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

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

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

Hypothesis: Hector can now know whether Philip is thirsty.


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

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

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

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


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

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

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

Hypothesis: Jeff can now know that Cora can know that someone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Douglas does not know whether Douglas'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) ] (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) ] [ ! (~(1&2&3)) ] [ ! ~(Agentb knows whether 2) ] 0

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Meagan's card is shown to Meagan. Nathaniel's card is shown to Meagan. Lawrence's card is shown to Meagan. Lawrence's card is shown to Nathaniel. Lawrence's card is shown to Lawrence. 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) ] Agenta knows whether (Agentb knows that (~(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: Meagan can now know whether Nathaniel can know that nobody picked a red card.


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

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

Hypothesis: Bertha can now know that Kenneth 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). Brittni's card is shown to James. Thomas's card is shown to James. Ashleigh's card is shown to Ashleigh. Thomas's card is shown to Ashleigh. James's card is shown to Brittni. Ashleigh's card is shown to Thomas. It is publicly announced that someone picked a red card.

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

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

Hypothesis: James can now know whether Brittni can know that nobody picked a red card.


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

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


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

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

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

Hypothesis: Al can now know whether Shawanda can know that Shawanda picked a red card.


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

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

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

Hypothesis: Cynthia can now know whether Ashley can know that Ashley 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 Daniel's forehead is muddy. It is publicly announced that Laura knows whether someone's forehead is muddy.

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

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


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

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

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

Hypothesis: Debbie can now know that Debbie 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.

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

Hypothesis: Naomi can now know that Mable can know that Theodora'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 Elise is thirsty. It is publicly announced that Alfred is not thirsty. It is publicly announced that Lowell does not know whether Elizabeth is thirsty. It is publicly announced that Elise does not know whether Lowell is thirsty.

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

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

Hypothesis: Lowell can now know whether Elise can know that Lowell 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 Dorothy does not know whether or not everyone is thirsty. It is publicly announced that Joshua does not know whether or not everyone is thirsty. It is publicly announced that Susan knows whether Susan is thirsty.

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

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

Hypothesis: Susan can now know that everyone is thirsty.


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

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

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

Hypothesis: Melba can now know whether Timothy is thirsty.


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

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

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

Hypothesis: Janie can now know whether Jaime 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 Scott's forehead is not muddy.

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

Hypothesis: James can now know that Dwight can know whether 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 not everyone's forehead is muddy. It is publicly announced that Ernest knows whether Marisol's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Phillip can now know whether Harold 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 Dennis does not know whether William is thirsty. It is publicly announced that Michele knows whether or not nobody is thirsty. It is publicly announced that William does not know whether Michele is thirsty.

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

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

Hypothesis: William can now know whether Michele can 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 Gregory'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 ] Agentc knows whether (Agentd knows whether 4)

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

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

Hypothesis: Melissa can now know whether Rebecca can know whether Rebecca'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 Justin does not know whether Floyd is thirsty. It is publicly announced that David knows whether someone is thirsty. It is publicly announced that someone is thirsty. It is publicly announced that Jonna 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)) ] [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] (Agenta knows whether (~(1|2|3|4)))

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

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


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

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

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

Hypothesis: Donna can now know that Michael can know that Michael 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 not everyone is thirsty. It is publicly announced that Christopher does not know whether Mark 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 whether 1) ] (Agenta knows that (1|2|3|4))

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

Hypothesis: Mark can now know that someone is thirsty.


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

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

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

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

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

Hypothesis: Jamal can now know that George 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 Danny knows whether or not nobody is thirsty. It is publicly announced that Anthony does not know whether Dennis is thirsty.

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

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

Hypothesis: Dennis can now know whether Anthony can know whether Dennis 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 Alice does not know whether Betty is thirsty. It is publicly announced that Alice 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)) ] Agentd knows whether (Agentc knows whether 4)

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

Hypothesis: Norman can now know whether Brittany can know whether Norman is thirsty.


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

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

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

Hypothesis: John can now know whether Barbara 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 not everyone's forehead is muddy.

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

Hypothesis: Stella can now know that Rosemary can know whether Stella'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 Rosetta's forehead is muddy. It is publicly announced that Rosetta's forehead is muddy. It is publicly announced that Jane's forehead is muddy.

