Yellow,Green,Red,Blue

[wolfgang]

An unknown number of prisoners( more than 14)were told a day before their execution, that they are going to be blind folded, and a colored paper(either.. yellow,green,red or blue) will be glued on forehead of each one of them( no one can see his own paper), then they should inter a big hall one by one to make 4 raws in this order( Yellow,Green,Red,Blue)i.e. a raw of Yallows, a raw of Greens etc.,each one will be unfolded when he inters the hall.No one of them is allowed to arrenge the others,each one should himslf choose where to stay, once he did,he can not change his place( only the first 2 prisoners can change their minds only once).Any kind of comunication between them is not allowed. If any one of them stands in a wrong raw, all of them will die!

All colores were used.

They should make a plane which saves all of them, can you help them?

[bonanova]

I can add something which may help you... The raws may face the door or the opposit direction!

Continuing with the community solve approach, can we clarify:

1. Must all the persons in a particular row face the same direction?
2. If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication?
3. Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place?

[bonanova]

Borrowing from araver's last post, let me suggest a framework for solving the four-color case.

• P1 enters and stands facing the door.

  No generality is lost - he has no information to act on.

• P2 enters and stands

  [in front/back of, to the left/right of] P1

  [facing toward/away from the door]

  based on the color of P1.

  Enough possibilities [and more] to tell P1 his color.

• P3 enters and stands

  [in line with/beside P1 and P2]

  [closer/farther from the door, or left/right or P1 and P2, as appropriate]

  [facing the same/opposite direction as P1]

  based on the colors and positions of P1 and P2.

• P1 then

  [does/does not alter the direction he is facing]

  [does/does not alter his alignment with P2 and P3]

  based on the colors of P2 and P3.

• P4 enters and stands so that he

  [forms a square/ell/tee together with P1 P2 and P3]

  [faces same/opposite direction as P1]

  based on the positions and colors of P1, P2 and P3.

• P2 then sets

  [does/does not alters his position and/or orientation]

  based on the colors and positions of P1, P3 and P4.

• No one in the room can move after this point.

  The others must know where to stand and which direction to face

  [somehow, haven't figured that part out yet]

  based on what they see when they enter.

That's a framework.

Are there enough permutations of the above variables to spell out to the next to enter what he needs to know?

Let's assume the puzzle is solvable [see my sig] and find a way to signal the needed information forward in each case.

What role does 14 play in all of this?

Cap'n, I'm with you on that ... no clue yet.

As I said a few posts back, this is one heck of a puzzle.

Hey araver, LTNS!

[bonanova]

Another point to clarify.

Must the rows be parallel, and in particular Y-G-R-B order?

OP seems to suggest this condition.

+-------------------------+

| |

| Y Y Y Y Y . . . |

| G G G . . . . . |

Door R R R R R R . . |

| B B B B B . . . |

| |

| |

+-------------------------+

Or could the rows e.g. all start at the middle of the room, and grow outward toward the four walls, in any sequence?

Say Yellow to the north, Green to the south, Red to the east and Blue to the west?

[wolfgang]

I can add something which may help you... The raws may face the door or the opposit direction!

Continuing with the community solve approach, can we clarify:

1. Must all the persons in a particular row face the same direction?
3. If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication?
5. Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place?

During making raws...some of them can be..back to back (in any raw)...but by reaching the last prisoner,they all shoud be at the same direction

[wolfgang]

Another point to clarify.

Must the rows be parallel, and in particular Y-G-R-B order?

OP seems to suggest this condition.

+-------------------------+

| |

| Y Y Y Y Y . . . |

| G G G . . . . . |

Door R R R R R R . . |

| B B B B B . . . |

| |

| |

+-------------------------+

Or could the rows e.g. all start at the middle of the room, and grow outward toward the four walls, in any sequence?

Say Yellow to the north, Green to the south, Red to the east and Blue to the west?

Yes...the raws should be as you mentioned in your diagram,but all should face one direction

[Prime]

If the crooks who have already formed 4 neat rows start turning around in their places upon the entry of the next criminal -- that should be deemed as a form of communication by the warden. If they stay put until the newcomer takes his place, he has 3/4 chance of spoiling that neat formation.

If you arrange randomly, say, 16 men in 4x4 formation, no consecutive change of which way they face can guaranty thus newly formed rows' order and same colors in each row.

[bonanova]

Another point to clarify.

