logical deduction - Art Accident at Airport!

There was an accident at the airport! A hanging art installation fell and landed on some passengers. In addition to tearing up all their tickets and mixing up their luggage, six people sustained head injuries, causing them to have amnesia. They each remember conflicting bits and pieces of where they were from and where they were going. As the airport's logician, you have gathered reports from each of them, and determined that each person is misremembering something.

Exactly one clue from each person's list is false.

Names: Mary; Jeff; Allan; Mike; Rebecca; Katie

Origins: Orlando, FL; Chicago, IL; Los Angeles, CA; Dallas, TX; Denver, CO; New York, NY

Destinations: San Francisco, CA; Charlotte, NC; Las Vegas, NV; Phoenix, AZ; Houston, TX; Miami, FL

Luggage Colors: Black; Brown; Red; Green; Blue; White

Airlines: Delta; United; American; Southwest; Frontier; JetBlue

Concourses: A; B; C; D; E; F

---

1. I was just meeting my college friend Katie before heading to my gate in concourse F.


2. Either Mike or the person flying out of concourse F flew United. Mike is not flying out of concourse F.


3. The black bag, which wasn't mine, was going to Houston.


4. Jeff was headed to concourse A or B.


5. I came from Dallas.

1. The person from Denver does not fly out of concourse F.


2. Either Allan or the person from Chicago is flying out of concourse B. Allan is not from Chicago.


3. I am flying from Orlando to Miami.


4. The person flying to Phoenix did not fly frontier.


5. I have a green bag.

1. The person flying from concourse F has either a black or white bag.



2. The person with the brown suitcase is not flying out of concourse B.


3. The person from Los Angeles has either a white or blue bag.


4. American Airlines doesn't fly out of concourse D.


5. I always fly with JetBlue.

1. I had a blue bag, and I was flying United to Houston.


2. The person heading to concourse C has a green bag.


3. Only one passenger's origin and destination are in the same state, and that person flies Delta.



4. Allan is not flying Southwest.


5. Rebecca is flying out of Gate E27.

1. Whoever has the green bag is flying to Vegas on either Delta or JetBlue.


2. I am flying to Charlotte from Denver.


3. The black bag came from neither Orlando nor LA.


4. Whoever is flying from New York to Phoenix needs to get to gate D14 soon before Southwest closes the door!


5. Mary is going to Vegas.

1. Whoever is flying Delta does not have a white bag.


2. Nobody with a black bag is flying American Airlines.


3. I never went to college with Mary and my luggage is brown.


4. Whoever had a blue bag is flying Southwest.


5. My origin and destination are both in the same state.

---

Here it is in Excel, with conditional formatting set up to change cell color if you input X and O.

Per Marius' suggestion, here is the solving grid as a Google Doc.

Consider the following:

1. 
   

   Third person references to an unnamed person could also refer to the person making the recollection (for example, Mary saying "The person with a green suitcase" doesn't preclude Mary from having a green suitcase).

   
   


2. 
   
   

   AND and OR between different statements of a clue work as boolean operators when evaluating a clue as true or false. For example: If a clue "Mary is from Dallas and Mike flew United" turns out to be false, then either Mary is not from Dallas or Mike is not flying United, or both.

   
   


3. 
   

   Implied statements, such as "Either Jeff or the person from Chicago..." (which implies that Jeff is not from Chicago), are also part of a clue's true or false value. If that clue is false, Jeff may or may not be from Chicago, as long as some part of that clue is false.

   
   


4. 
   

   This is a traditional logic grid, meaning no two people come from the same origin, are flying to the same destination, have the same luggage color, fly the same airline, or fly out of the same concourse.

   
   

Answer

Here is the solution progressed further thanks to the solver referenced in another answer.

First, notice that Mary 3 and Mike 1 cannot both be true: they disagree on the bag that went to Houston.

Lets assume that Mike 1 is true. Thus, Mary 3 is false, and Mary 1,2,4,5 are all true. Notice that Mary 1 being true means Katie 3 is false since they disagree about going to college. This makes Katie 1,2,4,5 all true. But we have a contradiction because Katie 4 and Mike 1 disagree on the airline blue bag flew with.

Thus, we know that Mike 1 is false and Mike 2,3,4,5 are all true.

Now notice that both Katie 5 and Jeff 3 cannot both be true: one of them must be false because of Mike 3 being true.

Lets assume that Katie 5 is true, so Jeff 3 is false. Thus, Jeff 1,2,4,5 are true. Since Jeff 5 is true, and Mike 2 is true, we know Jeff flies out of concourse C. This makes Mary 4 false, so Mary 1,2,3,5 are true. Since Mary 1 is true, again, we know Katie 3 is false, which makes Katie 1,2,4,5 true.

Alternatively, lets assume Jeff 3 is true, so Katie 5 is false. Thus, Katie 1,2,3,4 are true. But Katie 3 contradicts Mary 1, so Mary 1 is false and Mary 2,3,4,5 are true. Since Mary 4 is true, and Mike 2 is true, we know Jeff 5 cannot be true. Thus, Jeff 1,2,3,4 are true.

In both cases, Mary 2,3,5 are true, Jeff 1,2,4 are true, and Katie 1,2,4 are true.

