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The following is an exercise from the book: Reasoning about Knowledge by R.Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi.

I have been trying for some time to crack it but still unable to find a simple approach to solve it. Any suggestions are appereciated.

Consider a game which is played with a deck consisting of four aces and four eights. There are three players. Six cards are dealt out, two to each player. The remaining two cards are left face down. Without looking at the cards, each of the players raises them up to his or her forehead, so that the other two players can see them but he or she cannot. Then all of the players take turns trying to determine which cards they’re holding (they do not have to name the suits). If a player does not know which cards he or she is holding, the player must say so. Of course, it is common knowledge that none of you would ever lie, and that all players are perfect reasoners.

Show that there exists a situation where only one of the players will be able to determine what cards he or she holds, and the other two will never be able to determine what cards they hold, no matter how many rounds are played.

It is relatively simple to see that at least one player will detarmine her/his own cards the as it seems key cases are:

$AA,88,A8$ - third player determines his/her cards since first and second players can nonot determine theyr cards $AA,88$ are impossible.

$A8,88,A8$ - first player determines his/her own cards on second move.

$A8,A8,A8$ - second player determines his/her cards on second move.

All other cases are reducible to the three to see that at least one will determine his/her cards at some move.

But I can not find the case when the remaining two can not determine their cards in any number of moves. I don't see a structure which gives this.

It seems that A8,88,A8 and A8,AA,A8 are such situations that second and third player can not decide which cards they have, am I right?

Can players 1 and 2 distinguish $AA,88,A8$ from $AA, AA, 88$?

The following is an exercise from the book: Reasoning about Knowledge by R.Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi.

I have been trying for some time to crack it but still unable to find a simple approach to solve it. Any suggestions are appereciated.

Consider a game which is played with a deck consisting of four aces and four eights. There are three players. Six cards are dealt out, two to each player. The remaining two cards are left face down. Without looking at the cards, each of the players raises them up to his or her forehead, so that the other two players can see them but he or she cannot. Then all of the players take turns trying to determine which cards they’re holding (they do not have to name the suits). If a player does not know which cards he or she is holding, the player must say so. Of course, it is common knowledge that none of you would ever lie, and that all players are perfect reasoners.

Show that there exists a situation where only one of the players will be able to determine what cards he or she holds, and the other two will never be able to determine what cards they hold, no matter how many rounds are played.

It is relatively simple to see that at least one player will detarmine her/his own cards the as it seems key cases are:

$AA,88,A8$ - third player determines his/her cards since first and second players can nonot determine theyr cards $AA,88$ are impossible.

$A8,88,A8$ - first player determines his/her own cards on second move.

$A8,A8,A8$ - second player determines his/her cards on second move.

All other cases are reducible to the three to see that at least one will determine his/her cards at some move.

But I can not find the case when the remaining two can not determine their cards in any number of moves. I don't see a structure which gives this.

It seems that $A8,88,A8$ and $A8,AA,A8$ are such situations that second and third player can not decide which cards they have, am I right?

Can players 1 and 2 distinguish $AA,88,A8$ from $AA, AA, 88$?

The following is an exercise from the book: Reasoning about Knowledge by R.Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi.

I have been trying for some time to crack it but still unable to find a simple approach to solve it. Any suggestions are appereciated.

Consider a game which is played with a deck consisting of four aces and four eights. There are three players. Six cards are dealt out, two to each player. The remaining two cards are left face down. Without looking at the cards, each of the players raises them up to his or her forehead, so that the other two players can see them but he or she cannot. Then all of the players take turns trying to determine which cards they’re holding (they do not have to name the suits). If a player does not know which cards he or she is holding, the player must say so. Of course, it is common knowledge that none of you would ever lie, and that all players are perfect reasoners.

Show that there exists a situation where only one of the players will be able to determine what cards he or she holds, and the other two will never be able to determine what cards they hold, no matter how many rounds are played.

It is relatively simple to see that at least one player will detarmine her/his own cards the as it seems key cases are: AA,88,A8 - third player determines his/her cards since first and second players can nonot determine theyr cards AA,88 are impossible.

A8,88,A8 - first player determines his/her own cards on second move.

A8,A8,A8 - second player determines his/her cards on second move.

All other cases are reducible to the three to see that at least one will determine his/her cards at some move.

But I can not find the case when the remaining two can not determine their cards in any number of moves. I don't see a structure which gives this.