Must the rows be parallel, and in particular Y-G-R-B order?

OP seems to suggest this condition.

+-------------------------+

| |

| Y Y Y Y Y . . . |

| G G G . . . . . |

Door R R R R R R . . |

| B B B B B . . . |

| |

| |

+-------------------------+

Or could the rows e.g. all start at the middle of the room, and grow outward toward the four walls, in any sequence?

Say Yellow to the north, Green to the south, Red to the east and Blue to the west?

Yes...the raws should be as you mentioned in your diagram,but all should face one direction

"All" in one line? Or "All" in the whole room?

Say the the persons in the red line face the door.

Can the persons in the green line face away from the door?

This is like nailing jello to a tree.

[bonanova]

I can add something which may help you... The raws may face the door or the opposit direction!

Continuing with the community solve approach, can we clarify:

1. Must all the persons in a particular row face the same direction?
3. If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication?
5. Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place?

During making raws...some of them can be..back to back (in any raw)...but by reaching the last prisoner,they all shoud be at the same direction

But only the first two prisoners can move.

This means the seventh prisoner, say can come into a row back to back with someone else in that row, then at a later time switch his direction so that at the end all in that row have the same direction. So as long as a prisoner stays in the same position, he can flip his direction back and forth during the formation of rows.

That sounds like communication.

Another piece of jello fell off the nail...

[wolfgang]

I can add something which may help you... The raws may face the door or the opposit direction!

Continuing with the community solve approach, can we clarify:

1. Must all the persons in a particular row face the same direction?
3. If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication?
5. Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place?

During making raws...some of them can be..back to back (in any raw)...but by reaching the last prisoner,they all shoud be at the same direction

But only the first two prisoners can move.

This means the seventh prisoner, say can come into a row back to back with someone else in that row, then at a later time switch his direction so that at the end all in that row have the same direction. So as long as a prisoner stays in the same position, he can flip his direction back and forth during the formation of rows.

That sounds like communication.

Another piece of jello fell off the nail...

Yes....in my OP I said....each prisoner should choose his raw and once he did,he is not allowed to change his mind,but turning around himself was not mentioned

[bushindo]

I can add something which may help you... The raws may face the door or the opposit direction!

Continuing with the community solve approach, can we clarify:

1. Must all the persons in a particular row face the same direction?
3. If prisoners in say the Yellow row can face in differing directions does that constitute [illegal] communication?
5. Can the first two prisoners choose the time that they [legally] alter their position, at any time up to the last prisoner takes his place?

During making raws...some of them can be..back to back (in any raw)...but by reaching the last prisoner,they all shoud be at the same direction

But only the first two prisoners can move.

This means the seventh prisoner, say can come into a row back to back with someone else in that row, then at a later time switch his direction so that at the end all in that row have the same direction. So as long as a prisoner stays in the same position, he can flip his direction back and forth during the formation of rows.

That sounds like communication.

Another piece of jello fell off the nail...

Yes....in my OP I said....each prisoner should choose his raw and once he did,he is not allowed to change his mind,but turning around himself was not mentioned

This seems to imply that turning around (spinning or rotating) is allowed, even though switching rows isn't allowed. Isn't that a contradiction of the requirement of the OP, which I list below

An unknown number of prisoners( more than 14)were told a day before their execution, that they are going to be blind folded, and a colored paper(either.. yellow,green,red or blue) will be glued on forehead of each one of them( no one can see his own paper), then they should inter a big hall one by one to make 4 raws in this order( Yellow,Green,Red,Blue)i.e. a raw of Yallows, a raw of Greens etc.,each one will be unfolded when he inters the hall.No one of them is allowed to arrenge the others,each one should himslf choose where to stay, once he did,he can not change his place( only the first 2 prisoners can change their minds only once). Any kind of comunication between them is not allowed. If any one of them stands in a wrong raw, all of them will die!

All colores were used.

They should make a plane which saves all of them, can you help them?

Communication is defined as 'the imparting or exchanging of information or news', so by any reasonable interpretation, turning-in-place is considered a violation of the rules. If communication is allowed, then the puzzle becomes trivial. In fact, we don't even need to allow the first 2 prisoners to change their minds if we allow them to communicate.

[bonanova]

Agree that now it's not difficult.

But the first two might need change row locations.

Suppose the first three prisoners were [1] all red, [2] all different colors.

