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

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

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

Hypothesis: Margaret can now know whether Barbara can know whether Margaret 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 Ty does not know whether Heidi is thirsty.

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

Hypothesis: Ty can now know whether Cynthia can know whether Heidi 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 Alison knows whether someone's forehead is muddy. It is publicly announced that Michael does not know whether or not everyone's forehead is muddy.

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

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

Hypothesis: Michael can now know whether Alison can know whether Alison'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 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 3)

Hypothesis: Cindy can now know that Cindy'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 Lana knows whether or not nobody's forehead is muddy. It is publicly announced that Colleen's forehead is muddy.

Hypothesis: Geneva can now know that Colleen'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 Michele does not know whether Blake is thirsty. It is publicly announced that not everyone is thirsty.

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

Hypothesis: Michele can now know whether someone is thirsty.


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

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

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

Hypothesis: Jennifer can now know that Jennifer 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 Stephen's forehead is not muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Margaret knows whether Stephen'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) ] [ ! (~(1&2&3)) ] [ ! (Agenta knows whether 3) ] Agenta knows whether (Agentb knows that 3)

Hypothesis: Margaret can now know whether Lonnie can know that Stephen's forehead is muddy.


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

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

Hypothesis: Andrea can now know that Bret 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 Clara is thirsty. It is publicly announced that Willian 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) ] [ ! 3 ] [ ! (Agenta knows whether (~(1|2|3|4))) ] Agenta 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) ] [ ! 3 ] [ ! (Agenta knows whether (~(1|2|3|4))) ] 0

Hypothesis: Willian can now know whether Adam 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 Allison's forehead is not muddy. It is publicly announced that Allison's forehead is not muddy. It is publicly announced that Norman knows whether Allison's forehead is muddy.

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

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

Hypothesis: Vickie can now 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). Bonita's card is shown to Bonita. Bonita's card is shown to Susan. Susan's card is shown to Susan. Susan's card is shown to Brenda. Bonita's card is shown to Lee. It is publicly announced that someone picked a red card. It is publicly announced that Bonita does not know whether Lee picked a red card.

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

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

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


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

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

Hypothesis: Donna can now know whether Justin can 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 Paul knows whether Paul's forehead is muddy. It is publicly announced that Paul'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) ] Agentc knows that (Agenta knows that 2)

Hypothesis: Carter can now know that Paul can know that Duane'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 or not nobody's forehead is muddy.

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

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


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

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

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

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


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

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

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

Hypothesis: Ann can now know that Cathy can know whether Cathy picked a red card.


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

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

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Rebecca's card is shown to Crystal. Crystal's card is shown to Rebecca. Francis's card is shown to Rebecca. Crystal's card is shown to Francis. Francis's card is shown to Francis. 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:2 Agentb:1,3 Agentc:1,3 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] Agenta knows that (Agentb knows whether 2)

Hypothesis: Crystal can now know that Rebecca can know whether Rebecca picked a red card.


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

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

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

Hypothesis: Sarah can now know that Lemuel can know whether Lemuel picked a red card.


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

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

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

Hypothesis: Shayne can now know that Lottie can know that David 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 Diane's forehead is not muddy.

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

Hypothesis: Lauren can now know that Jeanne'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 Jane knows whether Gary'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) ] (Agentd knows that 1)

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


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

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

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

Hypothesis: Larry can now know that Tanya can know whether Hal 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 Mamie does not know whether Mamie's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Dorothy knows whether Dorothy 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) ] (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) ] [ ! (Agenta knows whether 1) ] 0

Hypothesis: Bradley can now know that nobody is thirsty.


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

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))

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

Hypothesis: Sang 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 Yolanda knows whether someone is thirsty. It is publicly announced that Joleen does not know whether Eugenia is thirsty. It is publicly announced that Mario is thirsty.

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

Hypothesis: Mario can now know that everyone is thirsty.


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

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 (Agentb knows that 2)

Hypothesis: Elenor can now know whether Jerry can know that Jerry's forehead is muddy.


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

Hypothesis: Josie 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 Mark 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) ] [ ! (Agentb knows whether (1|2|3|4)) ] (Agentd knows whether 4)

Hypothesis: Mariano can now know whether Mariano'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 someone is thirsty. It is publicly announced that Patricia does not know whether Brenda is thirsty. It is publicly announced that Patricia does not know whether Brenda is thirsty. It is publicly announced that Patricia does not know whether Brenda is thirsty.

