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 Rodney knows whether Cruz'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) ] Agentb knows that (Agentc knows that (~(1|2|3|4)))

Hypothesis: Silas can now know that George 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 Louie does not know whether Sandra is thirsty. It is publicly announced that Mario does not know whether or not everyone is thirsty.

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

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

Hypothesis: Brenda can now know whether Mario can know that Sandra 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 Patricia knows whether someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Gary knows whether Heather'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 4) ] Agentb knows that (Agentd knows that (~(1|2|3|4)))

Hypothesis: Mary can now know that Heather 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 Jamison knows whether or not nobody is thirsty. It is publicly announced that Jamison knows whether Jamison is thirsty.

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

Hypothesis: Maria can now know that Jamison can know that everyone is thirsty.


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

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

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

Hypothesis: Stella can now know whether Shannon can know that Shannon's forehead is muddy.


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

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

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

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


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

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

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

Hypothesis: Donald can now know whether Adam 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 Rhonda'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 ] Agentc knows that (Agenta knows that 3)

Hypothesis: Rhonda can now know that Felicia can know that Rhonda'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 Jasmine knows whether Anne's forehead is muddy. It is publicly announced that Anne does not know whether or not everyone's forehead is muddy. It is publicly announced that Jasmine knows whether Jesus's forehead is muddy. It is publicly announced that Jasmine knows whether Jeffery's forehead is muddy.

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

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

Hypothesis: Jeffery 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 Marcia does not know whether Christopher is thirsty. It is publicly announced that Christopher does not know whether or not everyone is thirsty.

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

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

Hypothesis: Christopher can now know that Joan can know whether someone is thirsty.


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

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

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

Hypothesis: Martha can now know that Harvey picked a red card.


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

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

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

Hypothesis: Ronald can now know that Brenda can know whether or not nobody picked a red card.


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

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

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

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


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

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

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

Hypothesis: Debra can now know whether Carmen can know whether Debra is thirsty.


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

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

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


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

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

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

Hypothesis: Willie can now 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 someone's forehead is muddy.

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

Hypothesis: Scott can now know whether Maria can know that Ernie'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 Scott does not know whether or not everyone's forehead is muddy. It is publicly announced that Sheri knows whether someone's forehead is muddy. It is publicly announced that Scott does not know whether or not everyone's forehead is muddy.

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

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

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


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

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

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

Hypothesis: David can now know whether James can know whether James 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 Brittney knows whether John's forehead is muddy. It is publicly announced that John knows whether Brittney'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) ] [ ! (Agentc knows whether 2) ] Agenta 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) ] [ ! (Agentb knows whether 3) ] [ ! (Agentc knows whether 2) ] 0

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


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

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

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

Hypothesis: Deborah can now know that David can know that Wilma is thirsty.


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

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

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

Hypothesis: Cindy 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 Elaine knows whether or not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Brenda knows whether or not nobody's forehead is muddy. It is publicly announced that Brenda'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) ] [ ! (Agentc knows whether (1&2&3)) ] [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] [ ! ~(1) ] (Agentb knows whether 2)

Hypothesis: Patricia can now know whether Patricia'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 Farah knows whether or not nobody is thirsty.

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

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

Hypothesis: Joseph can now know whether Alan can know whether James is thirsty.


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

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

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

Hypothesis: Dong can now know whether Dorothy can 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 Hector knows whether someone's forehead is muddy. It is publicly announced that Naomi's forehead is not muddy.

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

Hypothesis: Naomi can now know whether Naomi'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 Cynthia knows whether someone's forehead is muddy.

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

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

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


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

Hypothesis: Virginia can now know whether Virginia is thirsty.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Christian knows whether Jaime is thirsty. It is publicly announced that Christian knows whether Jaime is thirsty. It is publicly announced that Jaime knows whether Christian is thirsty. It is publicly announced that Christian 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)) ] Agentb knows that (Agenta knows whether 1)

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

Hypothesis: Jaime can now know that Christian can know whether Christian 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 Robert's forehead is not muddy.

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Sharon 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&2&3&4)) ] Agentc knows that (Agenta knows that 4)

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

Hypothesis: Mary can now know that Pasquale can know that Gloria 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 Patricia is not thirsty.

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

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

Hypothesis: Steven can now know whether Alex 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 Jenifer knows whether Jenifer's forehead is muddy. It is publicly announced that Amy knows whether Vicky'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) ] [ ! (Agentd knows whether 1) ] (Agentc knows that 4)

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

Hypothesis: Maryann can now know that Amy'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. Chad's card is shown to Charles. Robin's card is shown to Adrianne. It is publicly announced that someone picked a red card. It is publicly announced that Adrianne does not know whether Adrianne picked a red card.

