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 Carmelita knows whether someone's forehead is muddy. It is publicly announced that Carmelita'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)) ] [ ! 1 ] (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) ] [ ! (Agenta knows whether (1|2|3)) ] [ ! 1 ] 0

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joann knows whether Dorthy'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) ] (Agenta knows that 3)

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

Hypothesis: Dorthy can now know that Courtney'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 Charlotte's forehead is muddy. It is publicly announced that Shannon knows whether Charlotte's forehead is muddy.

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

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


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

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

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

Hypothesis: Jill can now know that Janet can know whether Tonya 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 Virginia knows whether Hector is thirsty.

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

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

Hypothesis: Hector can now know whether Virginia can know that someone is thirsty.


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

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

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


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

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

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

Hypothesis: Hugo can now know that Cynthia can know whether Hugo 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 Diana's forehead is not muddy.

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

Hypothesis: John can now know whether Diana can know that nobody's forehead is muddy.


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

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

Hypothesis: Robert can now know whether Charles can 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 Patti knows whether or not nobody's forehead is muddy. It is publicly announced that Lucile'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) ] [ ! (Agentc knows whether (~(1|2|3))) ] [ ! ~(2) ] (Agentb knows whether 2)

Hypothesis: Lucile can now know whether Lucile'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 Erin does not know whether Erin's forehead is muddy. It is publicly announced that Harold knows whether someone's forehead is muddy. It is publicly announced that Harold knows whether Maria'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 (1|2|3)) ] [ ! (Agentb knows whether 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) ] [ ! (Agentb knows whether (1|2|3)) ] [ ! (Agentb knows whether 3) ] 0

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


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

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

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

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


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

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

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

Hypothesis: Shawna can now know that Angela picked a red card.


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

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

Hypothesis: Matthew can now know whether Tom can know that Lawrence picked a red card.


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

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

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

Hypothesis: Lynn can now know that someone is thirsty.


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

Hypothesis: Sandra can now know that Sandra'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 Anna'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) ] [ ! ~(2) ] Agentb knows that (Agenta knows that 3)

Hypothesis: Anna can now know that Joe can know that Harry'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 John does not know whether Rosa is thirsty. It is publicly announced that Thomas does not know whether Rosa 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) ] [ ! ~(Agentb knows whether 3) ] (Agenta knows that 2)

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

Hypothesis: John can now know that Thomas is thirsty.


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

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

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


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

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

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


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

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

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

Hypothesis: Julie can now know that Ana can know whether Jesus picked a red card.


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

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

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

Hypothesis: John 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 Virginia does not know whether or not everyone's forehead is muddy. It is publicly announced that Kari knows whether Virginia's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Timothy can now know whether Rodolfo 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 someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

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


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

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

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


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

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

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

Hypothesis: Gayle can now know whether Amy can know that Gayle 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 Warren is not thirsty.

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

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

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


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Joyce can now know that 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 not everyone's forehead is muddy.

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


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

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

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


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

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

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

Hypothesis: Josie can now know whether Nick 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 John does not know whether Thomas is thirsty. It is publicly announced that Alice does not know whether Thomas is thirsty.

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

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

Hypothesis: Alice can now know that Thomas is thirsty.


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

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

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

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


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

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

Hypothesis: David can now know that Michelle can know whether or not 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). David's card is shown to David. Katie's card is shown to David. Katie's card is shown to Katie. 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) ] (Agentb knows that (1&2&3))

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

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


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

Hypothesis: Apolonia can now know that nobody 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 Carla does not know whether Carla's forehead is muddy. It is publicly announced that John does not know whether John's forehead is muddy.

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

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

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


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

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


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

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

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

Hypothesis: Richard can now know whether Fannie is thirsty.


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

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

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

Hypothesis: Yong can now know that Robert can know whether Robert'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 John does not know whether John's forehead is muddy. It is publicly announced that John does not know whether John's forehead is muddy.

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

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

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


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

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

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


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

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

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

Hypothesis: Luis can now know that Bennett 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). Melissa's card is shown to Melissa. Melissa's card is shown to David. It is publicly announced that someone picked a red card. It is publicly announced that David does not know whether or not everyone picked a red card.

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

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

Hypothesis: David can now know that David 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 Elsa does not know whether Audrey is thirsty. It is publicly announced that Juanita does not know whether Audrey is thirsty. It is publicly announced that Elsa is not thirsty.

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

Hypothesis: Roland can now know whether Audrey is thirsty.