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

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

Hypothesis: Brenda can now know that everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Michael is not thirsty. It is publicly announced that Michael does not know whether Jacqueline is thirsty. It is publicly announced that Robert does not know whether Richard is thirsty. It is publicly announced that Robert does not know whether Richard is thirsty.

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

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


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

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

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

Hypothesis: Lori 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 everyone's forehead is muddy.

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

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


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

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

Hypothesis: James can now know whether Brittany can know whether someone is thirsty.


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

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


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

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

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

Hypothesis: Melinda can now know whether Pamula can know that Pamula'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 Rose knows whether Rose is thirsty. It is publicly announced that Eric does not know whether Rose is thirsty.

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

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

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

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

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

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


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

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

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

Hypothesis: Chantel can now know whether Susan is thirsty.


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 Dennis. Xuan's card is shown to Dennis. Dennis's card is shown to Charles. Charles's card is shown to Xuan. Xuan's card is shown to Xuan. It is publicly announced that someone picked a red card. It is publicly announced that Xuan picked a red card. It is publicly announced that Dennis does not know whether or not everyone picked a red card. It is publicly announced that Dennis 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))) ] Agenta knows that (Agentc knows that 1)

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: Dennis can now know that Xuan can know that Dennis 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 Linda does not know whether Grady is thirsty.

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

Hypothesis: Jose can now know that Joseph 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 William knows whether William's forehead is muddy. It is publicly announced that Gary knows whether James's forehead is muddy.

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

Hypothesis: Jane can now know that William can know that James'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 Isaac knows whether or not everyone is thirsty.

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

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

Hypothesis: Jeff can now know that Isaac can know that Rafael 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 someone is thirsty. It is publicly announced that Christina does not know whether Arthur is thirsty.

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

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

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

Hypothesis: Marie 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 Cindy knows whether Cindy is thirsty. It is publicly announced that Cindy does not know whether Carla is thirsty.

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

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

Hypothesis: Cindy can now know whether Lynn can 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 Walter does not know whether Patsy is thirsty. It is publicly announced that Edmond does not know whether Walter is thirsty. It is publicly announced that Walter does not know whether or not everyone is thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: Edmond can now know that Edmond is thirsty.


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

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

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

Hypothesis: Lance can now know that Lance 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 Glenn knows whether Heather's forehead is muddy. It is publicly announced that Heather knows whether or not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Bianca knows 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)) ] (Agentc knows that (1&2&3))

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

Hypothesis: Nicholas can now know that everyone is thirsty.


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

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

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

Hypothesis: John can now know whether Kenneth can know whether John is thirsty.


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

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

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

Hypothesis: Rosemary can now know that Debbie can know that everyone 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 someone's forehead is muddy.

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

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

Hypothesis: Linda can now know whether Linda'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 Francisco knows whether or not everyone's forehead is muddy. It is publicly announced that Waylon'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)) ] [ ! 4 ] Agentd knows whether (Agenta knows whether (~(1|2|3|4)))

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


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

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

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

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


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

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

Hypothesis: Douglas can now know whether someone is thirsty.


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

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

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

Hypothesis: Jose can now know whether Tracey can know that someone picked a red card.


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

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

Hypothesis: William can now know that Freda can know that Robert'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 Stacy knows whether or not everyone is thirsty.

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

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

Hypothesis: Mai can now know whether Stacy can know whether Mai 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.

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


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

Hypothesis: Donald can now know that Howard'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 Donna's forehead is muddy. It is publicly announced that George knows whether Donna'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) ] [ ! 2 ] [ ! (Agentd knows whether 2) ] (Agentc knows whether 1)

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

Hypothesis: Ray can now know whether Stacy'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 not everyone's forehead is muddy. It is publicly announced that Daniel's forehead is not muddy. It is publicly announced that Martin's forehead is muddy.

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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Steven's card is shown to David. Claude's card is shown to Steven. David's card is shown to Claude. 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:2 Agentb:3 Agentc:1 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] (Agentc knows that 3)

Hypothesis: Claude can now know that Claude 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 Derek knows whether or not everyone's forehead is muddy.

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

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

Hypothesis: Dorothy 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 not everyone's forehead is muddy. It is publicly announced that Fletcher knows whether or not nobody's forehead is muddy. It is publicly announced that Kasey knows whether Derek's forehead is muddy.

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

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


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

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

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

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

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

Hypothesis: Genevieve can now know whether David can know that David'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 Chester's forehead is not muddy.

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

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


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

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

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


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

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

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

Hypothesis: Janet can now know whether Michael can know that someone's forehead is muddy.


