Logical Riddle: How Many Cars Do I Have?

[roxberry]

All but two of my cars are Fords, all but two of my cars are Toyotas and all but two of my cars are Hondas. How many cars do I have.

The obvious answer is three.

I'm arguing with some folks that are claiming two is an acceptable answer and they are using the rationale that it is fine to have zero Fords according to Boolean logic.

I don't have a problem with statements like "There are zero Fords in my garage", or anything I read up on logic since this argument, but once you claim "all but" two of your cars are Fords, your total amount of cars must be greater than two.

Folks are making statements like this:

"I have a technical background and have taken multiple courses on Boolean logic, so I have no problem with the statement "All of my cars are Fords except two," when the speaker has two cars that are not Fords. It is absolutely a true statement logically. Potentially misleading, but true."

Am I right or wrong?

 

[mathman]

To be precise you need to have two cars, and they both can be Kias.

 

[DaveC426913]

Since they're provided their rationale/assumptions in their answer, I don't see how you can categorically disqualify their answer.

 

[CRGreathouse]

All but two of my cars are Fords, all but two of my cars are Toyotas and all but two of my cars are Hondas. How many cars do I have.

T=f+t+h+o

T = f + 2
T = t + 2
T = h + 2

t + h + o = 2
f + h + o = 2
f + t + o = 2

t + h = f + h = f + t = 2 - o
t = f = h = 1 - o/2

So the correct real answers are 1 - x/2 each of Toyotas, Fords, and Hondas, and x other cars, for a total of 3 - x/2 cars. The correct nonnegative integer answers are 2 or 3 cars.

The obvious answer is three.

When I was asked that question (actually with pets instead of cars) in grade school, I thought the answer was 2. The answer 3 didn't occur to me (and I didn't look further, as I solved the question).

 

[peos69]

This is not a logic but a mathematical h.school question.
Assume the total No of your cars to be x.
Since all your cars except two are TOYOTAS we have x-2= toyotas
for the same reason.........x-2=fords
for the same reason.........x-2=hondas
HENCE x-2+x-2+x-2=t+f+h=total No of cars=x
3x-6=x=====>2x=6===>x=3

 

[DaveC426913]

This is not a logic but a mathematical h.school question.
Assume the total No of your cars to be x.
Since all your cars except two are TOYOTAS we have x-2= toyotas
for the same reason.........x-2=fords
for the same reason.........x-2=hondas
HENCE x-2+x-2+x-2=t+f+h=total No of cars=x
3x-6=x=====>2x=6===>x=3

Your logic is flawed. The "hence" line makes the assumption that these values are mutually exclusive.

By the same logic, your answer to this puzzle would be 4:

Two fathers and two sons go fishing. What is the least number of people in the boat?

 

[CRGreathouse]

Your logic is flawed. The "hence" line makes the assumption that these values are mutually exclusive.

I made that assumption, too; I don't think it's unwarrented. I do think that pesos' assumption that the values are exhaustive is unjustified, however.

 

[peos69]

By the same logic, your answer to this puzzle would be 4:

DO IT i am very curious to see how.
Phrases like "by the same logic" imply that we are using many logics here

 

[peos69]

:

Two fathers and two sons go fishing. What is the least number of people in the boat?

I DO NOT know you tell me it depends how you count

 

[peos69]

Your logic is flawed.

Tell me specifically which laws of logic i violated,otherwise you just making an impression for
those that they do not know about logic

 

[DaveC426913]

Tell me specifically which laws of logic i violated

"Laws" of logic?

I simply said your logic is flawed.

You are adding your values: x-2 + x-2 + x-2. This act of addition assumes that the values are mutually exclusive. This is your "law of logic" that you have made up, though you have not stated it explicitly.

By the same logic: 2 fathers + 2 sons = 4 people in the boat. This is not true. There are only three people in the boat. Again, the act of addition assumed the two adders (2 + 2) are mutually exclusive. They're not.


How many shirts do I have that go with black? 2
How many shirts do I have that go with white? 2

How many shirts do I have? 2+2=4?
No, I have 3. One black, one white, and one with black and white stripes.

etc.

 

[Focus]

By the same logic: 2 fathers + 2 sons = 4 people in the boat. This is not true. There are only three people in the boat. Again, the act of addition assumed the two adders (2 + 2) are mutually exclusive. They're not.