[CaptainEd]

OK, I think we are agreeing that we have to relax our interpretation of Wolfgang's proscription against communication.

The first two prisoners can establish the locations of all 4 colored rows, before the third one enters the room.

Then, the prisoners can inspect each arriving prisoner, rotate their positions to indicate the newcomer's color, and the newcomer can take his place in the appropriate row.

* Let's consider that there are (invisible) row numbers, from -4 to +4.
* Prisoner A enters and stands at 0, asserting his own color to be Yellow (regardless of his actual color, which he doesn't know). So at his right hand (position 1) should be Green, next (position 2) Red, further right (position 3) is Blue.
* Prisoner B enters, takes his position according to A's actual color. That is, B goes to the row A SHOULD be in = Row(Color(A)). Now A knows his own color and that of B.
* Prisoner A now changes the color framework by moving himself so that B is in the row B should be in. That is, A moves to Row(Color(A) + Row(B) - Color( B ) ). At this point, both prisoners know their own colors, and any observer can tell the new color framework, which may or may not be 0 = Yellow.

1) A is Green, B is Blue. A enters, establishes location of 0. B enters, sees that A is Green, so he moves to (Row(Color(A)) = 1. Now A knows A is Green and B is Blue. Now A moves to (Row(Color(A) + Row(B) - Color( B ) ) = (1+1-3) = -1. At this point, in absolute terms, A is in position -1, B is in position 1. However, to the remaining prisoners, A (visibly Green) is therefore at position 1 and B (visibly Blue) is at position 3. Anybody knows where Yellow and Red go.

2) A is Green, B is Green. A enters. B enters, moves to 1. A now knows that both A and B are Green, so A moves to (Row(Color(A) + Row(B) - Color( B ) )= (1+1-1)= 1 (ie. stands in the same row, directly behind or ahead of B). Now remaining prisoners can tell where Green is, therefore can infer where Red, Yellow and Blue are.

3) A is Green, B is Yellow. A enters. B enters, moves to 1. A now knows that A is Green and B is Yellow, moves to (Row(Color(A) + Row(B) - Color( B ) ) = (1 + 1 -0) = 2. Now remaining prisoners see B is Yellow and A is Green; they can easily infer where Red and Blue belong.


* If the prisoners were permitted to choose their order of arrival, then prisoner K looks at prisoners I and J, who just entered before him.
* Alternatively, consider the sequence of all prisoners in the Yellow row, followed by Red row, followed by Green row, followed by Blue row. Take the first two prisoners in this sequence. They are the ones who will tell prisoner K his color.

When prisoner K enters, two chosen prisoners tell him his color by facing him or facing away from him, giving two bits of information, therefore a number from 0 to 3. Which two chosen prisoners? We have a number of ways to establish a sequence; here are two ways:

Edited January 21, 2013 by CaptainEd

[CaptainEd]

Conveying position to next arrival. Consider the sequence of prisoners in rows Y, G, R, B. The first two of those prisoners will convey the newcomer's color to him.

Also, To match Bonanova's picture, I should say that Prisoner A, facing the door, starts in location 0, with his LEFT hand pointing to Position 1 = Green, further LEFT is Position 2 = Red, even further LEFT is Position 3 = Blue. The negative values are to his RIGHT.

[wolfgang]

Conveying position to next arrival. Consider the sequence of prisoners in rows Y, G, R, B. The first two of those prisoners will convey the newcomer's color to him.

Also, To match Bonanova's picture, I should say that Prisoner A, facing the door, starts in location 0, with his LEFT hand pointing to Position 1 = Green, further LEFT is Position 2 = Red, even further LEFT is Position 3 = Blue. The negative values are to his RIGHT.

OK, I think we are agreeing that we have to relax our interpretation of Wolfgang's proscription against communication.

The first two prisoners can establish the locations of all 4 colored rows, before the third one enters the room.

Then, the prisoners can inspect each arriving prisoner, rotate their positions to indicate the newcomer's color, and the newcomer can take his place in the appropriate row.

* Let's consider that there are (invisible) row numbers, from -4 to +4.

* Prisoner A enters and stands at 0, asserting his own color to be Yellow (regardless of his actual color, which he doesn't know). So at his right hand (position 1) should be Green, next (position 2) Red, further right (position 3) is Blue.

