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

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

Hypothesis: Jimmy can now know whether David can know that nobody picked a red card.


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

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

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

Hypothesis: Carl can now know that Antoinette picked a red card.


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

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

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

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

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

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


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

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

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

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


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

Hypothesis: Lucretia can now know whether Kelly can know whether Richard'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 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) ] [ ! (Agentc knows whether (~(1|2|3|4))) ] Agentb knows whether (Agenta knows that (1|2|3|4))

Hypothesis: Robin can now know whether Frances 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 Elisabeth does not know whether Elisabeth's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Randall can now know whether Randall'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 Jane knows whether Patrick'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) ] Agenta knows that (Agentc knows whether 2)

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

Hypothesis: Jane can now know that Patrick can know whether Judith's forehead is muddy.


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

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

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

Hypothesis: Alma can now know whether Alexandria can know that Alma picked a red card.


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

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

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

Hypothesis: Jennifer can now know that Jennifer 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 Victoria knows whether someone's forehead is muddy. It is publicly announced that Victoria knows whether Donald's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

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

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


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

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

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

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


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Raylene is thirsty. It is publicly announced that Raylene does not know whether Florence is thirsty.

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

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

Hypothesis: Raylene can now know that Florence can know whether 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 Carolyn 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 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) ] [ ! ~(Agentb knows whether (1&2&3)) ] 0

Hypothesis: Carolyn can now know whether Carol can know whether Carolyn'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 Barbara knows whether or not everyone is thirsty. It is publicly announced that Mary is not thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: Anthony can now 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 Lori does not know whether or not everyone is thirsty. It is publicly announced that not everyone is thirsty.

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

Hypothesis: Juanita can now know that Juanita 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 Jose knows whether Carrie'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 2) ] (Agentc knows that 4)

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

Hypothesis: Steven can now know that Robert'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 is thirsty. It is publicly announced that Irene knows whether Mildred is thirsty.

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

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

Hypothesis: Irene can now know whether Eric 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 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 ] Agenta knows whether (Agentb knows whether 3)

Hypothesis: Constance can now know whether Margaret can know whether 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 John knows whether Barbara's forehead is muddy.

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

Hypothesis: John can now know that John'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 Ann knows whether someone's forehead is muddy. It is publicly announced that Ann knows whether Ann's forehead is muddy. It is publicly announced that Sharon 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) ] (Agentc knows that 3)

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

Hypothesis: Sharon can now know that Sharon'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 Norma's forehead is muddy. It is publicly announced that Lillian knows whether someone's forehead is muddy. It is publicly announced that Norma knows whether or not nobody's forehead is muddy.

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

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


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

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

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

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


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

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

Hypothesis: Jim can now know that Shelley can know whether someone is thirsty.


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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Grace knows whether Merrill's forehead is muddy. It is publicly announced that Joanne 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) ] [ ! (~(1&2&3&4)) ] [ ! (Agenta knows whether 3) ] [ ! (Agentd knows whether (1|2|3|4)) ] (Agentb knows that 4)

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


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

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


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

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

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

Hypothesis: Martin can now know whether Ronald can know that Martin'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 Barbara does not know whether Mary is thirsty. It is publicly announced that Val does not know whether Mary is thirsty.

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

Hypothesis: Mary can now know that Mary is thirsty.


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

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

Hypothesis: Juan can now know that Preston can know that Juan'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 Carmen 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)) ] (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: Carmen can now know that Maria's forehead is muddy.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Brian can now know whether Margaret can know whether Brian is thirsty.


