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 Paul's forehead is not muddy.

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

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


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. Otis's card is shown to Otis. It is publicly announced that someone picked a red card. It is publicly announced that Robert knows whether or not nobody picked a red card. It is publicly announced that Otis does not know whether Robert picked a red card.

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

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

Hypothesis: Robert can now know that Otis can 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. It is publicly announced that Roger'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) ] [ ! 2 ] (Agentc knows that (1&2&3))

Hypothesis: John 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 everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

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


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

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

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

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


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

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

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


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

Hypothesis: Mark can now know that Leticia'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 Joseph does not know whether Albert is thirsty.

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

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

Hypothesis: Albert can now know that Jackie can know whether Joseph 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 Hong does not know whether Steven 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) ] Agentc knows that (Agenta knows whether 3)

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

Hypothesis: Jean can now know that Cindy can know whether Jean 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 Cassie knows whether Cassie's forehead is muddy.

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

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

Hypothesis: Cassie can now know whether Carolyn 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 not everyone's forehead is muddy. It is publicly announced that Joan's forehead is not muddy.

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

Hypothesis: Joan can now know that Romeo can know whether Steven'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 Nelda knows whether Nelda'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 3) ] Agentb knows that (Agenta knows that 3)

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

Hypothesis: Scott can now know that Ashley can know that Nelda'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 Ida knows whether Queen's forehead is muddy.

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

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

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

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

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


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

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

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

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


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

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


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Steven knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Timothy can now know that Steven'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 Richard knows whether Richard's forehead is muddy.

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

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

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


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

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

Hypothesis: Jeff can now know that Stewart can know that Flossie 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 everyone's forehead is muddy. It is publicly announced that James's forehead is muddy.

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


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

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

Hypothesis: Jaclyn can now know that Joyce can know that Hannah'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 Linette knows whether Linette's forehead is muddy.

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

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


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

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

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

Hypothesis: Maryalice 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 Yvette knows whether someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Donna knows whether Donna's forehead is muddy. It is publicly announced that Yvette knows whether Donna'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)) ] [ ! (1&2&3) ] [ ! (Agentb knows whether 2) ] [ ! (Agentc knows whether 2) ] Agentc knows that (Agentb knows whether 3)

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

Hypothesis: Yvette can now know that Donna can know whether Yvette'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 Bessie's forehead is not muddy.

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

Hypothesis: Ruth can now know whether Linda can know that nobody's forehead is muddy.


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

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

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

Hypothesis: Venice can now know whether Venice 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 Thomas knows whether Marjorie's forehead is muddy. It is publicly announced that Lawrence knows whether or not everyone's forehead is muddy.

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

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

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


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

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

Hypothesis: Gordon 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 Aaron knows whether Kenneth'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 4) ] Agentd knows whether (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) ] [ ! (Agentb knows whether 4) ] 0

Hypothesis: Kenneth can now know whether Aaron can know that Aaron's forehead is muddy.


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

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

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

Hypothesis: Nellie 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 Mark knows whether Mark's forehead is muddy. It is publicly announced that Michael knows whether or not everyone's forehead is muddy.

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

Hypothesis: Ryan can now know that Michael 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). Ruth's card is shown to Franklin. Jonathan's card is shown to Franklin. Jeremy's card is shown to Franklin. Jonathan's card is shown to Ruth. Jonathan's card is shown to Jeremy. Jeremy's card is shown to Jeremy. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Jonathan can now know whether Franklin can know that Jeremy picked a red card.


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

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

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

Hypothesis: Virginia can now know that Tom can know whether Virginia 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 Thomas knows whether or not everyone's forehead is muddy. It is publicly announced that Kimberly's forehead is not muddy.