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Mildred's forehead is not muddy. It is publicly announced that David does not know whether David's forehead is muddy. It is publicly announced that David 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) ] [ ! ~(Agentb knows whether 2) ] [ ! (Agentb knows whether (1&2&3)) ] 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) ] [ ! ~(1) ] [ ! ~(Agentb knows whether 2) ] [ ! (Agentb knows whether (1&2&3)) ] 0

Hypothesis: Mary can now know that Mildred can 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 Eva knows whether someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Louis's forehead is not muddy.

Hypothesis: Steven can now know that Eva 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 Catalina knows whether or not everyone's forehead is muddy. It is publicly announced that Martha knows whether or not everyone's forehead is muddy.

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

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


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

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

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

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


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

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

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

Hypothesis: Marvin can now know that Richard can know that Steve 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 Cheryl knows whether or not everyone is thirsty.

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

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

Hypothesis: Allen can now know whether Dick can know whether Allen 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 Eric does not know whether Richard is thirsty. It is publicly announced that Scott does not know whether or not everyone is thirsty. It is publicly announced that Scott does not know whether Richard is thirsty. It is publicly announced that Richard does not know whether or not everyone is thirsty.

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

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

Hypothesis: Eric can now know whether Richard can know that Richard is thirsty.


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

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


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

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


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

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

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

Hypothesis: Matthew can now know that Michael can know whether someone picked a red card.


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

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 1 ] Agenta knows whether (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: Mary can now know whether Luis can know that someone is thirsty.


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

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

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

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

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

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

Hypothesis: Oliver can now know whether Peter can know that nobody's forehead is muddy.


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

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

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

Hypothesis: Douglas can now know whether Jayna can know that someone picked a red card.


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

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

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

Hypothesis: Billy 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 someone's forehead is muddy.

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

Hypothesis: Gigi can now know whether William can know that Gigi'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.

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


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

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


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

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

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

Hypothesis: Patricia can now know that Jeremy 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 Louis knows whether Louis is thirsty.

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

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

Hypothesis: Addie can now know whether Louis is thirsty.


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

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

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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Charles's forehead is muddy. It is publicly announced that Troy does not know whether Troy'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) ] Agenta knows whether (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) ] [ ! 2 ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Marty can now know whether Troy can know that nobody's forehead is muddy.


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

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

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

Hypothesis: Cassandra can now know that Cheryl 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 Elizabeth knows whether someone is thirsty.

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

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

Hypothesis: Elizabeth can now know that Frank can know that Fernando 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 Kimberly knows whether or not nobody is thirsty. It is publicly announced that Lavon does not know whether or not everyone is thirsty.

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

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

Hypothesis: Teri can now know whether Lavon can know that everyone is thirsty.


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

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

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

Hypothesis: Ray can now know that Ray is thirsty.


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

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

Hypothesis: Zachery can now know that Mary can know whether Jonas 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 Wallace knows whether someone's forehead is muddy. It is publicly announced that Bradley does not know whether or not everyone's forehead is muddy. It is publicly announced that Wallace knows whether someone's forehead is muddy.

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

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


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Yvonne knows whether Tammy'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 3) ] (Agentd knows that 4)

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


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

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

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

Hypothesis: Robert can now know that everyone is thirsty.


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

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

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

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


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

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

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

Hypothesis: Lynn can now know that Sandra can know whether Lynn 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 Anita does not know whether Kathleen is thirsty. It is publicly announced that Patricia does not know whether Kathleen is thirsty. It is publicly announced that Kathleen does not know whether Anita is thirsty. It is publicly announced that Anita is thirsty.

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

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

Hypothesis: Patricia 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 everyone's forehead is muddy. It is publicly announced that Alfredo'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) ] [ ! 3 ] Agenta knows whether (Agentc knows whether (1|2|3))

Hypothesis: Joseph can now know whether Alfredo 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 Dale does not know whether or not everyone's forehead is muddy.

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

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

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

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

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


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

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

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

Hypothesis: Kristy can now know whether Josephine can know that nobody picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Beverlee does not know whether Beverlee'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 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) ] 0

Hypothesis: Frank can now know that Frank'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 Rosemarie knows whether Rosemarie is thirsty. It is publicly announced that Lynwood does not know whether or not everyone is thirsty.

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

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

Hypothesis: Megan can now know that Lynwood can know whether Lynwood is thirsty.


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

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

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

Hypothesis: Jonathan can now know whether Anthony 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 Gussie knows whether Federico's forehead is muddy.

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

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


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

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

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

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


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

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

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

Hypothesis: Wesley can now know that Steven can know that someone is thirsty.