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

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

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

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

Hypothesis: Eric can now know whether Esther can know that Eric's forehead is muddy.


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

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

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

Hypothesis: Shellie can now know that Christopher picked a red card.


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

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

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

Hypothesis: Aaron 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 is thirsty. It is publicly announced that Richard does not know whether Zelda is thirsty.

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

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

Hypothesis: Richard can now know that Linda can know that everyone is thirsty.


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

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

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

Hypothesis: Andres can now know that John can know whether someone is thirsty.


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

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

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

Hypothesis: Lou can now know that Barbara can know whether or not nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that 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 1)

Hypothesis: Leonie can now know that Gregory'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 Juanita does not know whether Juanita'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) ] Agentc knows that (Agenta knows that 3)

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

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


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

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

Hypothesis: Steven can now know that Jennifer 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 Lawrence knows whether Irma's forehead is muddy.

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

Hypothesis: Lawrence can now know whether someone'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 Modesto knows whether Shirley'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) ] [ ! (Agenta knows whether 2) ] [ ! (~(1&2)) ] (Agentb knows that 2)

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

Hypothesis: Shirley can now know that Shirley'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 Shannon knows whether someone's forehead is muddy. It is publicly announced that Shannon's forehead is not muddy.

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

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

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


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

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

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

Hypothesis: Mike can now know that Noel can know that Noel 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 Katie's forehead is muddy.

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

Hypothesis: Luis can now know that Joyce can know whether Joyce'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 everyone is thirsty. It is publicly announced that everyone is thirsty.

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

Hypothesis: Sonia can now know whether Mary 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). Roxie's card is shown to Roxie. Tamela's card is shown to Roxie. Roxie's card is shown to Hung. Tamela's card is shown to Hung. Roxie's card is shown to Tamela. Hung's card is shown to Tamela. 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 Roxie picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that Hung knows whether Tamela picked a red card.

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

Hypothesis: Hung can now know whether Roxie picked a red card.


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

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

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


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

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

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

Hypothesis: Clyde can now know that Jason can know that Jonathan 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 Robert does not know whether or not everyone is thirsty. It is publicly announced that Jerry is not thirsty. It is publicly announced that Jerry knows whether someone is thirsty.

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

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

Hypothesis: Jerry can now know whether Robert can know whether Robert is thirsty.


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

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

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


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

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

Hypothesis: Colin can now know that Gloria can know whether someone is thirsty.


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

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


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

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

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

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


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

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

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

Hypothesis: Margaret can now know that Margaret 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 Sharon's forehead is muddy. It is publicly announced that Sharon knows whether Irene's forehead is muddy.

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Timothy does not know whether Sergio is thirsty. It is publicly announced that Sergio does not know whether Glen is thirsty. It is publicly announced that Timothy does not know whether Sergio is thirsty. It is publicly announced that Timothy does not know whether Rachel 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) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agentd knows whether 1) ] Agentd knows that (Agentc knows that (1&2&3&4))

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

Hypothesis: Timothy can now know that Glen can know that everyone is thirsty.


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

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

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

Hypothesis: Shannon can now know whether Brandon can know that everyone 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 Kenneth knows whether someone's forehead is muddy. It is publicly announced that Kenneth's forehead is muddy. It is publicly announced that 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)) ] [ ! 2 ] [ ! (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)) ] [ ! 2 ] [ ! (1&2) ] 0

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


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


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

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

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

Hypothesis: Nicholas can now know that Kenneth can know that Raquel is thirsty.


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

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

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

Hypothesis: Sandra can now know that Jeffrey can know that Mathew 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 Sarah does not know whether Marcus is thirsty. It is publicly announced that David does not know whether or not everyone is thirsty. It is publicly announced that David does not know whether Marcus is thirsty. It is publicly announced that Marcus does not know whether or not everyone is thirsty.

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

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

Hypothesis: Kelly can now know that Sarah can know whether David is thirsty.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Carol can now know that John can know whether John is thirsty.


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

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

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

Hypothesis: Kathryn can now know whether Charles can know that Walter picked a red card.


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

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

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

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


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

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

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

Hypothesis: June can now know whether Ethel'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 Ellen 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)) ] (Agentd knows whether 1)

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

Hypothesis: Ellen can now know whether Jesse 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 Javier'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 (Agenta knows that 1)

Hypothesis: Janis can now know whether Harold can know that Harold's forehead is muddy.