Hypothesis: Kimberly can now know that Thomas'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 Teresa knows whether Teresa'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) ] [ ! (Agentc knows whether 3) ] [ ! (~(1&2&3&4)) ] Agentc knows that (Agentb knows whether 1)

Hypothesis: Teresa can now know that Anthony can know whether Olin'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 Lloyd knows whether Cheryl's forehead is muddy. It is publicly announced that Cheryl knows whether Cheryl's forehead is muddy. It is publicly announced that Lloyd knows whether Lloyd's forehead is muddy.

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

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

Hypothesis: Lloyd can now know that Cheryl'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 Carol knows whether someone's forehead is muddy.

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

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


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

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

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

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


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

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

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

Hypothesis: Eddie can now know that Craig 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 Morgan knows whether Ramonita's forehead is muddy. It is publicly announced that Sandra's forehead is not muddy.

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

Hypothesis: Lara can now know that Lara'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 Jack's forehead is not muddy. It is publicly announced that Jack's forehead is not muddy. It is publicly announced that Jack's forehead is not muddy.

Hypothesis: Kenneth can now know that Jack can know that Jill'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 William 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))) ] Agentb knows that (Agentc knows whether (1|2|3|4))

Hypothesis: Troy can now know that Carrie 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). Frank's card is shown to Frank. Linda's card is shown to Frank. Frank's card is shown to Linda. Linda's card is shown to Linda. Frank's card is shown to Michelle. Mabel's card is shown to Michelle. Michelle's card is shown to Mabel. Mabel's card is shown to Mabel. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Michelle can now know whether Frank can know that Linda picked a red card.


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

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

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

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

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

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


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

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

Hypothesis: William can now know that Norman can know whether or not everyone is thirsty.


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

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

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

Hypothesis: Louie can now know that Elbert can know that Louie 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 Ryan knows whether Stephen's forehead is muddy. It is publicly announced that Steven knows whether or not everyone's forehead is muddy.

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

Hypothesis: Harry 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). Byron's card is shown to Kenneth. Byron's card is shown to Joshua. Joshua's card is shown to Byron. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card.

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

Hypothesis: Joshua can now know whether Kenneth can know whether Byron 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 Laura knows whether Jessica's forehead is muddy. It is publicly announced that Javier knows whether Andrea'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 4) ] [ ! (Agentc knows whether 2) ] (Agentd knows that (1|2|3|4))

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

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


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

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

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

Hypothesis: Michael can now know whether Michael 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 Irene knows whether or not everyone's forehead is muddy. It is publicly announced that Connie's forehead is muddy.

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

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


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

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

Hypothesis: Rodney can now know that Brian can know whether Carol's forehead is muddy.


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

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

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

Hypothesis: Jason can now know whether Kathleen can know that Kathleen 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 Alejandro does not know whether Alejandro's forehead is muddy.

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

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

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


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

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


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Brenda knows whether or not nobody is thirsty. It is publicly announced that Erma does not know whether Brenda is thirsty. It is publicly announced that Ruthann does not know whether Erma is thirsty.

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

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

Hypothesis: Erma can now know whether Brenda 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 Susan knows whether or not nobody 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))) ] (Agenta knows that 2)

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

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

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

Hypothesis: Robert can now know whether Lois can know that Patricia's forehead is muddy.


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

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

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

Hypothesis: Debra can now know that Beulah picked a red card.


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

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

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

Hypothesis: Kathy can now know that Thomas can know that Thomas 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 William knows whether Zelda's forehead is muddy. It is publicly announced that Sandra knows whether or not nobody's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Peggy can now know whether Gary can know that Gary 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 Joseph knows whether or not everyone's forehead is muddy.

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

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

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


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

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

Hypothesis: Jacqulyn can now know that Frank can know whether Jacqulyn 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 Dennis knows whether Rose's forehead is muddy. It is publicly announced that Karen's forehead is muddy.

Hypothesis: Dennis 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). Benjamin's card is shown to Benjamin. Michael's card is shown to Michael. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Benjamin can now know whether Ivory 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 Dustin knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: Brian can now 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 Catherine knows whether Nellie's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

Hypothesis: Gordon 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 Gregory's forehead is not muddy. It is publicly announced that Phyllis knows whether someone's forehead is muddy.