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

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

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

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


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

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

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

Hypothesis: Donald can now know whether Christina can know whether Jeanne 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 everyone's forehead is muddy.

Hypothesis: Lynn can now know whether Sandra 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 Warren knows whether Hazel's forehead is muddy. It is publicly announced that Warren knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Clyde can now know whether Lillian'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 Mary does not know whether Celia is thirsty. It is publicly announced that Celia does not know whether Amanda is thirsty. It is publicly announced that Celia is thirsty. It is publicly announced that Celia knows whether someone is thirsty.

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

Hypothesis: Celia can now know that Elizabeth is thirsty.


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

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

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

Hypothesis: Jennifer can now know whether Daniel can know that nobody's forehead is muddy.


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

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

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

Hypothesis: Ruth can now know that Roy can know whether Ruth 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 Lois knows whether someone's forehead is muddy. It is publicly announced that Ruth does not know whether or not everyone's forehead is muddy.

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

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


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

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

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


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

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

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

Hypothesis: James can now know whether Holly can know that Holly 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 Judith does not know whether Judith's forehead is muddy. It is publicly announced that Judith'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) ] [ ! ~(Agentc knows whether 3) ] [ ! ~(3) ] Agenta knows that (Agentd knows that 4)

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


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

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

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

Hypothesis: Daniel can now know that Dorothy 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.

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 1)

Hypothesis: Bonnie can now know that Mary can know that Robert'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 knows whether someone's forehead is muddy. It is publicly announced that Cody does not know whether or not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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

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

Hypothesis: Charles can now know that James can 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 Helen's forehead is not muddy.

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

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

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


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

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

Hypothesis: Kathleen can now know whether Carl can know that Kathleen's forehead is muddy.


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

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

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

Hypothesis: Jeffrey can now know whether Margaret 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). James's card is shown to Angelina. Angelina's card is shown to James. James's card is shown to James. Joseph's card is shown to James. Angelina's card is shown to Joseph. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Joseph can now know that Joseph 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 John knows whether John's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

Hypothesis: John can now know that Pauline'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 Geneva knows whether Ryan'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: Geneva can now know that David 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 Sean does not know whether or not everyone is thirsty. It is publicly announced that Sean does not know whether Anita is thirsty. It is publicly announced that Troy does not know whether Anita is thirsty. It is publicly announced that not everyone is thirsty.

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

Hypothesis: Anita 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 Roy does not know whether Roy's forehead is muddy.

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

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

Hypothesis: Roy can now know that John can 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 Gregory knows whether Beth'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) ] Agentb knows that (Agenta 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: Beth can now know that Gregory can know that Gregory's forehead is muddy.


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

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

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

Hypothesis: Judith can now know that Corina can know whether or not everyone picked a red card.


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

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

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

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

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

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

Hypothesis: Matthew can now know whether Michael can know whether someone'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 Alvin knows whether someone's forehead is muddy. It is publicly announced that Teresa does not know whether or not everyone's forehead is muddy. It is publicly announced that Alvin knows whether someone's forehead is muddy.

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

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

Hypothesis: Teresa can now know whether Teresa'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 Lorraine does not know whether Lorraine's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Marina knows whether or not everyone's forehead is muddy. It is publicly announced that Marina's forehead is muddy.

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

Hypothesis: Lorraine can now know whether Thomas can know that Lorraine'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 Marylou is thirsty. It is publicly announced that Mozella does not know whether or not everyone is thirsty. It is publicly announced that John does not know whether Kevin is thirsty. It is publicly announced that Marylou knows whether or not everyone is thirsty.

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

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


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

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

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

Hypothesis: Richard can now know whether Anthony can know that Anthony 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 Tom does not know whether Phyllis is thirsty.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Cynthia can now know that Charles can know that Cynthia'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 Laura is thirsty. It is publicly announced that Laura does not know whether Matthew is thirsty. It is publicly announced that Robert does not know whether Matthew is thirsty.

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

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

Hypothesis: Jesus can now know that nobody is thirsty.


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

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

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

Hypothesis: Marco can now know whether Mark can know that Raymond 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 Donna 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)) ] Agentc knows whether (Agentb knows whether 1)

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

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


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

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

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

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


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

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

Hypothesis: Amber can now know whether Paul can know that Gabriel'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 Lola does not know whether Eugene is thirsty. It is publicly announced that Eva does not know whether Eugene is thirsty.

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

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

Hypothesis: Sonya can now know that Eugene 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 Linwood is not thirsty. It is publicly announced that not everyone is thirsty.