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

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

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


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

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

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

Hypothesis: Amy can now know that Paul can know that Paul 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 Thomas does not know whether Thomas's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

Hypothesis: Thomas can now know whether Thomas'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 Daniel does not know whether Patsy is thirsty. It is publicly announced that Daniel knows whether someone is thirsty. It is publicly announced that Trudy does not know whether Patsy is thirsty. It is publicly announced that Patsy does not know whether Daniel is thirsty.

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

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

Hypothesis: Trudy can now know that nobody is thirsty.


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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Krystal does not know whether Clyde is thirsty. It is publicly announced that Clyde 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)) ] Agenta 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) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] 0

Hypothesis: Krystal can now know that Maryetta 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 not everyone's forehead is muddy. It is publicly announced that Terry knows whether or not nobody's forehead is muddy.

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

Hypothesis: Don can now know that Victor can know whether Jose's forehead is muddy.


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

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

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


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

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

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


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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Joanne does not know whether Ronald is thirsty. It is publicly announced that Arthur does not know whether Ronald is thirsty. It is publicly announced that Arthur knows whether someone is thirsty.

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

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

Hypothesis: Ronald can now know that Joanne can know that Joanne 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 Judy's forehead is muddy. It is publicly announced that Judy knows whether Judy's forehead is muddy.

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

Hypothesis: Kimberley can now know whether Judy can know that someone's forehead is muddy.


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

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

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

Hypothesis: Susan can now know whether Alma can know that Christine 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 Jim's forehead is not muddy.

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

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


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

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


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

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

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

Hypothesis: Mindy can now know whether Bill can know whether Bill'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 Brett does not know whether or not everyone's forehead is muddy. It is publicly announced that Rosalyn's forehead is muddy.

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

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

Hypothesis: Brett 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 Kurt knows whether someone's forehead is muddy. It is publicly announced that Kurt'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 that (Agentb knows that (1|2|3))

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

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


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

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

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

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


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

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

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

Hypothesis: Arnold can now know that Lawrence 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 Heidi knows whether someone is thirsty. It is publicly announced that Virginia does not know whether Kelly 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 3) ] (Agentc knows that (1&2&3))

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

Hypothesis: Kelly can now know that everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Brian knows whether Douglas's forehead is muddy. It is publicly announced that Brian's forehead is not muddy. It is publicly announced that Nancy knows whether someone's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Florence can now know whether Olga can know whether Jessica picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ivette's card is shown to Howard. Jeanne's card is shown to Howard. Howard's card is shown to Ivette. Ivette's card is shown to Ivette. Steven's card is shown to Jeanne. It is publicly announced that someone picked a red card. It is publicly announced that Ivette 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 whether 3)

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: Ivette can now know whether Ivette picked a red card.


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

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

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

Hypothesis: Timothy can now know whether Juanita can know that everyone picked a red card.


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

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

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

Hypothesis: Beatrice can now know whether Juana 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 Martin 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 that (Agentb knows that 1)

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

Hypothesis: Martin can now know that Kenneth can know that Martin'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 Irving knows whether Irving is thirsty. It is publicly announced that someone is thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: Harry can now know whether someone is thirsty.


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

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

Hypothesis: Jane can now know whether Ashley can know that Kimberly'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 Mildred does not know whether Deborah is thirsty.

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

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

Hypothesis: Christopher can now know whether Deborah can know that Deborah is thirsty.


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

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


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

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

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

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


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

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

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

Hypothesis: Glenn can now know whether Carly can know that nobody picked a red card.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Evelyn can now know whether Mary can know that Evelyn is thirsty.


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

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

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

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


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

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

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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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

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


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

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

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


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

Hypothesis: Denise can now know that Leroy'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 Juan's forehead is not muddy.

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

Hypothesis: Frank can now know that Thomas can know that Frank'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 everyone's forehead is muddy. It is publicly announced that Sara knows whether someone's forehead is muddy.

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

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

Hypothesis: Virginia 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 someone is thirsty. It is publicly announced that not everyone is thirsty.

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

Hypothesis: Gerald can now know whether Joseph is thirsty.


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

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

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

Hypothesis: Oscar can now know that Earlene 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 Mildred 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) ] [ ! (Agentb knows whether (1|2|3|4)) ] (Agentc knows that 2)

Hypothesis: Julie can now know that Mildred'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 Alfredo is thirsty. It is publicly announced that Edna does not know whether or not everyone is thirsty.

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

Hypothesis: Toni can now know whether Ruth is thirsty.


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

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

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

Hypothesis: Roy can now know that Rose can know whether Roy picked a red card.