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

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

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


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

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

Hypothesis: Aaron can now know that Tiffany 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). Annie's card is shown to Annie. Edna's card is shown to Annie. Charles's card is shown to Charles. Danielle's card is shown to Edna. Charles's card is shown to Danielle. It is publicly announced that someone picked a red card. It is publicly announced that Edna does not know whether or not everyone picked a red card. It is publicly announced that Charles knows whether or not everyone picked a red card.

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

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

Hypothesis: Annie can now know whether Annie 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 Katie knows whether or not everyone's forehead is muddy. It is publicly announced that Victor's forehead is muddy.

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

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


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

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

Hypothesis: David can now know whether Kimberly can know that Lance'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 Bonnie knows whether Nicole's forehead is muddy. It is publicly announced that Anna'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) ] [ ! 3 ] (Agentb knows whether (~(1|2|3|4)))

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


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

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

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

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


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

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

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

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

Hypothesis: Dawn 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). Derrick's card is shown to Derrick. Vera's card is shown to Maria. Derrick's card is shown to Vera. Vera's card is shown to Vera. It is publicly announced that someone picked a red card.

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

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

Hypothesis: Maria can now know that Vera 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 Betty knows whether someone's forehead is muddy. It is publicly announced that Erik knows whether Hilario's forehead is muddy.

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

Hypothesis: Erik can now know that Hilario'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 Cathy knows whether Micheal is thirsty. It is publicly announced that Cathy knows whether Micheal is thirsty.

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

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

Hypothesis: Micheal can now know whether or not 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 Patricia knows whether or not everyone is thirsty.

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

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

Hypothesis: Roger can now know that James 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.

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

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


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

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

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


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

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

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

Hypothesis: Ana can now know that Anthony can know that Ana 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 Terry knows whether Terry's forehead is muddy.

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Michele 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) ] [ ! ~(Agenta knows whether 3) ] Agentd knows that (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) ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: Kenneth can now know that William can know that everyone is thirsty.


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


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Wanda does not know whether or not everyone's forehead is muddy. It is publicly announced that Ryan knows whether Ryan'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) ] (Agenta knows whether (1&2))

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

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


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

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

Hypothesis: Doris can now know that Debra can know that Debra'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 Johanna does not know whether Charles is thirsty. It is publicly announced that Barbara is not thirsty.

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

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

Hypothesis: Rose can now know that Charles can know whether Rose 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 Tessie's forehead is not muddy. It is publicly announced that Elizabeth does not know whether Elizabeth's forehead is muddy.

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

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

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


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 Virginia. Anthony's card is shown to Anthony. Anthony's card is shown to Brooke. Brooke's card is shown to Brooke. Brooke's card is shown to Nola. It is publicly announced that someone picked a red card.

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

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

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


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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Huey does not know whether Aaron is thirsty. It is publicly announced that Lewis does not know whether Robert is thirsty. It is publicly announced that Huey knows whether or not nobody is thirsty.

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

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

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


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

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


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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Elizabeth does not know whether Frank 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) ] (Agentd knows that 4)

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

Hypothesis: Lisa can now know that Lisa 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 Tessa knows whether Tessa is thirsty. It is publicly announced that Esperanza does not know whether or not everyone is thirsty.

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

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


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

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

Hypothesis: Kimberly can now know whether David can know that Barrett'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 Laurie does not know whether Linda is thirsty. It is publicly announced that Laurie does not know whether or not everyone is thirsty. It is publicly announced that Dennis knows whether Laurie is thirsty. It is publicly announced that Linda 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 3) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agentc knows whether (1&2&3)) ] Agenta knows that (Agentc knows whether (1&2&3))

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

Hypothesis: Dennis can now know that Linda can know whether or not everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Bonnie does not know whether Ruth 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 that (Agentd knows that (~(1|2|3|4)))

Hypothesis: Esther can now know that Thomas can know that nobody is thirsty.


