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

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

Hypothesis: William can now know whether Jan 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 Angel knows whether Susan's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

Hypothesis: Susan can now know whether Susan'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.

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

Hypothesis: Robert can now know that Steven'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 Mary does not know whether Mary's forehead is muddy. It is publicly announced that David does not know whether David's forehead is muddy. It is publicly announced that Mary 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 3) ] [ ! ~(Agenta knows whether 1) ] [ ! (Agentc knows whether (~(1|2|3|4))) ] Agentc knows that (Agentb knows that 4)

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

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


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

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

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

Hypothesis: Drew can now know whether James can 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 Kevin's forehead is muddy. It is publicly announced that Ana knows whether someone's forehead is muddy.

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

Hypothesis: Lelia can now know that Kevin can know that Lelia'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 Alana knows whether someone's forehead is muddy.

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

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

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


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

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

Hypothesis: Merry can now know that Justin can know that Justin'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 Fanny knows whether someone is thirsty.

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

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

Hypothesis: Fanny can now know that nobody is thirsty.


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

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

Hypothesis: Lexie can now know whether Lexie'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 Jay's forehead is not muddy.

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

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


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

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

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

Hypothesis: Thomas can now know whether Cheryl 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). Courtney's card is shown to Charles. Charles's card is shown to Courtney. Courtney's card is shown to Courtney. Courtney's card is shown to Douglas. Douglas's card is shown to Douglas. It is publicly announced that someone picked a red card. It is publicly announced that Douglas does not know whether or not everyone picked a red card.

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

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

Hypothesis: Douglas can now know that Charles can know that Charles 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 Darlene knows whether someone's forehead is muddy.

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

Hypothesis: Stacy can now know whether Ronald'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 Vera is thirsty. It is publicly announced that Vera does not know whether or not everyone is thirsty. It is publicly announced that Vera does not know whether or not everyone is thirsty. It is publicly announced that Maria 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) ] [ ! 2 ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agentc knows whether (1&2&3)) ] (Agentb knows whether 1)

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

Hypothesis: Vera can now know whether Evelyn 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 John knows whether someone's forehead is muddy.

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

Hypothesis: Craig can now know that John can know that Craig'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 Luanne knows whether Arthur'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) ] Agentd knows that (Agentc knows that 4)

Hypothesis: Nicholas can now know that Luanne can know that Nicholas'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 Barbra does not know whether David is thirsty. It is publicly announced that Carmen does not know whether Barbra is thirsty. It is publicly announced that Carmen does not know whether David is thirsty.

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

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

Hypothesis: David can now know that Carmen can know that Barbra is thirsty.


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Rosie knows whether Rosie 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 that (Agentb knows that 3)

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

Hypothesis: Rosie can now know that Timothy can know that Rosie 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 Robin does not know whether Salvador is thirsty. It is publicly announced that Robin does not know whether Beverly 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) ] Agentc knows that (Agentb knows whether (1&2&3))

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

Hypothesis: Beverly can now know that Robin can know whether or not everyone is thirsty.


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

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

Hypothesis: Kathryn can now know that Derrick can know whether Derrick's forehead is muddy.


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

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

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

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


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

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

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

Hypothesis: Mellisa can now know whether Hae 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 Carmen knows whether Carmen's forehead is muddy.

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

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

Hypothesis: Carmen can now know that Jackeline can know whether Carmen'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) ] (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) ] 0

Hypothesis: Faye can now know that nobody is thirsty.


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

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

Hypothesis: Dana can now know whether Andrew 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 Jamel knows whether Jamel'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 whether 2)

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: Chris can now know whether Chris'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 Richard knows whether Richard is thirsty. It is publicly announced that Maren does not know whether or not everyone is thirsty. It is publicly announced that Stephanie does not know whether or not everyone is thirsty. It is publicly announced that James does not know whether or not everyone is thirsty.

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

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

Hypothesis: Maren can now know that James can know whether or not everyone is thirsty.


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

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

Hypothesis: Craig can now know that Ricky'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 Albert does not know whether Shelley 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) ] Agentc knows whether (Agentd knows whether 2)

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

Hypothesis: Albert can now know whether Peter can know whether Carol 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 Charlotte does not know whether or not everyone is thirsty. It is publicly announced that Mattie knows whether Mattie is thirsty.

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

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

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


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

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

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

Hypothesis: Rebecca can now know whether Aileen is thirsty.


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

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

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

Hypothesis: Shelby can now know that Walter can know that Shelby 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 Trina knows whether someone's forehead is muddy.

