncion is thirsty. It is publicly announced that Asuncion does not know whether or not everyone is thirsty. It is publicly announced that Asuncion does not know whether Juliet 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 1) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! ~(Agenta knows whether 3) ] (Agenta knows whether (1&2&3))

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

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


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

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

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

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


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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Elsie does not know whether Dawn is thirsty. It is publicly announced that Chad knows whether Chad 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 4) ] Agentc knows whether (Agenta knows that (1|2|3|4))

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

Hypothesis: Matthew can now know whether Elsie can know that someone is thirsty.


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

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

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

Hypothesis: Gail can now know whether Thomas can 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 Beverly knows whether someone's forehead is muddy. It is publicly announced that Denise knows whether Denise's forehead is muddy.

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

Hypothesis: Beverly can now know that George'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 Kenneth. Kenneth's card is shown to Abel. Abel's card is shown to Yvonne. Yvonne's card is shown to Yvonne. Yvonne's card is shown to Katheryn. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Yvonne can now know whether Kenneth 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 Robert does not know whether or not everyone is thirsty. It is publicly announced that Robert does not know whether or not everyone is thirsty.

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

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

Hypothesis: Robert can now know whether Janice can know that everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Edgar does not know whether or not everyone's forehead is muddy. It is publicly announced that Suzanna'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)) ] [ ! 3 ] (Agenta knows whether (1|2|3|4))

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Eldon's card is shown to Patricia. Ethel's card is shown to Ethel. Eldon's card is shown to Ethel. Ethel's card is shown to Eldon. Eldon's card is shown to Eldon. It is publicly announced that someone picked a red card. It is publicly announced that Eldon picked a red card. It is publicly announced that Eldon does not know whether Patricia picked a red card. It is publicly announced that Eldon 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 2)

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: Patricia can now know that Ethel 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 Judy's forehead is muddy.

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

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


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

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

Hypothesis: Mark can now know that Jessie'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 Wanda's forehead is muddy. It is publicly announced that Wanda's forehead is muddy. It is publicly announced that Thomas's forehead is muddy.

Hypothesis: Wanda 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 not everyone's forehead is muddy. It is publicly announced that Lynn knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Lynn can now know that Edna can know that Lynn'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 Carolyn does not know whether Carolyn's forehead is muddy. It is publicly announced that Edward 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 2) ] [ ! (Agentc knows whether (1&2&3)) ] (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) ] [ ! ~(Agentb knows whether 2) ] [ ! (Agentc knows whether (1&2&3)) ] 0

Hypothesis: Donna can now know that Edward'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 Sharon knows whether or not nobody's forehead is muddy. It is publicly announced that Norman'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) ] (Agentc knows that (1|2|3))

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


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

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

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

Hypothesis: James can now know that Rachel can know whether Eun picked a red card.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Kerry can now know whether Earl can know that Earl 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 Lula knows whether Brenda's forehead is muddy.

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

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


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

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

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) ] [ ! (Agenta knows whether 3) ] [ ! (Agenta knows whether (1&2&3)) ] [ ! ~(3) ] 0

Hypothesis: Norma can now know that Matthew'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 Joanne does not know whether or not everyone is thirsty. It is publicly announced that David knows whether someone is thirsty. It is publicly announced that Dennis does not know whether David is thirsty.

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

Hypothesis: Joanne can now know that David is thirsty.


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

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

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

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


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

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

Hypothesis: Anthony can now know that Irma can know that nobody is thirsty.


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

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

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


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

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

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

Hypothesis: Gennie can now know whether Walter 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 not everyone is thirsty.

