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I did a slightly different method from everyone else, finding the order of colors first and the color of the door to freedom last, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue would not be able to be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.

I did a slightly different method from everyone else, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue would not be able to be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.

I did a slightly different method from everyone else, finding the order of colors first and the color of the door to freedom last, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue would not be able to be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.

2 added 14 characters in body
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I did a slightly different method from everyone else, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue cannotwould not be able to be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.

I did a slightly different method from everyone else, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue cannot be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.

I did a slightly different method from everyone else, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue would not be able to be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.

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I did a slightly different method from everyone else, so I guess I'll post it. I started out with the third requirement: G is 3 away from F (the "Freedom door").

... G _ _ F ... (one side or the other is filled in, not both)

White clearly must go between them. If white goes in the left slot...

... G W _ F ...

Then since white is in the middle, the remaining door would have to be to the left of G.

_ G W _ F

Then since yellow is next to F, we would get

_ G W Y F

And every door except the two on the end is assigned, so blue cannot be placed anywhere.

So, by elimination, white must go in the right slot, and since it is in the middle, the extra door must be to the right of the Freedom door.

G _ W F _

Now there is only one place for the yellow door to go...

G _ W F Y

The blue door is allowed to go in either remaining slot. The only remaining requirement is that R is to the left of F. If the Freedom door is R, then this requirement clearly cannot be met. So the R has to go in the slot that isn't F, and B has to be the Freedom Door.

G R W BF Y

Luckily, when I worked this out on paper I put the doors in the right order, but if I hadn't I would have reflected them horizontally at this stage to make R be on the left rather than on the right.