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

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

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


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

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

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

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


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

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

Hypothesis: Evelyn can now know that Antonia can know whether John'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 Steven 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) ] [ ! (Agenta knows whether (1&2&3)) ] (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&2&3)) ] 0

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


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

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

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

Hypothesis: Jessie can now know that Mildred can know whether someone is thirsty.


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

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

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


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

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

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


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

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

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

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


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

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

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

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


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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Julie'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 (Agentc knows that 2)

Hypothesis: Janice can now know that Julie can know that Janice'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 Frank does not know whether Sergio is thirsty.

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

Hypothesis: Lisa can now know that Sergio can know whether Frank 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 Billie'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) ] Agentc knows that (Agenta knows that (1&2&3))

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


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

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

Hypothesis: Irene can now know whether Dawn can know whether Irene'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 Faith knows whether someone is thirsty. It is publicly announced that Faith does not know whether Bill is thirsty.

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

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

Hypothesis: Faith can now know whether Bill can know that nobody is thirsty.


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

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Franklin's card is shown to Henry. Henry's card is shown to Franklin. Franklin's card is shown to Franklin. Franklin's card is shown to Roger. Roger's card is shown to Roger. It is publicly announced that someone 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 (Agenta knows whether (1|2|3))

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

Hypothesis: Roger can now know whether Henry can 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 Linda's forehead is muddy. It is publicly announced that Ismael knows whether someone's forehead is muddy. It is publicly announced that Ismael knows whether Erin's forehead is muddy. It is publicly announced that Ismael knows whether or not everyone's forehead is muddy.

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

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


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

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

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

Hypothesis: Kathryn can now know that William picked a red card.


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

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

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

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


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

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

Hypothesis: Taylor can now know that Dan can know that Dan'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 Jack knows whether Blanca is thirsty.

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

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

Hypothesis: Jack can now know whether Blanca can know that someone is thirsty.


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

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

Hypothesis: Charles can now know whether Billy can know whether Kenneth'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 Carol knows whether or not nobody's forehead is muddy.

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

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

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


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

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


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

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

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

Hypothesis: Steven can now know that Todd can know whether Steven'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 Jarrod is not thirsty.

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

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

Hypothesis: Aaron can now know that Jarrod can know that Jarrod 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 Sheila 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 (1&2&3&4)) ] (Agentd knows that 1)

Hypothesis: Leslie can now know that Willie'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 Nancy knows whether Crystal's forehead is muddy. It is publicly announced that Mary does not know whether Mary's forehead is muddy. It is publicly announced that Bobby knows whether Crystal'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) ] [ ! ~(Agentd knows whether 4) ] [ ! (Agentb knows whether 1) ] Agentb knows whether (Agentc 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 1) ] [ ! ~(Agentd knows whether 4) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Bobby can now know whether Nancy can know whether Bobby's forehead is muddy.


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

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

Hypothesis: Cruz can now know that Phillip can know that Barbara's forehead is muddy.


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

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 (Agentb knows whether 4)

Hypothesis: Dexter can now know whether Andrea can know whether Thomas's forehead is muddy.


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

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

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

Hypothesis: Stephen can now know that Alexandra can know whether Alexandra'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 Kathy is thirsty. It is publicly announced that Eileen 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 4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] (Agentc knows that (1&2&3&4))

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

Hypothesis: Eileen can now know that everyone is thirsty.


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

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

Hypothesis: Natalie can now know that Natalie'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 Margaret'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 ] Agentc knows that (Agentb knows whether 1)

Hypothesis: James can now know that Gloria can know whether Margaret'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 Richard does not know whether Richard's forehead is muddy. It is publicly announced that Richard'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) ] [ ! ~(1) ] Agenta knows that (Agentc knows whether (~(1|2|3|4)))

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


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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Thomas's card is shown to Thomas. Helen's card is shown to Alice. Thomas's card is shown to Helen. Helen's card is shown to Helen. 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.

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

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

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


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

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

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

Hypothesis: Alan can now know whether Donna can know that Alan 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 Derek's forehead is muddy.

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

Hypothesis: Michelle can now know that Edward can know that Michelle'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 Samuel knows whether Samuel is thirsty. It is publicly announced that Jennifer 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 2) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] (Agentb knows that 1)

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

Hypothesis: Samuel can now know that Antonio is thirsty.