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

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

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

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


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

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

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

Hypothesis: Evelyn can now know whether Evelyn 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 Christopher 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)) ] Agentb knows that (Agentc 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) ] [ ! (Agentb knows whether (1&2&3)) ] 0

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


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

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

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

Hypothesis: Keith can now know whether Charles can know whether Angela 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 Samuel does not know whether Samuel's forehead is muddy. It is publicly announced that Samuel does not know whether Samuel'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) ] [ ! ~(Agenta knows whether 1) ] (Agenta knows whether 3)

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Adrienne's card is shown to Barbara. Forrest's card is shown to Adrienne. Barbara's card is shown to Forrest. 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)) ] (Agentb knows that 1)

Hypothesis: Adrienne can now know that Barbara 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 Daniel does not know whether or not everyone's forehead is muddy. It is publicly announced that Kate'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)) ] [ ! 4 ] Agentd knows whether (Agentc knows whether 3)

Hypothesis: Kate can now know whether 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). Mike's card is shown to Mike. Mike's card is shown to Michael. Linda's card is shown to Michael. Michael's card is shown to Linda. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Mike can now know that Linda 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). Alex's card is shown to Alex. Josue's card is shown to Josue. Tony's card is shown to Josue. Alex's card is shown to Tony. 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 that 1)

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

Hypothesis: Alex can now know that Josue can know that Alex picked a red card.


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

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

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

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


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

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

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

Hypothesis: Scott can now know whether Alan'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 Lloyd does not know whether Donnell is thirsty. It is publicly announced that someone is thirsty. It is publicly announced that Mary is not thirsty. It is publicly announced that Donnell 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 3) ] [ ! (1|2|3|4) ] [ ! ~(2) ] [ ! (Agentc knows whether (1&2&3&4)) ] (Agentd knows that 3)

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

Hypothesis: Kevin can now know that Donnell 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 Meagan's forehead is not muddy. It is publicly announced that John's forehead is not muddy.

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


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

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

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

Hypothesis: Lawrence can now know that Julian can 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 Steve does not know whether Barbara is thirsty. It is publicly announced that Steve does not know whether Sylvia 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 3) ] Agenta knows that (Agentb 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) ] [ ! ~(Agentb knows whether 3) ] 0

Hypothesis: Barbara can now know that Steve can know that Barbara 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 Paul knows whether someone's forehead is muddy.

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

Hypothesis: Kathleen can now know whether Gloria can know that Lynda'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 Jessica's forehead is not muddy. It is publicly announced that Renee knows whether someone's forehead is muddy.

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

Hypothesis: Renee 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 Candace 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 (Agentb knows whether 1)

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

Hypothesis: Jean can now know whether Candace can know whether Jean'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 Margaret's forehead is muddy. It is publicly announced that Mark does not know whether Mark's forehead is muddy. It is publicly announced that Ryan knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Mark can now know that Ryan'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 Ronald knows whether or not nobody's forehead is muddy.

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

Hypothesis: Grace can now know that Raymond'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 Karen knows whether Karen is thirsty.

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

Hypothesis: Karen can now 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). Derek's card is shown to Phyllis. Phyllis's card is shown to Phyllis. Cathy's card is shown to Phyllis. Phyllis's card is shown to Cathy. Cathy's card is shown to Cathy. It is publicly announced that someone picked a red card.

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

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

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


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

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

Hypothesis: Theodore can now know that Loyd can know that Theodore'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 Renna does not know whether or not everyone's forehead is muddy. It is publicly announced that Kelly's forehead is muddy.

Hypothesis: Kelly 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 Bradley does not know whether or not everyone's forehead is muddy. It is publicly announced that Bradley does not know whether Bradley's forehead is muddy. It is publicly announced that Antonia's forehead is muddy. It is publicly announced that Antonia knows whether Bradley's forehead is muddy.

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

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

Hypothesis: Bradley can now know that Antonia'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 Christopher knows whether Bernice's forehead is muddy. It is publicly announced that Helene knows whether Robert's forehead is muddy.

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

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

Hypothesis: Bernice can now know whether Bernice'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 Shelley knows whether Thomas's forehead is muddy.

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

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

Hypothesis: Karen can now know whether Shelley 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). Ashley's card is shown to Brett. Brett's card is shown to Matthew. It is publicly announced that someone picked a red card. It is publicly announced that Ashley picked a red card.