* Prisoner B enters, takes his position according to A's actual color. That is, B goes to the row A SHOULD be in = Row(Color(A)). Now A knows his own color and that of B.

* Prisoner A now changes the color framework by moving himself so that B is in the row B should be in. That is, A moves to Row(Color(A) + Row(B) - Color( B ) ). At this point, both prisoners know their own colors, and any observer can tell the new color framework, which may or may not be 0 = Yellow.

1) A is Green, B is Blue. A enters, establishes location of 0. B enters, sees that A is Green, so he moves to (Row(Color(A)) = 1. Now A knows A is Green and B is Blue. Now A moves to (Row(Color(A) + Row(B) - Color( B ) ) = (1+1-3) = -1. At this point, in absolute terms, A is in position -1, B is in position 1. However, to the remaining prisoners, A (visibly Green) is therefore at position 1 and B (visibly Blue) is at position 3. Anybody knows where Yellow and Red go.

2) A is Green, B is Green. A enters. B enters, moves to 1. A now knows that both A and B are Green, so A moves to (Row(Color(A) + Row(B) - Color( B ) )= (1+1-1)= 1 (ie. stands in the same row, directly behind or ahead of B). Now remaining prisoners can tell where Green is, therefore can infer where Red, Yellow and Blue are.

3) A is Green, B is Yellow. A enters. B enters, moves to 1. A now knows that A is Green and B is Yellow, moves to (Row(Color(A) + Row(B) - Color( B ) ) = (1 + 1 -0) = 2. Now remaining prisoners see B is Yellow and A is Green; they can easily infer where Red and Blue belong.

When prisoner K enters, two chosen prisoners tell him his color by facing him or facing away from him, giving two bits of information, therefore a number from 0 to 3. Which two chosen prisoners? We have a number of ways to establish a sequence; here are two ways:

* If the prisoners were permitted to choose their order of arrival, then prisoner K looks at prisoners I and J, who just entered before him.

* Alternatively, consider the sequence of all prisoners in the Yellow row, followed by Red row, followed by Green row, followed by Blue row. Take the first two prisoners in this sequence. They are the ones who will tell prisoner K his color.

your method is successful with the first two prisoners, but you should find a very simple,uncomplicated way for the 3rd one,if you find it, the rest will be very easy and straight forward job.

[CaptainEd]

Channeling other denizens, as we all thought this would constitute "communication", hence illegal.

The first two prisoners are now standing in rows corresponding to their own colors, and anybody can tell where the other rows are.

The prisoners inspect Prisoner C's color, and two prisoners adopt individual rotations that combine to tell C his color.

Prisoner C looks at the first prisoner in the sequence (Yellow row,Green row, Red row, Blue row).

-- If that prisoner is facing the door, C has color Yellow or Green.

-- If that prisoner is facing away from the door, C has color Red or Blue.

Prisoner C looks at the second prisoner in the sequence.

--If that prisoner is facing the door, C has color Yellow or Red.

--If that prisoner is facing away from the door, C has color Green or Blue.

Prisoner C now takes his position in the correct row, and all prisoners face the door.

These same instructions are followed no matter how many prisoners have already arrived. They are all standing in their appropriately colored rows. The first two prisoners in the sequence (Yellow row, Green row, Red row, Blue row) adopt individual rotations to tell the new arrival his own color.

[bushindo]

your method is successful with the first two prisoners, but you should find a very simple,uncomplicated way for the 3rd one,if you find it, the rest will be very easy and straight forward job.

How about this *very easy and straight forward* method?

Let's index the prisoners by the order in which they enter the room. Let prisoner 1 be the communicator that signals the hat color of all subsequent prisoners as they enter. We can signal the hat color by degree of turning. Let a 90-degree clockwise turn be Yellow, 180-degree turn be Green, 270-degree clockwise (or 90-degree counterclockwise) turn be Red, and 360-degree turn (or not turning at all) be Blue.

Let's index the invisible rows by numbers -4 to 4. Prisoner 1 stands at row 0 and start signalling at the others prisoners as they enter. The subsequent prisoners can take on the corresponding row depending on prisoner 1's color in order that the rows form a (Yellow,Green,Red,Blue) sequence.

The great thing about the above is that prisoners 1 and 2 *do not* have to change chosen positions at all. This assumes that prisoner 1 can rotate *within sight* of entering prisoners (don't see anything prohibiting this in the OP).