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

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

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

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


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

Hypothesis: Janet 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 Stephanie knows whether someone's forehead is muddy.

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

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

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

Hypothesis: Joseph can now know that Susan can know that nobody is thirsty.


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

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

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

Hypothesis: Hope can now know that Hope 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 Felicia does not know whether James is thirsty. It is publicly announced that Felicia does not know whether Ruth is thirsty.

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

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

Hypothesis: Ruth can now know that James can know that Felicia 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 Vaughn knows whether or not nobody'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) ] [ ! (Agenta knows whether (~(1|2))) ] [ ! (~(1&2)) ] Agenta knows whether (Agentb knows that 1)

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

Hypothesis: Vaughn can now know whether Ingrid can know that Vaughn'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 Veronica does not know whether Veronica's forehead is muddy. It is publicly announced that Donald does not know whether or not everyone's forehead is muddy. It is publicly announced that Jenny knows whether Donald's forehead is muddy.

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

Hypothesis: Veronica can now know that Veronica'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 Carrie does not know whether Ross is thirsty. It is publicly announced that Carrie does not know whether Marian is thirsty. It is publicly announced that Ross does not know whether Marian is thirsty.

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

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

Hypothesis: Carrie can now know whether Carrie 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 Ronald knows whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1&2&3&4)) ] (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: Ronald can now know whether Colleen 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 Carlos does not know whether Carlos's forehead is muddy.

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

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

Hypothesis: Terrance can now know that Carlos can know that Jason'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 Brian does not know whether Brian's forehead is muddy.

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

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

Hypothesis: James can now know whether Brian can know whether James'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 Arturo does not know whether or not everyone is thirsty. It is publicly announced that Arturo does not know whether Ruth is thirsty. It is publicly announced that Ruth knows whether or not everyone is thirsty.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Anna can now know that Gregory can know whether or not everyone is thirsty.


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

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

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


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

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

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


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Rudy 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:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (~(1&2)) ] [ ! (Agenta knows whether (1&2)) ] [ ! (1|2) ] Agenta knows whether (Agentb knows whether (~(1|2)))

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


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

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

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

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


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

Hypothesis: Linda can now know whether Paul can know whether Kevin's forehead is muddy.


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

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

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


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

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

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

Hypothesis: Janette can now know that Jane is thirsty.


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

Hypothesis: Cecil can now know that Richard picked a red card.


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

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

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

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


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

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

Hypothesis: Rafaela can now know that Stephanie can know that Sonia's forehead is muddy.


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

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

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

Hypothesis: Ramon 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 Ruby knows whether or not everyone's forehead is muddy.

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

Hypothesis: Roy can now know whether Richard can know that nobody's forehead is muddy.


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

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

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

Hypothesis: Deborah can now know that nobody 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 Ashley knows whether Roger's forehead is muddy. It is publicly announced that Roger's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

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


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

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

Hypothesis: Ellen can now know whether Lou can know that Ellen's forehead is muddy.


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

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

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

Hypothesis: Victor can now know whether Nathaniel can know that Myrtle 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 Roderick is thirsty. It is publicly announced that Roderick does not know whether Ernest is thirsty.

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

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

Hypothesis: Ernest can now know whether Roderick can know whether someone is thirsty.


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

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

Hypothesis: Jordan can now know that Stanley is thirsty.


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

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

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

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


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

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

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

Hypothesis: Terri can now know whether Bruce can know that nobody is thirsty.


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

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

Hypothesis: Lula can now know that Louis can know that Annette'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 Rosemarie is not thirsty. It is publicly announced that Rosemarie does not know whether Juana is thirsty. It is publicly announced that Jesus knows whether Jesus is thirsty.

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

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

Hypothesis: Jesus can now know whether John can know that nobody is thirsty.


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

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

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

Hypothesis: Bobby can now know whether Bobby 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 Ada does not know whether or not everyone is thirsty. It is publicly announced that Ada does not know whether or not everyone is thirsty. It is publicly announced that Ada does not know whether or not everyone is thirsty. It is publicly announced that Ada does not know whether Laura is thirsty.

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

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

Hypothesis: Laura can now know that Ada can know that someone is thirsty.


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

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

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

Hypothesis: Gladys can now know whether Leonard 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 Geoffrey knows whether or not nobody is thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: Geoffrey can now know whether Jeffrey is thirsty.


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

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

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

Hypothesis: Emilio can now know that Emilio is thirsty.


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

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

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

Hypothesis: Lorenzo can now know that Carolyn picked a red card.