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

Hypothesis: Ashley can now 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 David knows whether Darcy's forehead is muddy. It is publicly announced that Darcy does not know whether Darcy's forehead is muddy. It is publicly announced that Darcy does not know whether Darcy's forehead is muddy. It is publicly announced that Darcy knows whether David's forehead is muddy.

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

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


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

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: Jan can now know that Tammy can know whether Tammy's forehead is muddy.


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

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

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

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


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

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

Hypothesis: Terry can now know whether Donald can know that Mark 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 Tamara is thirsty.

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

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

Hypothesis: Marc can now know whether Tamara can know that Marc 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 Robert knows whether someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1|2|3)) ] (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: Robert can now know that Edward'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 Curtis does not know whether Kathleen is thirsty.

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

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

Hypothesis: Steve can now know that 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 Antonio does not know whether Michelle is thirsty. It is publicly announced that Dawn does not know whether Antonio is thirsty. It is publicly announced that Antonio 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 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 3) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] [ ! (1|2|3|4) ] 0

Hypothesis: Dawn 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 not everyone's forehead is muddy. It is publicly announced that George knows whether or not nobody's forehead is muddy. It is publicly announced that Jerome knows whether Theresa'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) ] Agentb knows that (Agenta knows whether 3)

Hypothesis: George can now know that Frank can know whether Theresa's forehead is muddy.


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

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

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

Hypothesis: Jason can now 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). Clarence's card is shown to Clarence. Tameka's card is shown to Clarence. Sharon's card is shown to Sharon. Clarence's card is shown to Tameka. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Tameka can now know whether Clarence can know that someone 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 Steve knows whether Travis'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) ] (Agentd knows that (~(1|2|3|4)))

Hypothesis: John 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 is thirsty. It is publicly announced that Miquel does not know whether Steven is thirsty.

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

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

Hypothesis: Steven can now know whether Richard can know that Steven 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 Charles knows whether Tim's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

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

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

Hypothesis: Tim 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 William does not know whether Andrea is thirsty. It is publicly announced that Jacqueline is not thirsty.

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

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

Hypothesis: Raymond can now know whether Jacqueline can know that William 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 Glenn does not know whether Lori is thirsty.

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

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

Hypothesis: Lindsey can now know that Lori can know that everyone is thirsty.


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

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

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

Hypothesis: Hattie can now know that Linda'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 Nancy 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)) ] Agentb knows that (Agenta knows whether 3)

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

Hypothesis: Curtis can now know that Nancy can know whether Horace 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 Emily knows whether Lillie's forehead is muddy. It is publicly announced that Lillie's forehead is muddy.

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

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

Hypothesis: Emily can now know that Lillie 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's forehead is muddy. It is publicly announced that Ed does not know whether or not everyone's forehead is muddy. It is publicly announced that Carey knows whether or not nobody's forehead is muddy. It is publicly announced that Ed does not know whether Ed's forehead is muddy. It is publicly announced that Carey knows whether or not nobody'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))) ] [ ! ~(Agenta knows whether 1) ] [ ! (Agentb knows whether (~(1|2))) ] (Agentb knows whether 1)

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

Hypothesis: Carey can now know whether Ed'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 Robert knows whether someone's forehead is muddy. It is publicly announced that Beverly's forehead is not muddy.

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

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

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


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

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

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

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


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

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

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

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


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

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

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


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

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

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

Hypothesis: Glenn can now know whether Joann can know whether Joann 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 Mitchell does not know whether Mitchell's forehead is muddy. It is publicly announced that Lisa knows whether or not everyone's forehead is muddy.

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

Hypothesis: Mitchell can now know that Lisa can know whether someone's forehead is muddy.


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

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

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

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


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

Hypothesis: William can now know whether William'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 Richard's forehead is not muddy.

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

Hypothesis: Richard can now know that Beth can know that Richard'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 Donald does not know whether Donald's forehead is muddy.