[CaptainEd]

Good point, Bushindo! Prisoner 2 is the first one who knows both colors, so he adopts a position that makes Prisoner 1's location correct as well. And, following TSLF's suggestion of NEWS-coding, Prisoner 1 tells all arrivals their own color.

Very pretty!

[bonanova]

I'm taking a slightly stricter view, that NEWS coding is not available.
Wolfgang has allowed that prisoners can either face toward or away from the door.
That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color.

That presents a particular problem for P2 on initial entry, but I think it can be worked around.

[TimeSpaceLightForce]

Plan: Reflection

The 1st ex-convict stands in front of the door, do nothing

just stand there "still" with wide open eyes.

2nd one enters, look into the 1st ones eyes....see the

reflected color on his forehead.. take a row, so on...

[CaptainEd]

I'm taking a slightly stricter view, that NEWS coding is not available.

Wolfgang has allowed that prisoners can either face toward or away from the door.

That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color.

That presents a particular problem for P2 on initial entry, but I think it can be worked around.

So, my solution, spread out over 38, 39 and 41 solves this, right?

* P1 enters

* P2 sees P1's color, communicates it to P1 by standing where P1 should be

* P1 sees P2's color, moves to a place where both P1 and P2 are in the right place

* for all future entries, NorthWestmost two prisoners see entrant's color, face East or West to denote entrant's color, entrant goes there.

[bushindo]

I'm taking a slightly stricter view, that NEWS coding is not available.

Wolfgang has allowed that prisoners can either face toward or away from the door.

That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color.

That presents a particular problem for P2 on initial entry, but I think it can be worked around.

If only a binary signaling scheme is allowed, CaptainEd has shown that only prisoner 1 needs to change his mind. Even in the stricter NS view, there are some latitude to define a quadnary signaling scheme due to an implied necessary condition

the entering prisoners *must* be able to see the signaler rotate before taking their position. Otherwise, the last prisoner has no way to receive information about his hat color, and the puzzle is not solvable. Therefore, rotation *within sight* of entering prisoners is a necessary condition of the puzzle.

Since the entering prisoners are able to see rotation, it is then trivial to combine rotation speed, direction, and amount with NS coding to allow quadnary signals. (Again, I don't see anything in the OP prohibiting this; well, except for the 'no communication of any kind allowed', but we all know how that went). Once quadnary signals are allowed, no one needs to change position.

[bonanova]

I'm taking a slightly stricter view, that NEWS coding is not available.

Wolfgang has allowed that prisoners can either face toward or away from the door.

That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color.

That presents a particular problem for P2 on initial entry, but I think it can be worked around.

If only a binary signaling scheme is allowed, CaptainEd has shown that only prisoner 1 needs to change his mind. Even in the stricter NS view, there are some latitude to define a quadnary signaling scheme due to an implied necessary condition

the entering prisoners *must* be able to see the signaler rotate before taking their position. Otherwise, the last prisoner has no way to receive information about his hat color, and the puzzle is not solvable. Therefore, rotation *within sight* of entering prisoners is a necessary condition of the puzzle.

Since the entering prisoners are able to see rotation, it is then trivial to combine rotation speed, direction, and amount with NS coding to allow quadnary signals. (Again, I don't see anything in the OP prohibiting this; well, except for the 'no communication of any kind allowed', but we all know how that went). Once quadnary signals are allowed, no one needs to change position.

You're right.

NS signaling is enough.

Four easily distinguishable rotations: None, 180^ocw, 180^occw, and 360^occw.

And entering prisoners must be able to see the rotation.

I think we spent most of the time on this puzzle with a poor understanding of the conditions.

But it was fun to finally solve something possible.

It's just that the problem became teasing out the constraints that admitted a solution!

Bertrand was helpful in this case.

Thanks, Wolfgang.

[wolfgang]

I'm taking a slightly stricter view, that NEWS coding is not available.

Wolfgang has allowed that prisoners can either face toward or away from the door.

That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color.

That presents a particular problem for P2 on initial entry, but I think it can be worked around.

If only a binary signaling scheme is allowed, CaptainEd has shown that only prisoner 1 needs to change his mind. Even in the stricter NS view, there are some latitude to define a quadnary signaling scheme due to an implied necessary condition

the entering prisoners *must* be able to see the signaler rotate before taking their position. Otherwise, the last prisoner has no way to receive information about his hat color, and the puzzle is not solvable. Therefore, rotation *within sight* of entering prisoners is a necessary condition of the puzzle.