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

Hypothesis: Ernest can now know that Francesca can know whether Wanda 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 Janet'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 ] Agentc knows whether (Agentb knows that (~(1|2|3|4)))

Hypothesis: Janet can now know whether Jeffrey 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 not everyone's forehead is muddy. It is publicly announced that Peggy does not know whether Peggy'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)) ] [ ! ~(Agentc knows whether 3) ] (Agentb knows that 1)

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

Hypothesis: Tenisha can now know that 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 James does not know whether or not everyone is thirsty. It is publicly announced that James does not know whether Tina 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 whether 2) ] Agentb knows that (Agenta knows whether 1)

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

Hypothesis: Tina can now know that David can know whether David 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 Jamie is thirsty. It is publicly announced that Ronald does not know whether or not everyone is thirsty.

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

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

Hypothesis: Jamie can now know that everyone is thirsty.


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Donna can now know whether Paul can know that Paul picked a red card.


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

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

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

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


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

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

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

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


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

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

Hypothesis: Larry can now know that Samuel can know that Lucia'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 Clara does not know whether Mary is thirsty.

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

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

Hypothesis: Clara can now know whether Mary can know that Rhonda 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 Modesta's forehead is not muddy.

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

Hypothesis: Modesta can now know that Barbara can know that Rudolph's forehead is muddy.


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

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

Hypothesis: James can now know that Sharon can know that Danny'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 Maurine knows whether Elmer's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Elmer does not know whether Elmer'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 3) ] [ ! (1|2|3) ] [ ! ~(Agentc knows whether 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 3) ] [ ! (1|2|3) ] [ ! ~(Agentc knows whether 3) ] 0

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Jessica knows whether or not nobody 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 (Agentb knows that 3)

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

Hypothesis: Rebecca can now know whether Jessica can know that Crystal is thirsty.


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

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

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

Hypothesis: Deborah can now know whether Jack picked a red card.


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

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

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

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


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

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

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

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


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

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

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


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

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

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

Hypothesis: Elmer can now know that Mary can know that Linda picked a red card.


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

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

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

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


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

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

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


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

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

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

Hypothesis: Nadia can now know whether Brian 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 Robert does not know whether or not everyone's forehead is muddy.

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

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


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

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

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

Hypothesis: Michael can now know that Robert can know whether Robert 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 Timothy's forehead is not muddy. It is publicly announced that Mary knows whether someone's forehead is muddy.

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

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

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


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

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

Hypothesis: Joyce can now know whether Lori can know whether Kari 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 Bernard knows whether Bernard is thirsty. It is publicly announced that Albert knows whether or not everyone is thirsty.

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

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

Hypothesis: Albert can now know that everyone is thirsty.


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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Valerie knows whether Valerie'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) ] Agentc knows that (Agentd knows that 4)

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

Hypothesis: Valerie can now know that Margaret can know that Margaret'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 Thomas's forehead is muddy. It is publicly announced that Earl knows whether someone's forehead is muddy. It is publicly announced that Thomas 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 ] [ ! (Agentc knows whether (1|2|3)) ] [ ! (Agenta knows whether (~(1|2|3))) ] Agenta knows whether (Agentb knows that (~(1|2|3)))

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

Hypothesis: Thomas can now know whether Carolyn can know that nobody's forehead is muddy.


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

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

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

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


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

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

Hypothesis: Carol can now know that Tanya can know whether or not everyone 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 George knows whether or not nobody is thirsty.

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

Hypothesis: George can now know whether Robert 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 Kelvin does not know whether Ernest is thirsty. It is publicly announced that Sharon 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) ] [ ! ~(Agentd knows whether 1) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] Agenta knows whether (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) ] [ ! ~(Agentd knows whether 1) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] 0

Hypothesis: Ernest can now know whether Kelvin can know whether James is thirsty.


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

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

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

Hypothesis: Darnell can now know that Johnnie can know whether or not nobody picked a red card.


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

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

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


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

Hypothesis: Richard 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 Grace'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) ] [ ! ~(4) ] Agentc knows that (Agentb knows that 4)

Hypothesis: Juanita can now know that Johnny can know that Grace'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 Sarah 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)) ] Agenta knows whether (Agentd knows whether 2)

Hypothesis: Audrey can now know whether Anne can know whether Sarah'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 Seth knows whether someone's forehead is muddy.