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

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

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

Hypothesis: Lacey can now know whether Roseanne can know that Lacey 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 Lloyd is thirsty. It is publicly announced that Edwin 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) ] [ ! 3 ] [ ! (Agenta 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) ] [ ! 3 ] [ ! (Agenta knows whether (~(1|2|3|4))) ] 0

Hypothesis: Russell can now know that Raymond 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 Thomas knows whether Lisa's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 3) ] (Agentb knows whether 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: Norine can now know whether Lisa'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 Lester knows whether or not everyone's forehead is muddy.

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

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

Hypothesis: Lynda can now know that Lester can know that Jay's forehead is muddy.


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

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

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

Hypothesis: Aleida can now know that Crystal can know whether Crystal 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 Corinne does not know whether Corinne's forehead is muddy. It is publicly announced that Corinne knows whether or not everyone's forehead is muddy.

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

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

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


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

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

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


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

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

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

Hypothesis: Hassan can now know that Cory is thirsty.


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

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

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

Hypothesis: Tabitha can now know whether Robert 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 Tracey does not know whether or not everyone's forehead is muddy. It is publicly announced that Jeremy knows whether Tracey's forehead is muddy.

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

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

Hypothesis: Tracey can now know that Maria can know that Maria'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 Jon does not know whether or not everyone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

Hypothesis: Sharon can now 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 Robert's forehead is not muddy.

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

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


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

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

Hypothesis: Lynette can now know that Ronald can know whether James 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 Rose does not know whether Rose's forehead is muddy. It is publicly announced that Rose does not know whether Rose's forehead is muddy.

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

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

Hypothesis: Rose can now know that Nick can 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 someone is thirsty. It is publicly announced that Carrie does not know whether or not everyone is thirsty. It is publicly announced that not everyone is thirsty.

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

Hypothesis: Martin can now know whether Carrie can know whether someone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Elaine 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)) ] Agenta knows that (Agentd knows whether 4)

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

Hypothesis: Elaine can now know that Judith can know whether Judith 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 Lizette's forehead is not muddy. It is publicly announced that Lee knows whether or not everyone's forehead is muddy.

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

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


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

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

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


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Lyda's card is shown to Craig. Lyda's card is shown to Lyda. Richard's card is shown to Lyda. Craig's card is shown to Bruce. Bruce's card is shown to Bruce. Craig'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:2 Agentb:2,4 Agentc:1,3 Agentd:1 VALID? [ ! (1|2|3|4) ] (Agentc knows whether 4)

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

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


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

Hypothesis: Wilmer can now know that Jacob can know whether Wilmer's forehead is muddy.


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

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

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

Hypothesis: Ron can now know whether Rita can know whether Gerard's forehead is muddy.


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

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

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

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


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

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

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

Hypothesis: Louis can now know that Lettie can know whether George 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) ] Agentc knows that (Agentb knows that 1)

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


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

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

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

Hypothesis: Betty can now know whether Frances can know whether Debi 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 Jacqueline does not know whether Lindsey is thirsty. It is publicly announced that Jacqueline does not know whether Barbara 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) ] [ ! ~(Agentd knows whether 3) ] Agenta knows whether (Agentc knows that 2)

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

Hypothesis: Lindsey can now know whether Barbara can know that James is thirsty.


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

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

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

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


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

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

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

Hypothesis: Louis can now know whether Michael can know that Michael 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 Judith does not know whether or not everyone's forehead is muddy. It is publicly announced that Ebony knows whether or not nobody's forehead is muddy.

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

Hypothesis: Tom can now know that Ebony'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 Paul knows whether David's forehead is muddy.

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

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

Hypothesis: David 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 Matthew does not know whether Rick is thirsty.

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

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

Hypothesis: Patricia can now know that Lorie 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 Margery knows whether or not everyone's forehead is muddy. It is publicly announced that Michelle's forehead is muddy. It is publicly announced that Max's forehead is muddy.

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

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


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

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

Hypothesis: Darlene can now know that Jerry can know that 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 Steven does not know whether Morton is thirsty.

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

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

Hypothesis: Steven can now know whether Morton can know that nobody is thirsty.


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

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

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

Hypothesis: Bryan can now know that Bryan 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 Herbert is thirsty.

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

Hypothesis: Leslie can now know that Leslie 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 Michael knows whether Elizabeth'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) ] (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) ] [ ! (Agentc knows whether 1) ] 0

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


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

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

Hypothesis: Booker can now know whether Willard 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 Anthony knows whether someone's forehead is muddy.

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

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

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