Since the entering prisoners are able to see rotation, it is then trivial to combine rotation speed, direction, and amount with NS coding to allow quadnary signals. (Again, I don't see anything in the OP prohibiting this; well, except for the 'no communication of any kind allowed', but we all know how that went). Once quadnary signals are allowed, no one needs to change position.

You're right.

NS signaling is enough.

Four easily distinguishable rotations: None, 180^ocw, 180^occw, and 360^occw.

And entering prisoners must be able to see the rotation.

I think we spent most of the time on this puzzle with a poor understanding of the conditions.

But it was fun to finally solve something possible.

It's just that the problem became teasing out the constraints that admitted a solution!

Bertrand was helpful in this case.

Thanks, Wolfgang.

I want to thank you all....I was thinking like this:

After the first three prisoners took their right places,they all should be facing the door, the possible combinations would be:

Yellow,Green,Red,........

Yellow,Green,........,Blue

Yellow,.........,Red, Blue

.........,Green, Red, Blue

so when the 4th one enters, Let him to have X color..the X one standing in the raw will turn his face to the wall,and the new comer will stand in that raw facing the door....and when nobody turns to the wall so he should stand at the empty place facing the door.

when the four raws are made, each new comer will know where he belongs when the man with the same color turns his face toward the wall.

[Prime]

Turning around when new man enters is clearly a communication. And the OP said:

An unknown number of prisoners( more than 14)were told a day before their execution, that they are going to be blind folded, and a colored paper(either.. yellow,green,red or blue) will be glued on forehead of each one of them( no one can see his own paper), then they should inter a big hall one by one to make 4 raws in this order( Yellow,Green,Red,Blue)i.e. a raw of Yallows, a raw of Greens etc.,each one will be unfolded when he inters the hall.No one of them is allowed to arrenge the others,each one should himslf choose where to stay, once he did,he can not change his place( only the first 2 prisoners can change their minds only once).Any kind of comunication between them is not allowed. If any one of them stands in a wrong raw, all of them will die!

All colores were used.

They should make a plane which saves all of them, can you help them?

[bonanova]

I'm taking a slightly stricter view, that NEWS coding is not available.

Wolfgang has allowed that prisoners can either face toward or away from the door.

That is, NS (or EW, as I drew it) coding is available, and that requires two prisoners to signal a color.

That presents a particular problem for P2 on initial entry, but I think it can be worked around.

If only a binary signaling scheme is allowed, CaptainEd has shown that only prisoner 1 needs to change his mind. Even in the stricter NS view, there are some latitude to define a quadnary signaling scheme due to an implied necessary condition

the entering prisoners *must* be able to see the signaler rotate before taking their position. Otherwise, the last prisoner has no way to receive information about his hat color, and the puzzle is not solvable. Therefore, rotation *within sight* of entering prisoners is a necessary condition of the puzzle.

Since the entering prisoners are able to see rotation, it is then trivial to combine rotation speed, direction, and amount with NS coding to allow quadnary signals. (Again, I don't see anything in the OP prohibiting this; well, except for the 'no communication of any kind allowed', but we all know how that went). Once quadnary signals are allowed, no one needs to change position.

You're right.NS signaling is enough.Four easily distinguishable rotations: None, 180^ocw, 180^occw, and 360^occw.And entering prisoners must be able to see the rotation. I think we spent most of the time on this puzzle with a poor understanding of the conditions.But it was fun to finally solve something possible.It's just that the problem became teasing out the constraints that admitted a solution!Bertrand was helpful in this case. Thanks, Wolfgang.

I want to thank you all....I was thinking like this:After the first three prisoners took their right places,they all should be facing the door, the possible combinations would be:Yellow,Green,Red,........Yellow,Green,........,BlueYellow,.........,Red, Blue.........,Green, Red, Blueso when the 4th one enters, Let him to have X color..the X one standing in the raw will turn his face to the wall,and the new comer will stand in that raw facing the door....and when nobody turns to the wall so he should stand at the empty place facing the door.when the four raws are made, each new comer will know where he belongs when the man with the same color turns his face toward the wall.

The first three may not all be different.

What if all the prisoners are say red, except for the last three to make all the colors used?

Edit: ok, that's not a problem I guess, it just delays the row formation.