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

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


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

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

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

Hypothesis: Charlie can now know that Clinton can know that Charlie'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 Daniel's forehead is muddy. It is publicly announced that Daniel's forehead is muddy. It is publicly announced that Fernando's forehead is not muddy.

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


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

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

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


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

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

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

Hypothesis: Jonathan can now know whether Kevin 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 Allyson does not know whether Catherine is thirsty. It is publicly announced that Lucille 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) ] [ ! 2 ] (Agenta knows whether 3)

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

Hypothesis: Harry can now know whether Allyson is thirsty.


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

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

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

Hypothesis: Malcolm can now know that Christina 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 Catherine's forehead is muddy.

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

Hypothesis: Catherine can now know whether Jackie can know that Jackie's forehead is muddy.


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

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


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

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

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

Hypothesis: Kathy 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). Gary's card is shown to Charles. Charles's card is shown to James. James's card is shown to James. Charles's card is shown to Gary. It is publicly announced that someone picked a red card. It is publicly announced that Charles does not know whether James picked a red card.

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

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

Hypothesis: Charles can now know whether or not 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 Alan does not know whether or not everyone's forehead is muddy.

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

Hypothesis: Margaret 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). Leo's card is shown to Steven. Leo's card is shown to Leo. It is publicly announced that someone picked a red card. It is publicly announced that Steven does not know whether or not everyone picked a red card. It is publicly announced that Steven does not know whether Steven picked a red card. It is publicly announced that Steven does not know whether or not everyone picked a red card.

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

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

Hypothesis: Leo 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 Heather knows whether Nelson's forehead is muddy. It is publicly announced that Heather's forehead is muddy.

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

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


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

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

Hypothesis: Kimberly can now know whether William can know that William'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 Edward knows whether or not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] Agentc knows 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: Becky can now know whether Janie can know whether Edward'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 Scott knows whether Charles's forehead is muddy.

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

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

Hypothesis: Scott can now know that Lola 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). Jessie's card is shown to Patricia. Jessie's card is shown to Jessie. Jessie's card is shown to Alicia. It is publicly announced that someone picked a red card.

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

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

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


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

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

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

Hypothesis: Sheila can now know that Sharon 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 Wanda knows whether Wanda is thirsty. It is publicly announced that Janis does not know whether or not everyone is thirsty.

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

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


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Adam's card is shown to Idell. Idell's card is shown to Adam. Adam's card is shown to Adam. Lillian's card is shown to Adam. Idell's card is shown to Lillian. It is publicly announced that someone picked a red card. It is publicly announced that Lillian does not know whether or not everyone picked a red card. It is publicly announced that Adam knows whether Lillian 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 whether (1&2&3)) ] [ ! (Agentb knows whether 3) ] Agentb knows whether (Agentc knows whether 3)

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

Hypothesis: Adam can now know whether Lillian can know whether Lillian 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 Lisa does not know whether William is thirsty. It is publicly announced that William does not know whether or not everyone is thirsty. It is publicly announced that William does not know whether Marilyn is thirsty. It is publicly announced that William does not know whether Marilyn is thirsty.

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

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

Hypothesis: Marilyn can now know that Lisa can know whether Lisa 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 Bernice does not know whether Donald is thirsty.

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

Hypothesis: Bernice can now know whether Donald 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 Brian does not know whether or not everyone is thirsty. It is publicly announced that David knows whether someone is thirsty.

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

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

Hypothesis: Brian can now know whether someone is thirsty.