Hypothesis: Trina can now know that someone'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 Jean does not know whether or not everyone is thirsty. It is publicly announced that everyone is thirsty. It is publicly announced that Jean knows whether someone is thirsty. It is publicly announced that Jean knows whether Jean is thirsty.

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

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

Hypothesis: Frances can now know whether Jean can know that Frances 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 Richard does not know whether Richard's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that David knows whether someone's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

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


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

Hypothesis: Patricia can now know that Patricia'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 Evelyn knows whether someone's forehead is muddy.

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

Hypothesis: Gary can now know that Darrell can know whether Evelyn'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 everyone's forehead is muddy.

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

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


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

Hypothesis: Gertrude can now 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 Maribel is not thirsty.

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

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

Hypothesis: Kathy can now know that Thomas 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 Charles is thirsty.

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

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

Hypothesis: Myrna can now know that Charles can know whether Crystal 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 David does not know whether Dean is thirsty. It is publicly announced that Dean does not know whether Anne 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) ] [ ! ~(Agentc knows whether 4) ] (Agentb 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) ] [ ! ~(Agenta knows whether 3) ] [ ! ~(Agentc knows whether 4) ] 0

Hypothesis: Roger can now know whether Roger 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 Stephen does not know whether April is thirsty.

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

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

Hypothesis: Stephen can now know whether Barbara can know that someone is thirsty.


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

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

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


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

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

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


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

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

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

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


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

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

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

Hypothesis: Joshua 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 not everyone's forehead is muddy.

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

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

Hypothesis: Phillip can now know whether Thomas 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 Catherine's forehead is not muddy. It is publicly announced that Bonnie'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) ] [ ! ~(2) ] [ ! ~(1) ] Agenta knows whether (Agentc knows whether (1&2&3))

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


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

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

Hypothesis: Bonnie can now know that Jerry can know that 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). Scott's card is shown to Nina. Nina's card is shown to Nina. Nellie's card is shown to Nellie. It is publicly announced that someone picked a red card.

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

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

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


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

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

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

Hypothesis: Mary can now know whether Cheryl 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 Amy knows whether Kathleen's forehead is muddy. It is publicly announced that Amy knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Amy can now know whether Amanda's forehead is muddy.


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

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

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

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


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

Hypothesis: Eldon can now know that John'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 Rudy does not know whether or not everyone is thirsty.

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

Hypothesis: Rudy can now know whether Kathy can know whether Rudy is thirsty.


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

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

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

Hypothesis: Tyrell can now know whether Christy 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 Megan does not know whether Scott is thirsty.

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

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

Hypothesis: Megan can now know that Scott can know whether Scott 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 Lindsey is not thirsty.

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

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

Hypothesis: James 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 Sandi knows whether James's forehead is muddy. It is publicly announced that Anna knows whether or not everyone's forehead is muddy.

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

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

Hypothesis: Sandi can now know whether Anna'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 Dennis does not know whether Maria is thirsty. It is publicly announced that Maria does not know whether James is thirsty.

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

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

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


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

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

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


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

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

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

Hypothesis: Rita can now know whether Maria can know that Shirley 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 Richard does not know whether Roland is thirsty.

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

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

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


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Barbara 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 that 2)

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

Hypothesis: Brent can now know that Brent'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 Joy 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 whether (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 (1&2&3)) ] 0

Hypothesis: Heidi can now know whether Joy can know that Joy'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 Elton knows whether or not everyone is thirsty. It is publicly announced that Brandon is thirsty.

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

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

Hypothesis: Elton can now know that everyone is thirsty.


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

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

Hypothesis: Robert can now know whether Nichole can know whether Robin is thirsty.


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

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

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

Hypothesis: Lauren can now know whether Alicia can know whether Alicia picked a red card.


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

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

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

Hypothesis: Karen 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 Julius's forehead is not muddy. It is publicly announced that Susanne does not know whether Susanne's forehead is muddy. It is publicly announced that Susanne 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) ] [ ! ~(2) ] [ ! ~(Agentc knows whether 3) ] [ ! (Agentc knows whether (~(1|2|3))) ] Agenta knows whether (Agentb knows whether 2)

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

Hypothesis: Kyle can now know whether Julius can know whether Julius's forehead is muddy.


