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 Louis knows whether or not everyone's forehead is muddy. It is publicly announced that Frances's forehead is muddy.

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

Hypothesis: Frances can now know that Felix'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 Evelyn knows whether someone is thirsty. It is publicly announced that Toni knows whether or not nobody is thirsty. It is publicly announced that Lisa knows whether Lisa is thirsty. It is publicly announced that Toni does not know whether Evelyn 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) ] (Agentd knows whether (1&2&3&4))

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

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


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

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

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

Hypothesis: Clifford can now know whether Frankie can know that Frankie'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 James'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 whether (Agentb knows whether (1|2|3))

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


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

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

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

Hypothesis: Mary can now know that Mathew can know whether someone's forehead is muddy.


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

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

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

Hypothesis: Christine can now know whether Christine 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 Kim's forehead is not muddy.

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

Hypothesis: Marisa can now know that Kim can know whether Marisa'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. John's card is shown to Lucie. Lucie's card is shown to Lucie. Carol's card is shown to Carol. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Lucie can now know whether Carol picked a red card.


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

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

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

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


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

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

Hypothesis: Douglas can now know that Cindy can 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 Rosa does not know whether Rosa's forehead is muddy. It is publicly announced that Gilbert knows whether Rosa's forehead is muddy.

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

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

Hypothesis: Rosa can now know that Kim can know whether Rosa'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 Gregory knows whether Gregory's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: June can now know whether Eugene picked a red card.


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

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

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


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Clifton can now know that Quinn can know whether someone is thirsty.


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

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

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

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


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

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

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


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

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

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

Hypothesis: Gina can now know that Terry can know that nobody is thirsty.


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

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

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

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


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

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

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

Hypothesis: John can now know whether Michael can know whether Michael 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 Nola knows whether Nola is thirsty. It is publicly announced that Frances knows whether Frances is thirsty.

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

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

Hypothesis: Nola can now know that Frances can 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 Michael knows whether someone is thirsty.

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

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

Hypothesis: Kourtney can now know whether Michael can know whether Angela 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 Loretta's forehead is muddy.

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

Hypothesis: Marguerite can now know that Dennis can know whether someone's forehead is muddy.


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

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

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

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


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

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

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

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


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

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

Hypothesis: Brendon can now know that Billy can know that Brendon 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 Julie is not thirsty. It is publicly announced that Roberto does not know whether William is thirsty.

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

Hypothesis: Julie can now know that nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Thomas does not know whether Sally 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) ] Agentc knows whether (Agenta knows that 4)

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

Hypothesis: Jeffery can now know whether Thomas can know that Robert is thirsty.


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

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

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

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


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Linda knows whether or not nobody's forehead is muddy. It is publicly announced that Beulah'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) ] [ ! (Agentc knows whether (~(1|2|3|4))) ] [ ! ~(2) ] Agenta knows that (Agentc knows whether 4)

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


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Jeff can now know that Raymond can know whether Raymond is thirsty.


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

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

Hypothesis: Lester can now know that Tanya can know that Lester 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 Larry's forehead is not muddy.

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

Hypothesis: James can now know whether Larry can know that nobody's forehead is muddy.


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

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

Hypothesis: Travis can now know that Susan can know whether someone 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 Clare knows whether Clare'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) ] Agentd knows that (Agenta knows whether 1)

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

Hypothesis: Clare can now know that Monica can know whether Monica'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 Michael does not know whether Maria is thirsty. It is publicly announced that Maria is not thirsty. It is publicly announced that Betty does not know whether Michael is thirsty.

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

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

Hypothesis: Betty can now know whether Maria is thirsty.


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

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

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

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

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

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


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

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

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

Hypothesis: Tommie can now know that Tommie 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 Darlene knows whether Julia's forehead is muddy.

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

Hypothesis: Julia can now know that Denis can know that Tracey's forehead is muddy.


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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lorna'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 that (Agenta knows whether (1&2&3))

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


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

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

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

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


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

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

Hypothesis: Chad can now know that John can know whether Andrew's forehead is muddy.