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

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

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

Hypothesis: Helen can now know whether Albert'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 Troy does not know whether Troy's forehead is muddy. It is publicly announced that Josiah knows whether Josiah'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) ] [ ! (Agentc knows whether 3) ] Agentb knows whether (Agentc knows whether 2)

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

Hypothesis: Annie can now know whether Josiah can know whether Annie'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 Rose knows whether someone's forehead is muddy. It is publicly announced that Ethan knows whether Jennifer's forehead is muddy. It is publicly announced that Jennifer knows whether someone's forehead is muddy.

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

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

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

Hypothesis: Tina 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 John 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) ] [ ! 1 ] [ ! (1|2|3) ] (Agenta knows that 1)

Hypothesis: John can now know that John is thirsty.


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

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

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

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


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

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

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Daniel does not know whether Anna is thirsty. It is publicly announced that Lorenzo 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) ] [ ! 2 ] Agenta knows whether (Agentc knows whether (1|2|3|4))

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

Hypothesis: Victor can now know whether Daniel can know whether 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 Brian knows whether Micaela's forehead is muddy.

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

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

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


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

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

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

Hypothesis: Gregory can now know whether Robert can know that 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 Flora knows whether someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] (Agentc knows whether (~(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: Flora 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 Sandra's forehead is muddy. It is publicly announced that Sandra knows whether Mary'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) ] [ ! 3 ] [ ! (Agentc knows whether 1) ] [ ! (~(1&2&3)) ] Agenta knows that (Agentb knows whether 1)

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


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

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

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

Hypothesis: Katie can now know whether Christian can know that Katie is thirsty.


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

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

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

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


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

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

Hypothesis: Betty can now know that James'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 Douglas knows whether Richard's forehead is muddy.

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

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

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


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

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

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

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


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

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

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

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


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

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

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

Hypothesis: Luis can now know that 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 Irma'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) ] [ ! 2 ] Agenta knows that (Agentc knows that (1&2&3))

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


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

Hypothesis: James can now know that Nancy can know whether James 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 Arthur's forehead is not muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Ronda knows whether Arthur's forehead is muddy.

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

Hypothesis: Joseph can now know that Arthur can know that Joseph'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 James knows whether or not nobody's forehead is muddy.

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

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

Hypothesis: James can now know whether Katherine 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 John's forehead is muddy.

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

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

Hypothesis: Wesley can now know whether John'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 Jerry knows whether Frank'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) ] (Agentc knows that 3)

Hypothesis: Frank can now know that Frank'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 Zachery's forehead is not muddy.

Hypothesis: Leah can now know that Zachery'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 Lindsay 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) ] [ ! (Agenta knows whether (1|2|3|4)) ] (Agentc knows that 2)

Hypothesis: James can now know that Grace'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 Tony knows whether or not everyone's forehead is muddy. It is publicly announced that Rickie's forehead is not muddy.

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

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

Hypothesis: Tony can now know whether Gregory can know whether Gregory'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 James's forehead is muddy.

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

Hypothesis: Jason can now know that Crystal'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 Tim knows whether Frederica's forehead is muddy.

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

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

Hypothesis: Frederica can now know that Thomas can know that Frederica'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 Julie knows whether Julie's forehead is muddy.

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

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


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jeffrey does not know whether Jeffrey'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 (Agentd 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) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Preston can now know whether Vita can know whether Jeffrey's forehead is muddy.


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

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

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

Hypothesis: Thomas can now know that Wanda can know whether Wanda 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 Cathy knows whether Cathy's forehead is muddy. It is publicly announced that Jessica knows whether Cathy's forehead is muddy.

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

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

Hypothesis: Cathy can now know that Kelly can know that Kelly'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 Cherry 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) ] [ ! (Agenta knows whether (1&2&3&4)) ] Agentc knows that (Agentd knows that 4)

Hypothesis: Joseph can now know that Anthony can know that Anthony'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 Arturo does not know whether Lisa is thirsty. It is publicly announced that Lisa is not thirsty. It is publicly announced that Elizabeth does not know whether Arturo is thirsty.

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

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

Hypothesis: Elizabeth can now know that Elizabeth is thirsty.