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

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

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

Hypothesis: Kellee can now know whether or not 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). Daniel's card is shown to Terrell. Terrell's card is shown to Christi. Christi'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:1 Agentc:2 VALID? [ ! (1|2|3) ] Agenta knows that (Agentb knows whether 2)

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

Hypothesis: Terrell can now know that Christi can know whether Christi 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 Annie does not know whether Annie'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 (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 3) ] 0

Hypothesis: Annie can now know that Gary 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 Sally knows whether or not everyone is thirsty. It is publicly announced that Roger knows whether or not nobody is thirsty.

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

Hypothesis: Sally can now know that nobody is thirsty.


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

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

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

Hypothesis: Jan can now know whether Aaron 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.

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Marie's forehead is muddy. It is publicly announced that Franklin knows whether Kimberly'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) ] Agentc knows whether (Agentb knows that 3)

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

Hypothesis: Franklin can now know whether Marie can know that Franklin'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 Carolyn knows whether Willie's forehead is muddy.

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

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


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

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

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

Hypothesis: Donald can now know that Kathryn can know whether someone is thirsty.


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

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

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

Hypothesis: Debra can now know that Mark 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). Latoya's card is shown to Peggy. Peggy's card is shown to Latoya. It is publicly announced that someone picked a red card. It is publicly announced that Alan does not know whether Peggy picked a red card. It is publicly announced that Alan does not know whether Alan picked a red card.

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

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

Hypothesis: Alan can now know whether Peggy can know whether Latoya picked a red card.


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

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

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

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


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

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

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

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

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

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


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

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

Hypothesis: Sheila can now know whether Jeffrey can know that Josephine 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 is not thirsty. It is publicly announced that someone is thirsty.

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

Hypothesis: Leona can now know that Wanda can know whether Leona 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 Theresa's forehead is not muddy. It is publicly announced that Christopher does not know whether Christopher's forehead is muddy. It is publicly announced that Christopher 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))) ] 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) ] [ ! ~(2) ] [ ! ~(Agentc knows whether 3) ] [ ! (Agentc knows whether (~(1|2|3))) ] 0

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


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

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

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

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


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

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

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

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


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 (Agentb knows whether 1)

Hypothesis: Jerry can now know whether Sheila can know whether Jerry'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 (Agentb knows that 3)

Hypothesis: Anthony can now know that Jimmy can know that Ruth'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 Frank does not know whether Frank's forehead is muddy. It is publicly announced that Frank'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) ] [ ! ~(1) ] Agentb knows that (Agentd knows whether 4)

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


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

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

Hypothesis: Tabitha can now know that Jeanette can know that Patrice is thirsty.


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

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

Hypothesis: Holly can now know that Harry can know whether Olinda'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 Antionette is thirsty. It is publicly announced that Christina 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 ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agentb knows that 2)

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

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

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

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

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


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

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

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

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


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

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

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 (1|2|3|4)) ] [ ! (Agentd knows whether 2) ] 0

Hypothesis: Maurice can now know that Hilton can know that Hilton'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 Greg does not know whether Donna is thirsty. It is publicly announced that Nick knows whether or not everyone is thirsty.

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

Hypothesis: Donna can now know that Dominique can know whether Donna 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 Leo does not know whether Virginia is thirsty. It is publicly announced that Leo 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 4) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] Agenta knows that (Agentc knows whether 4)

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

Hypothesis: Thomas can now know that Michael can know whether Virginia 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 Emily knows whether someone's forehead is muddy.

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

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

Hypothesis: Anthony can now know that Emily'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 Valerie's forehead is muddy.

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

Hypothesis: Albert can now know that Valerie can know that Helen'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 Pearl does not know whether Richard is thirsty. It is publicly announced that Richard does not know whether Pearl is thirsty. It is publicly announced that Pearl does not know whether Richard is thirsty.

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

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

Hypothesis: George can now know that Richard can know that George is thirsty.


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

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

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

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

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

Hypothesis: Desmond can now know whether Desmond'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 Douglas knows whether Douglas's forehead is muddy.

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

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

Hypothesis: Douglas can now know that Nancy 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 Gerald does not know whether Eva 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) ] Agentb knows that (Agenta knows that 3)

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

Hypothesis: Gerald can now know that Jessica can know that Mary 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 Paul 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)) ] (Agentc knows that 3)

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

Hypothesis: George can now know that George is thirsty.


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

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

Hypothesis: Christopher can now know that Christopher'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 Virgil is thirsty.

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

Hypothesis: Virgil can now know whether John 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 Bonita does not know whether Tammy is thirsty.

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

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

Hypothesis: Catherine can now know that Bonita can know whether or not everyone is thirsty.