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Thurman does not know whether Thurman's forehead is muddy. It is publicly announced that Dorothy 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) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] (Agenta knows that (1&2&3&4))

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

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


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

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

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

Hypothesis: Holly can now know whether Joseph can know that Cheryl 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 Kevin's forehead is not muddy.

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

Hypothesis: Bryant can now know whether Kevin can know that Bryant'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 Kris is not thirsty. It is publicly announced that Gearldine does not know whether Madeline is thirsty.

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

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

Hypothesis: Madeline can now know that everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Doris does not know whether Doris'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) ] (Agentc 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) ] [ ! ~(Agentb knows whether 2) ] 0

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


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

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

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

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


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

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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Lucas is thirsty. It is publicly announced that James knows whether Lucas 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 whether (~(1|2|3)))

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

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


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Zachary does not know whether or not everyone's forehead is muddy. It is publicly announced that Juana knows 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)) ] [ ! (Agentb knows whether (1&2)) ] Agentb knows whether (Agenta knows whether (1&2))

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

Hypothesis: Juana can now know whether Zachary 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 Tatiana knows whether Robert's forehead is muddy. It is publicly announced that Tatiana'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 (Agentb knows that (1|2))

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

Hypothesis: Robert can now know whether Tatiana 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 Marie knows whether or not nobody's forehead is muddy. It is publicly announced that Marie knows whether Marie's forehead is muddy. It is publicly announced that Derek's forehead is muddy. It is publicly announced that Dora knows whether Derek's forehead is muddy.

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

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 (~(1|2|3|4))) ] [ ! (Agentb knows whether 2) ] [ ! 1 ] [ ! (Agentc knows whether 1) ] 0

Hypothesis: Marie can now know whether Dora can know that Derek'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 Gloria does not know whether Gloria's forehead is muddy. It is publicly announced that Stewart'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) ] [ ! 3 ] (Agentd knows that 3)

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

Hypothesis: Gloria can now know that Stewart'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 Carrie knows whether Elbert'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 4) ] Agentb knows whether (Agenta knows that 2)

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

Hypothesis: Debra can now know whether Alice can know that Debra's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Alissa's card is shown to Alissa. Lisa's card is shown to Alissa. Alissa's card is shown to Lisa. 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 Alissa picked a black card.

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

Hypothesis: Lisa can now know that Alissa 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 Barbara knows whether or not everyone's forehead is muddy. It is publicly announced that Marina does not know whether Marina's forehead is muddy. It is publicly announced that Jack knows whether Frank's forehead is muddy. It is publicly announced that Marina does not know whether Marina's forehead is muddy.

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

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

Hypothesis: Jack 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 Michael 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))) ] (Agenta knows that (1&2))

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

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


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

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

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

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


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

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

Hypothesis: Olin can now know whether Elva can know whether Elva is thirsty.


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

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

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

Hypothesis: Thomas can now know that Michael picked a red card.


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

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

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

Hypothesis: Amy can now know that Angela can know that Angela is thirsty.


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

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

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

Hypothesis: Leonora 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.

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 that 2)

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


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

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

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

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


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

Hypothesis: Jeremy can now know that Robert can know that everyone 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 Charles does not know whether Kathleen is thirsty. It is publicly announced that Theresa is not thirsty.

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

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

Hypothesis: Theresa can now know whether someone is thirsty.


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

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

Hypothesis: Evelyn can now know that Christopher 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 Rashida knows whether Rashida'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 that (~(1|2|3|4)))

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

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


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

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

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

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


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

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 (Agenta knows whether (~(1|2|3|4)))

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


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

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

Hypothesis: Ethel can now know whether Nancy can know that Nancy'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 Faith knows whether Faith's forehead is muddy. It is publicly announced that Michael knows whether Faith's forehead is muddy. It is publicly announced that Faith knows whether Joan'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) ] Agentb knows that (Agentd knows that 3)

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

Hypothesis: Joan can now know that Fred can know that Faith's forehead is muddy.