There is gramps, father and his son? Wow nice riddle. I'll be asking people this for a while

 

[CRGreathouse]

Tell me specifically which laws of logic i violated

You and I both 'misused' the inclusion-exclusion principle. As I wrote above, I think this is justified for this problem -- I don't think that any cars are both Fords and Toyotas. But truly, the total is

T(f) + T(h) + T(t) + T(o) - T(fh) - T(ft) - T(fo) - T(ht) - T(ho) - T(to) + T(fht) + T(fho) + T(fto) + T(hto) - T(fhto)

and we simply assumed that T(fh) = T(ft) = T(fo) = T(ht) = T(ho) = T(to) = 0.

 

[DaveC426913]

I don't think that any cars are both Fords and Toyotas.

That's not the problem. The problem is that the groups (eg. the group that is "all except 2 are Toyotas") can refer to the same vehicle multiple times. You'd be adding a single vehicle multiple times.

 

[DaveC426913]

I DO NOT know you tell me it depends how you count

Three. (A boy, a middle-aged man and an elderly man.) 2 fathers, 2 sons.

See?

 

[CRGreathouse]

That's not the problem. The problem is that the groups (eg. the group that is "all except 2 are Toyotas") can refer to the same vehicle multiple times. You'd be adding a single vehicle multiple times.

I don't know at all what you're talking about. To me, this statement means:

t + h + f + o = t + 2

where t is the number of Toyotas, h is the number of Hondas, f the number of Fords, and o is the number of non-Toyota, non-Honda, non-Ford cars. Does this means something else to you?

 

[DaveC426913]

I don't know at all what you're talking about. To me, this statement means:

t + h + f + o = t + 2

where t is the number of Toyotas, h is the number of Hondas, f the number of Fords, and o is the number of non-Toyota, non-Honda, non-Ford cars. Does this means something else to you?

I propose for the sake of argument that we have two cars; they are Kias.

There is a group called "all cars that are not Toyotas". It has 2 members.
There is a group called "all cars that are not Fords". It has 2 members.
Do you add these two groups? No. You'd be adding the same cars more than once, arriving at the wrong number.

This is the mistake peos69 has made where he says:

HENCE x-2+x-2+x-2 = ...

 

[LukeD]

All but two of my cars are Fords, all but two of my cars are Toyotas and all but two of my cars are Hondas. How many cars do I have.

The obvious answer is three.

I'm arguing with some folks that are claiming two is an acceptable answer and they are using the rationale that it is fine to have zero Fords according to Boolean logic.

I don't have a problem with statements like "There are zero Fords in my garage", or anything I read up on logic since this argument, but once you claim "all but" two of your cars are Fords, your total amount of cars must be greater than two.

Folks are making statements like this:

"I have a technical background and have taken multiple courses on Boolean logic, so I have no problem with the statement "All of my cars are Fords except two," when the speaker has two cars that are not Fords. It is absolutely a true statement logically. Potentially misleading, but true."

Am I right or wrong?

With the way you stated the question, I would say that 2 cars is a valid answer.

It's similar to the statement "All of my children are on the moon". Since I don't have any children, they're all on the moon.
Or another one: "All horses except for the ones that aren't blue are actually unicorns". There aren't any blue horses (to my knowledge), so it's certainly true that every horse that's blue is a unicorn.
In mathematics, it is generally taken that whenever you say "all members of group X have property P", it is automatically true if X doesn't have any members.

The fact that this is valid is actually very useful in proving things.
If you show that everything everything in a group of objects (we'll call them S) except for some of them (we'll call them T) satisfy a certain property, and then you show that nothing can satisfy that property, then you've shown that the only things in your original group of objects are those that you said didn't satisfy that property (i.e., that S = T)

Of course, you're free to interpret the question however you want. Most mathematicians though would interpret it as allowing for either 2 or 3 as an answer, and this was probably the intent of the person who posed the question. However, there'd be nothing wrong with you posing the question and saying "oh, by the way, 'all but' means that there is at least one that satisfies the requirement". It's just that if you intend the question like that, you should clarify what you mean by "all but" because otherwise many people will think that it's perfectly acceptable for "all but" to be 0.