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

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

Hypothesis: Inez can now know whether Donald 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 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) ] [ ! (1|2|3) ] Agentc knows whether (Agenta knows whether (1&2&3))

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

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

Hypothesis: Robert 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) ] Agenta knows that (Agentb knows whether (~(1|2|3)))

Hypothesis: Eric can now know that Lucy 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). Larry's card is shown to Steven. Steven's card is shown to Larry. Steven's card is shown to Sharon. Larry's card is shown to Sharon. Sharon's card is shown to Sharon. It is publicly announced that someone picked a red card. It is publicly announced that Larry does not know whether or not everyone picked a red card.

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

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

Hypothesis: Sharon can now know that Larry can know whether someone picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Rachel does not know whether Patricia is thirsty. It is publicly announced that Rachel does not know whether Patricia is thirsty.

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

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

Hypothesis: Patricia can now know that Rachel can know whether Patricia 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 Edward does not know whether Daniel is thirsty. It is publicly announced that Daniel is not thirsty.

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

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

Hypothesis: Vickie can now know whether Edward can know whether Erick 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 Melanie does not know whether or not everyone is thirsty. It is publicly announced that Michael knows whether Michael is thirsty. It is publicly announced that Melanie does not know whether or not everyone is thirsty.

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

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

Hypothesis: Lee can now know whether Michael can know that Michael 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 Geneva does not know whether Geneva's forehead is muddy. It is publicly announced that Geneva does not know whether Geneva'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) ] (Agentb knows that (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: Mildred 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 Reuben knows whether Reuben's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

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

Hypothesis: John can now know that Reuben'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 Edwin does not know whether Fredricka is thirsty. It is publicly announced that Nancy does not know whether Edwin is thirsty.

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

Hypothesis: Fredricka can now know that Charles can know that Edwin 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 Donald does not know whether Donald's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Donald knows whether Cassandra's forehead is muddy. It is publicly announced that Donald 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) ] [ ! (1|2|3) ] [ ! (Agenta knows whether 3) ] [ ! (Agenta knows whether (~(1|2|3))) ] Agenta 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) ] [ ! (1|2|3) ] [ ! (Agenta knows whether 3) ] [ ! (Agenta knows whether (~(1|2|3))) ] 0

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


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

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

Hypothesis: Norbert can now know whether George 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 everyone is thirsty. It is publicly announced that Athena knows whether Pamela is thirsty.

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

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

Hypothesis: Athena can now know whether Derek can know whether Mae 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 Erica is thirsty. It is publicly announced that someone is thirsty.

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

Hypothesis: Erica can now know that everyone is thirsty.


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

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

Hypothesis: Roberta can now know that Sherry can know that Daniel'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 Milissa is thirsty.

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

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

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


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

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

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

Hypothesis: John can now 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 Philip knows whether or not nobody's forehead is muddy. It is publicly announced that Kelvin's forehead is not muddy.

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

Hypothesis: Philip 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). Nikki's card is shown to Walter. Nikki's card is shown to Nikki. Christine's card is shown to Nikki. Nikki's card is shown to Christine. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Walter can now know whether Walter 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 Josephine knows whether or not nobody is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (~(1|2|3))) ] (Agenta knows that (~(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: Doris can now 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 Laura's forehead is not muddy.

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

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


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

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

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


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

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

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

Hypothesis: Frank can now know whether Nancy can know that Corinne 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 Stella knows whether or not everyone's forehead is muddy.

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

Hypothesis: Jeffrey 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 Robert is not thirsty. It is publicly announced that David does not know whether Trevor is thirsty. It is publicly announced that Trevor does not know whether David is thirsty.

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

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

Hypothesis: Robert can now know whether David can know that nobody is thirsty.


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

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

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


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Matthew can now know that everyone is thirsty.


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

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

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


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

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

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

Hypothesis: Francis can now know whether Richard can know that Francis'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 Denice does not know whether Doris is thirsty. It is publicly announced that Stella does not know whether Doris is thirsty. It is publicly announced that Denice knows whether Denice is thirsty.

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

Hypothesis: Denice can now know that nobody 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 not everyone is thirsty. It is publicly announced that Kevin is not thirsty.