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

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

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

Hypothesis: Jennifer can now know that Miguel can know whether Victor picked a red card.


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

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

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

Hypothesis: Norma can now know that Francis can know that Wayne picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Richard knows whether Sharon'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) ] (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 3) ] 0

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


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

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

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

Hypothesis: Gary can now know that Charles is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). June's card is shown to June. Mary's card is shown to June. June's card is shown to Mary. Mary's card is shown to Mary. It is publicly announced that someone picked a red card. It is publicly announced that Mary picked a black card. It is publicly announced that June 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))) ] Agenta knows whether (Agentb knows whether (1&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: June can now know whether Mary can know whether or not everyone picked a red card.


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

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

Hypothesis: Thelma can now know that Carlos can know whether someone 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 Jessica 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)) ] (Agentb knows whether 2)

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

Hypothesis: Jessica can now know whether Jessica'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 Modesta knows whether Jason'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) ] (Agenta knows that (1&2&3))

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

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


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

Hypothesis: Teodoro can now know that Candi can know whether Teodoro'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 George knows whether or not nobody is thirsty. It is publicly announced that Garret does not know whether George is thirsty. It is publicly announced that George 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 whether 1) ] [ ! (Agenta knows whether (1&2&3&4)) ] (Agentc knows that 4)

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

Hypothesis: Richard can now know that Garret 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 Bonnie knows whether someone's forehead is muddy.

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

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

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


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

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

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

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

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

Hypothesis: Doris can now know whether Damian 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). Charles's card is shown to Charles. Stanley's card is shown to Eugenio. It is publicly announced that someone picked a red card. It is publicly announced that Charles does not know whether Eugenio picked a red card.

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

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

Hypothesis: Stanley can now know that Charles picked a red card.


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

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

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

Hypothesis: Charlotte can now know whether Michelle can know whether Manuel picked a red card.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Raymond's card is shown to Raymond. Raymond's card is shown to Jonathan. It is publicly announced that someone picked a red card. It is publicly announced that Jonathan 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 VALID? [ ! (1|2) ] [ ! (Agentb knows whether (1|2)) ] [ ! (1|2) ] (Agenta knows that (1&2))

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

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


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

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

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

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

Hypothesis: Michaele can now know whether Gerald can know whether David'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 Paul knows whether Travis's forehead is muddy. It is publicly announced that Paul does not know whether or not everyone's forehead is muddy. It is publicly announced that Travis knows whether or not everyone's forehead is muddy. It is publicly announced that Travis knows whether Paul's forehead is muddy.

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

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

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


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

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

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

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

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


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

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 that (Agenta knows that 4)

Hypothesis: Doreen can now know that Philip can know that Jeremy's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Michelle does not know whether Raymond 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 2) ] Agentb knows whether (Agentc knows that 3)

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

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


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

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

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

Hypothesis: Donald can now know that Karen 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). Yaeko's card is shown to Yaeko. Sandra's card is shown to Eunice. Eunice's card is shown to Sandra. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Eldridge can now know that 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). Arlene's card is shown to Michael. Timothy's card is shown to Michael. Timothy's card is shown to Timothy. It is publicly announced that someone picked a red card. It is publicly announced that Arlene does not know whether Arlene picked a red card.

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

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

Hypothesis: Arlene can now know that Michael can know whether Timothy picked a red card.


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

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

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

Hypothesis: Paulene can now know whether Carlos picked a red card.


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

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

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

Hypothesis: Dana can now know whether Dana 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 Robert does not know whether Robert's forehead is muddy.

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

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

Hypothesis: Traci can now know whether Robert'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 Raymond is not thirsty.

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

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

Hypothesis: Kenneth can now know that Kenneth is thirsty.


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

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

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

Hypothesis: James can now know that Barry can know that someone is thirsty.


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

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

Hypothesis: Tabetha can now know that someone'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 John knows whether Maria's forehead is muddy. It is publicly announced that Maria's forehead is muddy.

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

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

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


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

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

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

Hypothesis: George can now know that Sasha'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. It is publicly announced that Jennifer knows whether Jennifer is thirsty.

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

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

Hypothesis: Carla can now know that Carla 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 Shirley 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) ] [ ! (Agentb knows whether (1|2|3|4)) ] Agentd knows that (Agentb knows whether (1|2|3|4))

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

Hypothesis: Alonzo can now know that Shirley can know whether someone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Erwin does not know whether Clara 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 2) ] Agentc knows whether (Agenta knows that (1|2|3))

Hypothesis: Erwin can now know whether Martha can know that someone is thirsty.


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

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

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

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