28 replies to this topic

[araver]

Spoiler for

Without anyone contacting anyone else, send 4 of them one at a time to the door leading outside. There is a scanner connected to this door that will detect who is infected and who is clean. When a person reaches the scanner, it will set off an alarm if they are infected. Of the 4 people sent to the scanner, 1 or 2 will set off the alarm. Here are the possibilities for the two infected:

p1, p2 (2 doors)
p1, p3 (3 doors)
p1, p4 (4 doors)
p1, p5 (4 doors)
p2, p3 (3 doors)
p2, p4 (4 doors)
p2, p5 (4 doors)
p3, p4 (4 doors)
p3, p5 (4 doors)
p4, p5 (3 doors)

Once you have found out who the two infected individuals are, gas their rooms and send one of the non-infected persons to inject them with the serum from their own room.

Min doors opened: 4
Max doors opened: 6
Avg doors opened: 5.5
Risk of spreading infection: 0.0%

Indeed you found a loophole.

Spoiler for

I forgot about using the outdoor filter alarm

It is not what I intended with the puzzle, but I need to think on an alternate setting that would allow for a full indoor solution.

[bushindo]

More clarification:

Suppose that an infected member is in a room with a non-infected unconscious member, presumably he would need an excuse to touch the unconscious person. What would the infected member say about the result of his scan?

What happens when you connect two conscious crewmember together, one of which is infected, and the other being non-infected?

[araver]

More clarification:

Suppose that an infected member is in a room with a non-infected unconscious member, presumably he would need an excuse to touch the unconscious person. What would the infected member say about the result of his scan?

He would need an excuse. Saying that the scan is positive and the unconscious needs to be cured is one such excuse.

What happens when you connect two conscious crewmember together, one of which is infected, and the other being non-infected?

Odds are the infected one will manage to infect the other without you figuring it out. It's not 100% certain but it's the worst case scenario you can expect (as I imagine that once the second is infected he will not say he was touched by the other).

[Molly Mae]

Spoiler for

It's pretty safe to say that you can't trust two people together. There might be a better way, but I'd
1. Gas everyone except person #1.
2. Have person #1 deliver serum to the central hub.
3. Have person #1 return to his room.
4. Gas everyone except person #2.
5. Have person #2 deliver serum to the central hub.
...
...
...

After the fifth person delivers his serum, have him inject himself with one from every pile.

...
32(?). Have person #5 return to his room.
33. Gas everyone except person #1.
34. Have person #1 enter the central hub and inject himself with with serum from every pile.
35. Have person #1 return to his room.
36. Gas everyone except person #2.
...
...
...

It's a bit lengthy, but no two people will ever come into contact with each other. Further, they'll all have taken legitimate serum.

While this might not be the best way, it at least leads us to a worthwhile conclusion.

[willbrownsf]

Spoiler for What if...

Instead of knocking one crew member unconscious you left them both conscious, had them suit up in space gear, leaving one forearm exposed. You tell member 1 & member 2 to inject each other with the cure while in front of a camera with the hand inside the space suit. If someone tries to use the other hand then you know they are infected and can quarantine both of them for the time being. You then have member 3 & member 4 perform the same procedure, with the same action taken if one tries anything funny. This leaves member 5, who judging from the actions of the other members, you can deduce whether he is infected or not. He would be able to cure both teams if they failed the test and he could be cured by members who passed the test.
If everything goes better than expected it would be possible for only 3 doors to be opened, at most only 4 doors opening would be needed.

Edited by willbrownsf, 20 December 2010 - 07:45 PM.

[bushindo]

He would need an excuse. Saying that the scan is positive and the unconscious needs to be cured is one such excuse.


Odds are the infected one will manage to infect the other without you figuring it out. It's not 100% certain but it's the worst case scenario you can expect (as I imagine that once the second is infected he will not say he was touched by the other).

Well, if that's the case, then here's a strategy,

Spoiler for outline

The key point of this strategy is that we reduce the puzzle into a previously solved problem where we need to determine the position of 2 random answerers from a group of 4 (2 random answerers + 2 truth tellers). It is known that we can narrow the possibilities into 2 configurations. Having a 5th crew member will then allow us to narrow these 2 configuration down into a single possibility.

Some general concepts are as follows. Let the non-infected members be called truth tellers, and let the infected members be called random answerers. Let the crew members be called A, B, C, D, and E. We'll let E be called a place holder. Within the group A, B, C, and D, we can determine the status of any member using another member. For instance, suppose that we want to assess the status of A using B. We then simply gas E and tell A to come and disinfect E if necessary. That essentially copies A's status onto E. We then gas E again and tell B to come, scan E, and tell us the results. If B is a truth teller, he'll tell us the true status of A. If B is random answerer, then his answer could be anything*. Note that we are not changing the status of A, B, C, or D.

The steps goes as follows
1) At the beginning, let A come and scan E first, and report the results. If E is already infected before A comes in, then we will have 3 truth tellers and 1 random answerer among A, B, C, and D. This is a trivial situation, as we know we can readily narrow down the single random answerer with this puzzle. The non-trivial situation is where we have two random answerers among A, B, C, and D.

2) It has been shown that given two random answerers in a group of 4, we can only narrow the possibilities down to 2 possible configurations, which are mirror images of one another. One possibility will state that A is non-infected, and the other combination will state that A is infected. If we encounter this situation, then we'll know that E was non-infected.

3) Simply go back and see what A reported on the first scan on E at the beginning. If A reported that E was non-infected, then A is non-infected (truth teller) as well. If A reported infected (in order to come and infect E), then we know that A is infected. We now know the infect/non-infected status all crew members.