It seems that A8,88,A8 and A8,AA,A8 are such situations that second and third player can not decide which cards they have, am I right?

The following is an exercise from the book: Reasoning about Knowledge by R.Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi.

I have been trying for some time to crack it but still unable to find a simple approach to solve it. Any suggestions are appereciated.

Consider a game which is played with a deck consisting of four aces and four eights. There are three players. Six cards are dealt out, two to each player. The remaining two cards are left face down. Without looking at the cards, each of the players raises them up to his or her forehead, so that the other two players can see them but he or she cannot. Then all of the players take turns trying to determine which cards they’re holding (they do not have to name the suits). If a player does not know which cards he or she is holding, the player must say so. Of course, it is common knowledge that none of you would ever lie, and that all players are perfect reasoners.

Show that there exists a situation where only one of the players will be able to determine what cards he or she holds, and the other two will never be able to determine what cards they hold, no matter how many rounds are played.

It is relatively simple to see that at least one player will detarmine her/his own cards the as it seems key cases are:

$AA,88,A8$ - third player determines his/her cards since first and second players can nonot determine theyr cards $AA,88$ are impossible.

$A8,88,A8$ - first player determines his/her own cards on second move.

$A8,A8,A8$ - second player determines his/her cards on second move.

All other cases are reducible to the three to see that at least one will determine his/her cards at some move.

But I can not find the case when the remaining two can not determine their cards in any number of moves. I don't see a structure which gives this.

It seems that A8,88,A8 and A8,AA,A8 are such situations that second and third player can not decide which cards they have, am I right?

Can players 1 and 2 distinguish $AA,88,A8$ from $AA, AA, 88$?

The following is an exercise from the book: Reasoning about Knowledge by R.Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi.

I have been trying for some time to crack it but still unable to find a simple approach to solve it. Any suggestions are appereciated.

Consider a game which is played with a deck consisting of four aces and four eights. There are three players. Six cards are dealt out, two to each player. The remaining two cards are left face down. Without looking at the cards, each of the players raises them up to his or her forehead, so that the other two players can see them but he or she cannot. Then all of the players take turns trying to determine which cards they’re holding (they do not have to name the suits). If a player does not know which cards he or she is holding, the player must say so. Of course, it is common knowledge that none of you would ever lie, and that all players are perfect reasoners.

Show that there exists a situation where only one of the players will be able to determine what cards he or she holds, and the other two will never be able to determine what cards they hold, no matter how many rounds are played.

It is relatively simple to see that at least one player will detarmine her/his own cards the as it seems key cases are: AA,88,A8 - third player determines his/her cards since first and second players can nonot determine theyr cards AA,88 are impossible.

A8,88,A8 - first player determines his/her own cards on second move.

A8,A8,A8 - second player determines his/her cards on second move.

All other cases are reducible to the three to see that at least one will determine his/her cards at some move.

But I can not find the case when the remaining two can not determine their cards in any numebr of moves. I don't see a structure which gives this.

The following is an exercise from the book: Reasoning about Knowledge by R.Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi.

I have been trying for some time to crack it but still unable to find a simple approach to solve it. Any suggestions are appereciated.

Consider a game which is played with a deck consisting of four aces and four eights. There are three players. Six cards are dealt out, two to each player. The remaining two cards are left face down. Without looking at the cards, each of the players raises them up to his or her forehead, so that the other two players can see them but he or she cannot. Then all of the players take turns trying to determine which cards they’re holding (they do not have to name the suits). If a player does not know which cards he or she is holding, the player must say so. Of course, it is common knowledge that none of you would ever lie, and that all players are perfect reasoners.

Show that there exists a situation where only one of the players will be able to determine what cards he or she holds, and the other two will never be able to determine what cards they hold, no matter how many rounds are played.

It is relatively simple to see that at least one player will detarmine her/his own cards the as it seems key cases are: AA,88,A8 - third player determines his/her cards since first and second players can nonot determine theyr cards AA,88 are impossible.

A8,88,A8 - first player determines his/her own cards on second move.

A8,A8,A8 - second player determines his/her cards on second move.

All other cases are reducible to the three to see that at least one will determine his/her cards at some move.

But I can not find the case when the remaining two can not determine their cards in any number of moves. I don't see a structure which gives this.

It seems that A8,88,A8 and A8,AA,A8 are such situations that second and third player can not decide which cards they have, am I right?