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

Hypothesis: Kevin can now know whether Kevin 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 James does not know whether Gayle is thirsty. It is publicly announced that James knows whether or not everyone is thirsty.

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

Hypothesis: Gayle can now know that Jessica can know that Jessica is thirsty.


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

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

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


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

Hypothesis: Angelica can now know whether Janet can know that Tracie's forehead is muddy.


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

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

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

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


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

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

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

Hypothesis: Iris can now know that Lillie can know whether Marceline 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 Ricky does not know whether Patrick is thirsty.

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

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

Hypothesis: Patrick can now know whether Tiffany can know that Tiffany 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 Marjorie knows whether Roy's forehead is muddy. It is publicly announced that Marjorie's forehead is muddy.

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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that John 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) ] [ ! (Agentc knows whether (~(1|2|3))) ] Agentb knows that (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) ] [ ! (Agentc knows whether (~(1|2|3))) ] 0

Hypothesis: Shelley can now know that John can know whether Shelley'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 does not know whether Mozell is thirsty. It is publicly announced that Dorothy does not know whether Monica 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) ] (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: Monica can now know that Hilary 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 Robert is not thirsty.

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

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

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


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

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

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

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


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

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

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

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


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

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

Hypothesis: Richard can now know whether Michael can know that Richard'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 Mary knows whether Mary is thirsty. It is publicly announced that Mary does not know whether Marlene is thirsty.

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

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

Hypothesis: Mary can now know that Marlene 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 Tommie does not know whether Minerva is thirsty.

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

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

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


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

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

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

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


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

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

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

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


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

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

Hypothesis: June can now know whether Martin can know whether Martin'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 Mari's forehead is not muddy.

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

Hypothesis: Mari can now know whether Anthony can know that Mari'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 Janice does not know whether Megan 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) ] (Agentb knows whether 4)

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

Hypothesis: Michael can now know whether Janice is thirsty.


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

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

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

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


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 Jessica. Doris's card is shown to Jessica. Charlotte's card is shown to Charlotte. Jessica's card is shown to Doris. Charlotte's card is shown to Doris. Doris's card is shown to Doris. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Doris can now know whether Jessica 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 Yvonne does not know whether Alfred is thirsty. It is publicly announced that Andrew does not know whether Yvonne is thirsty. It is publicly announced that not everyone is thirsty.

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

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

Hypothesis: Alfred can now know whether Yvonne can know whether Yvonne 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 Angela'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 ] (Agentc knows that 2)

Hypothesis: Lucille can now know that Bessie'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 knows whether Daniel is thirsty. It is publicly announced that Daniel is thirsty.

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

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

Hypothesis: Daniel can now know that Andrew 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 Bonnie's forehead is muddy. It is publicly announced that Bonnie knows whether someone's forehead is muddy. It is publicly announced that Frank's forehead is not muddy.

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

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

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


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

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

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

Hypothesis: Paul can now know that Virginia can know that Paul 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 Clinton knows whether or not everyone's forehead is muddy. It is publicly announced that Clinton knows whether Warren's forehead is muddy. It is publicly announced that Clinton knows whether someone's forehead is muddy.

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

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

Hypothesis: Clinton can now know that Warren 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 not everyone is thirsty. It is publicly announced that Alan knows whether Alan is thirsty. It is publicly announced that Alan knows whether someone is thirsty.

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

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

Hypothesis: Mattie can now 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 someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

Hypothesis: Ryan can now 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 Alejandro does not know whether Alejandro's forehead is muddy. It is publicly announced that Alejandro knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Don can now know whether Amy can know whether Amy'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 Thomas's forehead is not muddy.

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

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

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


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

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

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

Hypothesis: Eric can now know whether Vera can know whether or not 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 Barbara knows whether someone's forehead is muddy.

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

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

Hypothesis: Harold can now know that Barbara'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) ] Agenta knows whether (Agentb knows whether 3)

Hypothesis: Joseph can now know whether Brian can know whether Tiffany's forehead is muddy.