* Note: this solution is non-optimal. This is because random answerers aren't completely random. The key fact here is that when an infected person (random answerer) is asked about a non-infected person (truth teller), he will tell a lie. I only used that fact once in this solution, because the more general framework with the completely random answerers is already solved. Solving for an optimal solution may require something along the following line of this post .

Edited by bushindo, 20 December 2010 - 07:59 PM.

[araver]

Spoiler for

It's pretty safe to say that you can't trust two people together. There might be a better way, but I'd
1. Gas everyone except person #1.
2. Have person #1 deliver serum to the central hub.
3. Have person #1 return to his room.
4. Gas everyone except person #2.
5. Have person #2 deliver serum to the central hub.
...
...
...

After the fifth person delivers his serum, have him inject himself with one from every pile.

...
32(?). Have person #5 return to his room.
33. Gas everyone except person #1.
34. Have person #1 enter the central hub and inject himself with with serum from every pile.
35. Have person #1 return to his room.
36. Gas everyone except person #2.
...
...
...

It's a bit lengthy, but no two people will ever come into contact with each other. Further, they'll all have taken legitimate serum.

While this might not be the best way, it at least leads us to a worthwhile conclusion.

I like the quarantine-type solution yet it has one small flaw: you cannot be sure that an infected person actually injects himself with something. (low quality video and all).

Spoiler for What if...

Instead of knocking one crew member unconscious you left them both conscious, had them suit up in space gear, leaving one forearm exposed. You tell member 1 & member 2 to inject each other with the cure while in front of a camera with the hand inside the space suit. If someone tries to use the other hand then you know they are infected and can quarantine both of them for the time being. You then have member 3 & member 4 perform the same procedure, with the same action taken if one tries anything funny. This leaves member 5, who judging from the actions of the other members, you can deduce whether he is infected or not. He would be able to cure both teams if they failed the test and he could be cured by members who passed the test.
If everything goes better than expected it would be possible for only 3 doors to be opened, at most only 4 doors opening would be needed.

Same flaw as in another strategy above: you cannot trust the serum from a room where an infected person was in as it might have already been tampered with. And think of an extreme case of tampering: it might actually contain blood samples from infected people.
1) Non-infected people will not refuse a serum they know it's been tampered it if you pair-up infected people.
2) If you pair an infected with a non-infected, you either give each other serum from the other's room (in which case you cure the infected and potentially infect the other one due to previous tampering) or give each serum from his own room (therefore not curing the infected at all).
3) Non-paired member can still be infected.

Edited by araver, 20 December 2010 - 08:44 PM.

[araver]

Well, if that's the case, then here's a strategy,

Spoiler for outline

The key point of this strategy is that we reduce the puzzle into a previously solved problem where we need to determine the position of 2 random answerers from a group of 4 (2 random answerers + 2 truth tellers). It is known that we can narrow the possibilities into 2 configurations. Having a 5th crew member will then allow us to narrow these 2 configuration down into a single possibility.

Some general concepts are as follows. Let the non-infected members be called truth tellers, and let the infected members be called random answerers. Let the crew members be called A, B, C, D, and E. We'll let E be called a place holder. Within the group A, B, C, and D, we can determine the status of any member using another member. For instance, suppose that we want to assess the status of A using B. We then simply gas E and tell A to come and disinfect E if necessary. That essentially copies A's status onto E. We then gas E again and tell B to come, scan E, and tell us the results. If B is a truth teller, he'll tell us the true status of A. If B is random answerer, then his answer could be anything*. Note that we are not changing the status of A, B, C, or D.

The steps goes as follows
1) At the beginning, let A come and scan E first, and report the results. If E is already infected before A comes in, then we will have 3 truth tellers and 1 random answerer among A, B, C, and D. This is a trivial situation, as we know we can readily narrow down the single random answerer with this puzzle. The non-trivial situation is where we have two random answerers among A, B, C, and D.

2) It has been shown that given two random answerers in a group of 4, we can only narrow the possibilities down to 2 possible configurations, which are mirror images of one another. One possibility will state that A is non-infected, and the other combination will state that A is infected. If we encounter this situation, then we'll know that E was non-infected.

3) Simply go back and see what A reported on the first scan on E at the beginning. If A reported that E was non-infected, then A is non-infected (truth teller) as well. If A reported infected (in order to come and infect E), then we know that A is infected. We now know the infect/non-infected status all crew members.

* Note: this solution is non-optimal. This is because random answerers aren't completely random. The key fact here is that when an infected person (random answerer) is asked about a non-infected person (truth teller), he will tell a lie. I only used that fact once in this solution, because the more general framework with the completely random answerers is already solved. Solving for an optimal solution may require something along the following line of this post .

I admit, while creating the problem I was thinking of building on the same previous problems. Therefore your solution is very similar to what I had in mind.
I do not know if a different type of strategy exists (different i.e. exploits a different aspect of the problem).

As I got carried away with writing the setting for the problem as a contamination scenario and forgot to check for loopholes, I also have to credit Jobe17 for finding an alternative solution.

[bushindo]

I admit, while creating the problem I was thinking of building on the same previous problems. Therefore your solution is very similar to what I had in mind.
I do not know if a different type of strategy exists (different i.e. exploits a different aspect of the problem).

As I got carried away with writing the setting for the problem as a contamination scenario and forgot to check for loopholes, I also have to credit Jobe17 for finding an alternative solution.

Just read Jobe17's solution. Wish I thought of that. Thanks for the superbly constructed puzzle. I really enjoy it. It was devilishly challenging/fun to work through.