So notice here that the set Mary 1, Katie 5, and Jeff 5 is true or the set Mary 4, Katie 3, and Jeff 3 is true, but not both.

Since Mike 3 is true, we know that either Katie or Jeff fly Delta - no one else can.

So lets plug the following into the solver:

• Mary 3, 5


• Jeff 1, 4


• Mike 2, 4, 5


• Katie 1, 2, 4


• The inference from Mike 3 that only Katie and Jeff can fly Delta

And you get this.

Lets assume Jeff 5 is true. If Rebecca 1 and Rebecca 5 are also true, then both Mary and Jeff would be flying out of Vegas, so one of them must be false.

Either way, Rebecca 2, 3, 4 are true.

So, plug the following into the solver:

• Jeff 5, Katie 5 (eliminate invalid origin/destination), Mary 1


• Rebecca 2, 3, 4


• Katie files delta due to Mike 3

And you will have this. Notice that Jeff and Katie have limited options. In fact, if Jeff doesn't fly to Vegas, then both he and her end up with trips within the same state; a violation of Mike 3. Thus we must add the fact that Jeff flies to Vegas.

Also, from Mary 2, we must infer that since Mike flies Southwest, the person out of concourse F must fly United.

Since Alan is now flying out of Chicago, Jeff 2 implies this is concourse B.

After plugging all this in, we see Since Rebecca 5 is false, Rebecca 1 must be true. But neither Delta nor JetBlue are options for the green bag as can be seen in the solver.

Therefore, we must conclude our original assumption of Jeff 5 being true is false. Therefore Jeff 3, Katie 3, and Mary 4 are true, and Jeff 5, Katie 5, and Mary 1 are false.

Lets revert back to our previous solver and add these clues:

• Jeff 3, Katie 3, Mary 4


• Since Jeff is the one who is taking a trip within a state, he flies with Delta due to Mike 3


• Since Jeff is not not Allan, nor flying out of Chicago, for Jeff 2 to be true, Jeff cannot be on concourse B. This places him on concourse A.

You end up with this.

Looking at Rebecca 4, it is quite the load of info. If we assume it is true, then this ends up being Mike. Since he is not on concourse F, we know the United flight must be based on Mary 2. This gets us here.

Now lets look at Allan's clues. If Allan 2 is true, then it puts Katie on concourse F, making Allan 1 is false because she has a brown bag. Thus the rest of Allan's statements must be true. Allan 3 puts the white bag out of LA, and we get this.

Looking at Rebecca's clues again, if 2 is true, then by putting her out of Denver, we would have her going to Houston since they are both tied to a black suit case. This means Rebecca 2 is false and 1,3, and 5 are true. Rebecca 1 puts Mary on JetBlue going to Vegas with the green luggage. But then a second statement of Allans (Allan 5) is false.

Therefore, the original assumption of Rebecca 4 being true was false. This makes Rebecca 1, 2, 3, and 5 true.

So add Rebecca 2, 3, and 5 to the solver. Because Mary is now out of Vegas, Rebecca 1 means she has the green bag and flies JetBlue.

This makes Allan 5 false, so Allan 1,2,3,4 are all true and we get here.

We now know all the false statements:

• Mary 1


• Jeff 5


• Allan 5


• Mike 1


• Rebecca 4


• Katie 5

The rest are true. Of these, there are a few that we haven't been able to fully realize in the truth table. There are no obvious things that can be done in the truth table. These are:

• Mary 2


• Jeff 2,


• Allan 1, 3


• Mike 3, !Mike 1

We have to make sure truth (or falsehood) of these statements is maintained.

Lets try Allan on concourse B. This puts Mike on F, which makes him fly United from Mary 2.

Now it we try Mike coming out of Chicago, then the puzzle is solved(1)! Turns out if we try Mike out of New York, solved(2)!

So now lets try Allan on F instead. From Jeff 2, we know that Chicago must be out of concourse B. Since this puts F on America, Mike must fly United from Mary 2. This yields another solution(3)

So, there are 3 solutions to this puzzle based on these clues.

1:

Mary    Dallas      Las Vegas       Green   JetBlue     C
Jeff    Orlando     Miami           Red     Delta       A
Allan   Los Angeles Phoenix         White   American    B
Mike    Chicago     Houston         Black   United      F
Rebecca Denver      Charlotte       Blue    Southwest   E
Katie   New York    San Francisco   Brown   Frontier    D

2:

Mary    Dallas      Las Vegas       Green   JetBlue     C
Jeff    Orlando     Miami           Red     Delta       A
Allan   Los Angeles Phoenix         White   American    B
Mike    New York    Houston         Black   United      F
Rebecca Denver      Charlotte       Blue    Southwest   E
Katie   Chicago     San Francisco   Brown   Frontier    D

3:

Mary    Dallas      Las Vegas       Green   JetBlue     C
Jeff    Orlando     Miami           Red     Delta       A

Allan   Los Angeles Phoenix         White   American    F
Mike    Chicago     Houston         Black   United      B
Rebecca Denver      Charlotte       Blue    Southwest   E
Katie   New York    San Francisco   Brown   Frontier    D