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 Donna. Annie's card is shown to Donna. William's card is shown to William. It is publicly announced that someone picked a red card.

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

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

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


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

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

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


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

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

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

Hypothesis: Melvin can now know that Dorothy 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 Michael does not know whether Robert is thirsty.

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

Hypothesis: Rachel can now know that Robert can know whether or not nobody is thirsty.


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

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

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

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


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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Thomas's forehead is muddy. It is publicly announced that Sidney knows whether Jimmy'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 ] [ ! (Agentc knows whether 1) ] Agenta knows whether (Agentb knows whether (1&2&3))

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

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


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

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

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

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


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

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

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

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


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

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: Louise can now know that Alicia can know whether Alicia'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 Tony knows whether or not nobody is thirsty. It is publicly announced that Tony does not know whether Brandy is thirsty. It is publicly announced that Danny does not know whether or not everyone is thirsty. It is publicly announced that Tony does not know whether or not everyone is thirsty.

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

Hypothesis: Brandy can now know whether Tony 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 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 whether (Agentb knows that 1)

Hypothesis: Roger can now know whether Randall can know that Beau'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 Barbara'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 ] Agenta knows that (Agentb knows that (~(1|2|3)))

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


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

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

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

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


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

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

Hypothesis: Michael can now know that Michael'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 Edwin knows whether someone is thirsty.

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

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

Hypothesis: Martha can now know that Edwin can know whether Edwin 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 Willie knows whether Jason's forehead is muddy.

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

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


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

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

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

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


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

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

Hypothesis: Richard 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). Sarah's card is shown to Lori. Sarah's card is shown to Sarah. 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))

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

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


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

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

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

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


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

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

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


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

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

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


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

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

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

Hypothesis: Jorge can now know that 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). Maria's card is shown to Maria. Darlene's card is shown to Darlene. 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 Maria does not know whether Darlene picked a red card.

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

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

Hypothesis: Maria can now know that Darlene can know whether Maria picked a red card.


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

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

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

Hypothesis: Joe can now know whether Ernie can know that someone's forehead is muddy.


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

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

Hypothesis: Daniel can now know whether Phyllis 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 James knows whether Gayle's forehead is muddy.

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

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


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

Hypothesis: Ronald can now know whether Hector can know whether Hector'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 Delphia 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))) ] Agentb knows that (Agentc knows that 3)

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

Hypothesis: Amy can now know that Stacey can know that Stacey'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 Florence knows whether Thomas's forehead is muddy.

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

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

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


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

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agentb knows whether (1&2)) ] (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: Caroline can now know that everyone's forehead is muddy.


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

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

Hypothesis: Blanche can now know whether Glenda can know whether Blanche 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 Ellen does not know whether Maria is thirsty. It is publicly announced that Ernesto does not know whether or not everyone is thirsty. It is publicly announced that Ellen does not know whether Joyce is thirsty. It is publicly announced that Maria 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) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] [ ! ~(Agentb knows whether 4) ] [ ! (Agenta knows whether (1&2&3&4)) ] Agentb 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) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] [ ! ~(Agentb knows whether 4) ] [ ! (Agenta knows whether (1&2&3&4)) ] 0

Hypothesis: Ellen can now know whether Joyce can know that Ernesto 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 Beverly knows whether Jesse's forehead is muddy. It is publicly announced that Robert'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) ] [ ! (Agentc knows whether 4) ] [ ! ~(2) ] (Agentc knows whether 2)

Hypothesis: Beverly can now know whether Robert'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 Joseph knows whether Gerald's forehead is muddy. It is publicly announced that Joseph's forehead is muddy.

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

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

Hypothesis: Joseph can now know that Joseph'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 Wilfred knows whether Wilfred's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Sara 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). Anna's card is shown to Anna. John's card is shown to Anna. Anna's card is shown to John. Anna's card is shown to Joshua. Joshua's card is shown to Joshua. Anna's card is shown to Sheena. It is publicly announced that someone picked a red card.