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

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

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

Hypothesis: Austin can now know whether German can know whether Austin 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 Juan does not know whether Theresa is thirsty. It is publicly announced that Theresa is not thirsty. It is publicly announced that Theresa is not thirsty. It is publicly announced that Juan 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) ] [ ! ~(1) ] [ ! ~(1) ] [ ! (Agentb knows whether (1&2&3)) ] Agentb knows that (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) ] [ ! ~(1) ] [ ! ~(1) ] [ ! (Agentb knows whether (1&2&3)) ] 0

Hypothesis: Juan can now know that Theresa can know that Theresa 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 Kathy is thirsty. It is publicly announced that Todd does not know whether Robert is thirsty. It is publicly announced that Todd does not know whether Kathy is thirsty.

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

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

Hypothesis: Edward can now know that nobody is thirsty.


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

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

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

Hypothesis: David can now know whether Levi 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 Joanne does not know whether Joseph is thirsty. It is publicly announced that Joanne does not know whether Joseph 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&2))

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

Hypothesis: Joseph can now know that Joanne can know whether or not everyone is thirsty.


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

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


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

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

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

Hypothesis: Virginia can now know that someone is thirsty.


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

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

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

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


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

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


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. Ashley's card is shown to Patricia. Patricia's card is shown to Ashley. Ashley's card is shown to Ashley. Patricia's card is shown to Ray. Robert's card is shown to Ray. Ray's card is shown to Robert. Robert'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:1,2 Agentb:1,2 Agentc:1,4 Agentd:3,4 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows that (1|2|3|4))

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

Hypothesis: Patricia can now know whether Ashley can know that 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 Marquerite does not know whether or not everyone is thirsty. It is publicly announced that Diane knows whether or not everyone is thirsty.

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

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

Hypothesis: Diane can now know whether Diane is thirsty.


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

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

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

Hypothesis: Martha can now know that someone is thirsty.


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

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

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

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


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

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

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

Hypothesis: James can now know that Jose 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). Alexander's card is shown to Alexander. Paul's card is shown to Alexander. Paul's card is shown to Paul. Raymond's card is shown to Paul. Paul's card is shown to Raymond. It is publicly announced that someone picked a red card.

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

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

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


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

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

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

Hypothesis: Mary can now know that Jay 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 David does not know whether Linda is thirsty. It is publicly announced that Pam is thirsty. It is publicly announced that Linda is thirsty. It is publicly announced that Pam is thirsty.

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

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

Hypothesis: Linda can now know whether Pam can know that someone is thirsty.


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

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

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

Hypothesis: Derek can now know that Derek 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 Jill is not thirsty.

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

Hypothesis: Sally can now know whether someone is thirsty.


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

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

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

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


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

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

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


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

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

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

Hypothesis: Jeffrey can now know that Frank 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 Karen knows whether Karen's forehead is muddy. It is publicly announced that Tiffany knows whether Henry's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Jonathan can now know whether Virginia can know that Hope picked a red card.


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

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

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

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


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

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

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

Hypothesis: Brenda can now know that Brenda 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 Katherine knows whether Wendy's forehead is muddy.

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

Hypothesis: Wendy can now know whether Minerva 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 Michael knows whether Michael'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) ] Agenta knows that (Agentb knows whether (~(1|2|3)))

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

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


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

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

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


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

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

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

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

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

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

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


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

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

Hypothesis: Joseph can now know that 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 Dana knows whether Teresa'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) ] Agentb knows whether (Agentd knows that 2)

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

Hypothesis: Bill can now know whether Valerie can know that Bill'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 Berniece knows whether Craig's forehead is muddy.

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

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

Hypothesis: Berniece can now know that Craig can know whether John'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 Jerry does not know whether Jerry's forehead is muddy. It is publicly announced that Gloria 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 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: Jerry can now know that Gloria can know whether Gloria'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 Naomi does not know whether or not everyone is thirsty. It is publicly announced that Naomi does not know whether or not everyone is thirsty. It is publicly announced that Naomi does not know whether Tracey is thirsty. It is publicly announced that Naomi knows whether or not nobody is thirsty.

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

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

Hypothesis: Tracey can now 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 Pamela knows whether Ronald's forehead is muddy.

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

Hypothesis: Pamela can now know whether Ronald can know that everyone's forehead is muddy.


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

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

Hypothesis: Gregory can now know whether Anthony'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 Joshua'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) ] [ ! 4 ] Agentb knows that (Agenta knows whether 2)

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


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

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

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


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

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

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

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

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

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

Hypothesis: Katherine can now know that Kim can know whether Kim's forehead is muddy.