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

Hypothesis: Linwood can now know whether Tanya can know whether Tanya is thirsty.


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

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

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

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


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

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

Hypothesis: Ruth can now know that Janice can know that Ruth 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 Laura does not know whether Johnny is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 1) ] (Agentc knows that (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: Nohemi can now know that everyone is thirsty.


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

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

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

Hypothesis: Holly can now know whether Logan 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 everyone is thirsty.

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

Hypothesis: Shane can now know whether Clara 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 Brian does not know whether Ollie is thirsty. It is publicly announced that Ollie does not know whether Brian is thirsty. It is publicly announced that Julia does not know whether or not everyone is thirsty. It is publicly announced that Ollie does not know whether or not everyone is thirsty.

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

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

Hypothesis: Ollie can now know that Ollie 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 Beverly's forehead is not muddy. It is publicly announced that Joseph knows whether or not everyone's forehead is muddy.

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

Hypothesis: Jeff 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's forehead is muddy. It is publicly announced that Jose knows whether Jose's forehead is muddy.

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

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


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

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

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

Hypothesis: Kelly can now know that Tracy can 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 Barbara does not know whether Guadalupe is thirsty.

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

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

Hypothesis: Barbara can now know that Patricia can know that nobody is thirsty.


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

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

Hypothesis: Michele can now know whether James is thirsty.


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

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

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

Hypothesis: Pamela can now know that Robert can know that everyone is thirsty.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Matthew 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) ] (Agenta knows that (~(1|2)))

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

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


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

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

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

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


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Denise knows whether or not nobody'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) ] [ ! (Agentc knows whether (~(1|2|3))) ] [ ! (1&2&3) ] (Agentb knows whether 2)

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


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

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

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

Hypothesis: Sharon can now know whether Larry 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 Matthew knows whether or not everyone is thirsty.

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

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

Hypothesis: Matthew can now know whether Amy can know whether Diana 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 Laura knows whether Alice's forehead is muddy. It is publicly announced that Laura's forehead is muddy.

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

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

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


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Myrtle does not know whether Cody is thirsty. It is publicly announced that Myrtle does not know whether Cole 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) ] [ ! ~(Agentb knows whether 4) ] (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) ] [ ! ~(Agentb knows whether 4) ] 0

Hypothesis: Cody can now know that Dawn is thirsty.


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

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

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

Hypothesis: Frank can now know whether Antoinette picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Linda knows whether someone is thirsty. It is publicly announced that Abraham does not know whether Clarence is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Linda knows whether Linda 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)) ] [ ! ~(Agentd knows whether 1) ] [ ! (~(1&2&3&4)) ] [ ! (Agentc knows whether 3) ] Agentd knows that (Agentb knows whether (1|2|3|4))

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

Hypothesis: Abraham can now know that Tommie 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.

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

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

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


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

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

Hypothesis: Ruth can now know whether Eugene can know whether Eugene'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 Dianne knows whether someone is thirsty. It is publicly announced that everyone is thirsty.

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

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

Hypothesis: Christopher can now know that Dianne is thirsty.


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

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

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

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

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

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

Hypothesis: Vincent can now know whether Pat can know whether Pat'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 Viola does not know whether Patrick is thirsty. It is publicly announced that Viola does not know whether Jeffrey is thirsty.

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

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

Hypothesis: Patrick can now know that Viola 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). Charles's card is shown to Gladys. Kenneth's card is shown to Charles. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Kenneth can now know whether Gladys can know whether Ray picked a red card.


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

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

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

Hypothesis: Vernon can now know that Audrey picked a red card.


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

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

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

Hypothesis: Samuel can now know that Mark picked a red card.


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Lester can now know whether Anthony can know whether Donnie 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 Peggy knows whether Eugene's forehead is muddy. It is publicly announced that Peggy's forehead is not muddy. It is publicly announced that Luisa knows whether Peggy'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) ] (Agenta knows whether 3)

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

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


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

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

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

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


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

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

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

Hypothesis: Thomas can now know whether Bruce 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 Robin does not know whether or not everyone's forehead is muddy. It is publicly announced that Robin knows whether someone's forehead is muddy.

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

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

Hypothesis: Robin can now know whether Pamela'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 Claude does not know whether Debby is thirsty. It is publicly announced that Genevieve does not know whether Claude is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that not everyone is thirsty.

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

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

Hypothesis: Genevieve can now know that everyone is thirsty.


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

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

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

Hypothesis: James can now know that everyone is thirsty.