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

Hypothesis: Anna can now know whether or not 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 Thomas's forehead is not muddy. It is publicly announced that Mary knows whether Ruben's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Melissa can now know whether William can know that everyone picked a red card.


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

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

Hypothesis: Michael can now know that Felix'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 Peter knows whether Edward's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

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

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

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


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

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

Hypothesis: Keith can now know whether Rebecca can know whether Rebecca'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 Michael knows whether someone's forehead is muddy. It is publicly announced that Michael knows whether someone's forehead is muddy. It is publicly announced that Richard knows whether or not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

Hypothesis: Richard 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). Daniel's card is shown to Daniel. Myrna's card is shown to Daniel. Myrna's card is shown to Kristin. Myrna's card is shown to Joseph. Joseph's card is shown to Myrna. Myrna's card is shown to Myrna. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Myrna can now know whether Daniel can know that Myrna 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 David does not know whether or not everyone's forehead is muddy. It is publicly announced that Barbara knows whether David's forehead is muddy.

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

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


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Natalie knows whether someone is thirsty. It is publicly announced that Sandra knows whether or not nobody is thirsty. It is publicly announced that Sandra does not know whether or not everyone is thirsty.

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

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

Hypothesis: Sandra can now know that Natalie can 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 Susan knows whether Susan's forehead is muddy.

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

Hypothesis: Rebecca can now know that James can know that Arden'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 Wendy knows whether someone's forehead is muddy. It is publicly announced that Blythe knows whether Blythe's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Wendy knows whether someone's forehead is muddy.

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

Hypothesis: Blythe 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). Rebekah's card is shown to George. Joshua's card is shown to George. George's card is shown to Joshua. 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|2|3))

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

Hypothesis: George 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 someone is thirsty. It is publicly announced that Louis does not know whether Barbara is thirsty. It is publicly announced that Barbara does not know whether Louis 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 3) ] [ ! ~(Agentc knows whether 2) ] Agenta knows that (Agentc knows whether 2)

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

Hypothesis: Bobby can now know that Barbara can know whether Louis 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 Mary knows whether Mary's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Laura can now know that Pedro can know whether Pedro'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 Tracey does not know whether Tracey's forehead is muddy. It is publicly announced that Lori knows whether someone's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

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


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

Hypothesis: Rita 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). Patricia's card is shown to Robert. Nikki's card is shown to Robert. It is publicly announced that someone picked a red card. It is publicly announced that Patricia knows whether Nikki picked a red card. It is publicly announced that Patricia picked a red card.

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

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

Hypothesis: Patricia can now know that Patricia 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 Matthew does not know whether or not everyone is thirsty.

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

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

Hypothesis: Matthew can now know that Wayne can know that Matthew 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 Lottie's forehead is not muddy.

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

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

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


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

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

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


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

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

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

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


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

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

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

Hypothesis: Gerald can now know that Leanne can know whether Gerald'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 Annie's forehead is muddy. It is publicly announced that Barbara does not know whether Barbara's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Eva can now know that Phillip 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 Henry does not know whether Henry'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) ] Agentd knows that (Agentb knows whether (1&2&3&4))

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

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


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

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

Hypothesis: Olga can now know whether Betty 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 Irene does not know whether Abby is thirsty. It is publicly announced that Abby does not know whether Leslie is thirsty. It is publicly announced that Leslie is thirsty.

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

Hypothesis: Leslie can now 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). Larry's card is shown to Robert. Robert's card is shown to Larry. Terry's card is shown to Larry. It is publicly announced that someone picked a red card. It is publicly announced that Terry knows whether Robert picked a red card.

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

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

Hypothesis: Terry can now know that Robert 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 Gerald 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)) ] Agentb knows that (Agenta knows whether 3)

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

Hypothesis: Rich can now know that Gerald can know whether Scott is thirsty.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: William can now know whether William 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 Phyllis knows whether or not everyone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Ida knows whether Ida's forehead is muddy.

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

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


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 Althea. Althea's card is shown to Althea. Helen's card is shown to Althea. Althea's card is shown to Marlene. Helen's card is shown to Marlene. Althea's card is shown to Helen. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Christopher can now know whether Helen picked a red card.