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

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

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

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


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

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

Hypothesis: Michael can now know that Michael'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 Marcus does not know whether or not everyone is thirsty. It is publicly announced that Robert does not know whether Maria is thirsty. It is publicly announced that Maria does not know whether or not everyone is thirsty. It is publicly announced that Maria 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 2) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether (1&2&3)) ] (Agenta 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&2&3)) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agentb knows whether (1&2&3)) ] 0

Hypothesis: Robert can now know whether Marcus is thirsty.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Eugene can now know that Jim can know whether Noemi 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 Lindsey does not know whether Lindsey's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lindsey knows whether Candice's forehead is muddy. It is publicly announced that Lindsey 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))) ] Agentb knows whether (Agentc knows whether (~(1|2|3)))

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

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


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

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

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


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

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

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

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


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

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

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

Hypothesis: Bruce can now know that Jill can know whether someone picked a red card.


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

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

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

Hypothesis: Kimberly can now know whether Joseph can know that everyone picked a red card.


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

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

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


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

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

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

Hypothesis: Audrey can now know whether Donna is thirsty.


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

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

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

Hypothesis: Alisa 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). Brandon's card is shown to Felicia. Ruth's card is shown to Felicia. Felicia's card is shown to Felicia. It is publicly announced that someone picked a red card. It is publicly announced that Brandon knows whether Ruth picked a red card. It is publicly announced that Felicia picked a black card. It is publicly announced that Brandon knows whether Brandon picked a red card.

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

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

Hypothesis: Brandon can now know whether Felicia 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 Bernadine knows whether or not everyone is thirsty. It is publicly announced that Tim knows whether Tim is thirsty.

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

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

Hypothesis: Bernadine can now know that Tim 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 Carolyn's forehead is not muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Carolyn'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) ] [ ! (1|2|3) ] [ ! ~(1) ] Agenta knows whether (Agentc knows whether (1&2&3))

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


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

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

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

Hypothesis: Luis can now know that everyone is thirsty.


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

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

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

Hypothesis: Lawrence can now know whether Apryl's forehead is muddy.


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

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

Hypothesis: Nicole can now know that Sarah 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 Charles's forehead is not muddy. It is publicly announced that Thomas's forehead is muddy.

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

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


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

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

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

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


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

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

Hypothesis: Terrell can now know that Freddie 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 Jacklyn's forehead is not muddy.

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

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

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


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

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


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

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

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

Hypothesis: Nathan can now know whether Hallie 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 Gary does not know whether Richard is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Jeremy is thirsty. It is publicly announced that Gary knows whether Jeremy is thirsty.

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

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

Hypothesis: Gary can now know whether Gary is thirsty.


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

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

Hypothesis: Gary can now know whether Russell picked a red card.


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

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

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

Hypothesis: Ken can now know that Mitchell can know whether or not everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Richard knows whether or not nobody's forehead is muddy. It is publicly announced that Joel knows whether Val'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 that 4)

Hypothesis: Richard can now know that Hattie can know that Joel'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 Amy does not know whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agentb knows that 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: Floyd can now know that Maranda 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 Ronald does not know whether Norma is thirsty.

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

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

Hypothesis: Ryan can now know whether Norma 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 everyone's forehead is muddy. It is publicly announced that Gary knows whether Keith's forehead is muddy.

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

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

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

Hypothesis: Victor can now know that Mary can know whether Mary'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 Jose is thirsty. It is publicly announced that Jose does not know whether Bryan is thirsty.

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

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

Hypothesis: Bryan can now know that William is thirsty.


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

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

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

Hypothesis: Louis can now know that Lois 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). Roger's card is shown to Kathryn. Susan's card is shown to Kathryn. Kathryn's card is shown to Roger. Roger's card is shown to Roger. Kathryn's card is shown to Susan. It is publicly announced that someone picked a red card. It is publicly announced that everyone picked a red card. It is publicly announced that Susan knows whether Susan picked a red card. It is publicly announced that everyone picked a red card.

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

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


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

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

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

Hypothesis: Melissa can now know whether or not 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 Nathan's forehead is not muddy.

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


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

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

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

Hypothesis: Gordon can now know whether Vonda can know that James 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 Christina knows whether someone is thirsty. It is publicly announced that Sachiko 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)) ] [ ! 3 ] Agenta knows that (Agentb knows whether 1)

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

Hypothesis: William can now know that Christina can know whether William 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 William knows whether or 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)) ] 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)) ] 0

Hypothesis: William can now know that Robert can know that 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). Edwina's card is shown to John. John's card is shown to Janice. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card. It is publicly announced that Janice does not know whether or not everyone picked a red card.

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

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

Hypothesis: John can now know whether Edwina 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 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) ] Agentb knows whether (Agentc knows whether (1&2&3))

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