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

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

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

Hypothesis: Maria can now know that Jasmine can know whether 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 someone is thirsty.

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

Hypothesis: Carolyn can now know that Dennis can know that Carolyn 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 Trish does not know whether or not everyone is thirsty. It is publicly announced that Trish does not know whether Joe is thirsty. It is publicly announced that Seymour knows whether Seymour is thirsty.

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

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

Hypothesis: Joe can now know whether Trish 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 John knows whether John is thirsty.

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

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

Hypothesis: Edward can now know whether John can know that everyone is thirsty.


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

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

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

Hypothesis: Lee 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 Calvin knows whether someone is thirsty. It is publicly announced that Clifford does not know whether Calvin 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)) ] [ ! ~(Agenta knows whether 3) ] (Agentd knows that (1&2&3&4))

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

Hypothesis: Davina can now know that everyone is thirsty.


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

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

Hypothesis: Margaret can now know whether Aldo can know whether Aldo'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 Nancy'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 ] (Agentb knows that 2)

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

Hypothesis: Melinda can now know that Melinda'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 Martha knows whether Martha is thirsty. It is publicly announced that Scott does not know whether Martha is thirsty. It is publicly announced that someone is thirsty.

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

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

Hypothesis: Scott can now know that Scott is thirsty.


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

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

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

Hypothesis: Mary can now know whether Melvin can know whether Mary 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 Daisy does not know whether Daisy's forehead is muddy. It is publicly announced that Daisy does not know whether Daisy's forehead is muddy. It is publicly announced that Daisy's forehead is not muddy. It is publicly announced that Scott does not know whether Scott's forehead is muddy.

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

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

Hypothesis: Sarah can now know whether Daisy 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 Christine is thirsty.

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

Hypothesis: Henry can now know whether Shirl can know that everyone is thirsty.


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

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

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

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


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

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

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

Hypothesis: John 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 Phillip knows whether Shannon's forehead is muddy.

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

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

Hypothesis: Phillip can now know that Blanca can know whether Phillip'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 Timothy's forehead is muddy. It is publicly announced that Samuel knows whether someone's forehead is muddy. It is publicly announced that Samuel knows whether Rodney's forehead is muddy. It is publicly announced that Samuel knows whether or not everyone's forehead is muddy.

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

Hypothesis: Timothy can now know that Samuel can know whether someone's forehead is muddy.


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

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

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


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

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

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


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

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

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

Hypothesis: Victor can now know that Ines can know that Victor picked a red card.


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. Angel's card is shown to Angel. It is publicly announced that someone picked a red card.

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

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

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


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 Geraldine. Geraldine's card is shown to German. Robert's card is shown to Robert. It is publicly announced that someone picked a red card. It is publicly announced that Geraldine picked a black card.

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

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

Hypothesis: Robert can now know whether Geraldine 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 not everyone is thirsty. It is publicly announced that Beatrice knows whether Beatrice is thirsty.

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

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

Hypothesis: Lindsey 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 Barbara knows whether or not everyone's forehead is muddy. It is publicly announced that Gilbert does not know whether Gilbert's forehead is muddy. It is publicly announced that Barbara does not know whether Barbara's forehead is muddy.

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

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

Hypothesis: Jesus can now know whether Barbara can know that Jesus'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 Willie does not know whether Amy is thirsty. It is publicly announced that Amy is not thirsty. It is publicly announced that Sean does not know whether Willie is thirsty.

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

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

Hypothesis: Sean 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). Suzanne's card is shown to Natasha. Joyce's card is shown to Arthur. Suzanne's card is shown to Arthur. Natasha's card is shown to Joyce. 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 not everyone picked a red card. It is publicly announced that Joyce picked a black card.

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

Hypothesis: Natasha can now know that Arthur can know that Joyce picked a red card.


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Myrna does not know whether Melvin 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 that 1)

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

Hypothesis: Myrna can now know that Myrna is thirsty.


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

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

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

Hypothesis: Ida 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). Andrew's card is shown to Andrew. John's card is shown to John. Paul's card is shown to John. Paul's card is shown to Wayne. It is publicly announced that someone picked a red card.

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

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

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


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

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

Hypothesis: Shameka can now know whether Larry'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 Alicia's forehead is muddy.

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

Hypothesis: Francis can now know whether Douglas can know whether Francis'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 Teresa knows whether or not nobody's forehead is muddy.

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

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


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

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

Hypothesis: James can now know that Arnold is thirsty.


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

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

Hypothesis: Dale can now know that Patricia is thirsty.