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

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

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

Hypothesis: Sonya can now know that Eugene 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 Phillip's forehead is not muddy. It is publicly announced that Patricia's forehead is not muddy.

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


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

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

Hypothesis: Raymond can now know that Debra can know that Debra 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 Linda knows whether Linda's forehead is muddy. It is publicly announced that Trisha's forehead is not muddy. It is publicly announced that not everyone's forehead is muddy.

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

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


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

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

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


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

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

Hypothesis: Johnny can now know whether Johnny'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 Maggie does not know whether Maggie's forehead is muddy. It is publicly announced that Omar knows whether Omar's forehead is muddy.

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Duane knows whether Andrew's forehead is muddy. It is publicly announced that Andrew'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 2) ] [ ! 2 ] (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) ] [ ! (Agentd knows whether 2) ] [ ! 2 ] 0

Hypothesis: Juanita can now know that Andrew'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 Stacy is thirsty. It is publicly announced that Stacy is not thirsty.

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

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

Hypothesis: Stacy can now know whether Martha can know that Martha is thirsty.


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

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

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

Hypothesis: Marie 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's forehead is muddy. It is publicly announced that Maryellen does not know whether Maryellen's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

Hypothesis: Richard can now know whether Maryellen'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 Dawn does not know whether or not everyone is thirsty.

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

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

Hypothesis: Lawrence can now know whether Lynda can know that Dawn is thirsty.


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

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

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

Hypothesis: Miles can now know whether Elna 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 Richard knows whether Mary's forehead is muddy.

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

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

Hypothesis: Jeannie can now know that Mary can know that Mary'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 Brenda knows whether Markus's forehead is muddy.

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

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

Hypothesis: Brenda can now know whether Markus can know whether Markus'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 Lilliana is not thirsty.

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

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

Hypothesis: Larry can now know that someone is thirsty.


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

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

Hypothesis: Joan can now know that Frank can know that Heidi'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 Sharon does not know whether Sharon'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) ] Agentd knows whether (Agenta knows that 1)

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

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


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

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

Hypothesis: Philip can now know that William'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 Octavio'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 ] Agentb knows whether (Agentd knows that (~(1|2|3|4)))

Hypothesis: Anna can now know whether Ebony can 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). Prince's card is shown to Prince. Judith's card is shown to Judith. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Judith can now know whether Prince can know that Judith 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 Laurie knows whether Laurie's forehead is muddy.

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

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

Hypothesis: Laurie can now know that Earl can know whether Earl'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 Joseph does not know whether Kory is thirsty. It is publicly announced that Joseph knows whether or not everyone is thirsty. It is publicly announced that Joseph does not know whether Carlos is thirsty.

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

Hypothesis: Carlos can now know that nobody is thirsty.


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

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

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

Hypothesis: Jorge 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's forehead is muddy. It is publicly announced that Margie knows whether or not everyone's forehead is muddy. It is publicly announced that Sybil'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) ] [ ! (Agentb knows whether (1&2&3)) ] [ ! ~(1) ] (Agenta knows that (~(1|2|3)))

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

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


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

Hypothesis: Katy 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 is thirsty. It is publicly announced that Nicholas 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)) ] Agentd knows that (Agentb knows that 1)

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

Hypothesis: Shirley can now know that Irene can know that Nicholas is thirsty.


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

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

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

Hypothesis: Kirk can now know that Vincent can know whether Tracy picked a red card.


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

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

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

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


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

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

Hypothesis: Leila can now know that Catherine 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 Meghan does not know whether William is thirsty. It is publicly announced that John knows whether John is thirsty.

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

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

Hypothesis: William can now know that Meghan can know whether or not nobody is thirsty.


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

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

Hypothesis: Dawn can now know that Ruth can know that Karyn'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 Beth's forehead is muddy.

Hypothesis: Peggy can now know that Beth'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 someone is thirsty.

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

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

Hypothesis: Cathy can now know whether Anna can know that Anna is thirsty.