 

[D H]

Your logic is flawed. The "hence" line makes the assumption that these values are mutually exclusive.

CR did not make this assumption; he implicitly used the fact that Hondas are both non-Fords and non-Toyotas. He did make an assumption, however: He assumed that the only kinds of cars that can possibly be in the garage are Hondas, Fords, and Toyotas. The existence of Edsels, Kias, Chevys, etc. in other people's garages makes that assumption rather invalid.

 

[CRGreathouse]

I propose for the sake of argument that we have two cars; they are Kias.

There is a group called "all cars that are not Toyotas". It has 2 members.
There is a group called "all cars that are not Fords". It has 2 members.
Do you add these two groups? No. You'd be adding the same cars more than once, arriving at the wrong number.

This is the mistake peos69 has made where he says:

But peos didn't add (all cars that are not Toyotas) and (all cars that are not Fords); peos added (all cars, less all cars that are not Toyotas) and (all cars, less all cars that are not Fords).

 

[CRGreathouse]

CR did not make this assumption; he implicitly used the fact that Hondas are both non-Fords and non-Toyotas. He did make an assumption, however: He assumed that the only kinds of cars that can possibly be in the garage are Hondas, Fords, and Toyotas. The existence of Edsels, Kias, Chevys, etc. in other people's garages makes that assumption rather invalid.

I certainly did not assume that there are only Hondas, Fords, and Toyotas! I explicitly included a term "o" for all other types of car.

Are you confusing me with peos? (Perish the thought!)

 

[D H]

I certainly did not assume that there are only Hondas, Fords, and Toyotas! I explicitly included a term "o" for all other types of car.

My bad! That is what peos did, not you. You explicitly arrived at solutions of two or three cars in the garage.

Are you confusing me with peos? (Perish the thought!)

The thought never cross my mind, but apparently it did cross my fingers.

 

[CRGreathouse]

My bad! That is what peos did, not you. You explicitly arrived at solutions of two or three cars in the garage.


The thought never cross my mind, but apparently it did cross my fingers.

No problem, I was just confused until I realized that you might not be talking about me. I'll admit to assuming mutual exclusivity, though; a part-Toyota, part-Ford might cause problems for my analysis. (Though hopefully that would count under my real solution!)

 

[peos69]

Three. (A boy, a middle-aged man and an elderly man.) 2 fathers, 2 sons.

See?

adding two Nos in;
1) group theory can be any No
2) Riddle theory you scratch your head
3) in prop ability theory depends on the relation...e.t.c e.t.
4)But in our problem addition is within the natural Nos system and if
anybody can make the No of cars other than 3 let him/her do it here
and now

 

[peos69]

The thought never cross my mind, but apparently it did cross my fingers.

mine something else

 

[peos69]

It seems to me you guys you are highly knowledgeable in logic, boolean algebras,,godel stuff and so on .
yet i happened to notice that there is another post where the author asks for the proof or the miss proof that the empty set has a supremum.
so turn your knowledge there and solve the problem otherwise your talk it is all up in the air and when i say proof i mean proof and not just intuitive shots

 

[peos69]

I propose for the sake of argument that we have two cars; they are Kias.

There is a group called "all cars that are not Toyotas". It has 2 members.
There is a group called "all cars that are not Fords". It has 2 members.
Do you add these two groups? No. You'd be adding the same cars more than once, arriving at the wrong number.

This is the mistake peos69 has made where he says:

a cat in the corner therefor it rains

 

[LukeD]

It seems to me you guys you are highly knowledgeable in logic, boolean algebras,,godel stuff and so on .
yet i happened to notice that there is another post where the author asks for the proof or the miss proof that the empty set has a supremum.
so turn your knowledge there and solve the problem otherwise your talk it is all up in the air and when i say proof i mean proof and not just intuitive shots

done

Do you have other "challenges" that you want to pose to the math community to make sure that they're not just talk?

poes, please try to not put multiple posts in a row. Try to decide everything that you want to say before you write a post so that you don't write 5 in a row. If you think of something that you want to add a minute or two later, use the edit button. The chances are, whomever your posts are intended for will not have read the post in the 2 minutes between your posting and editing, so they probably will not miss anything.

 

[montoyas7940]

Three. (A boy, a middle-aged man and an elderly man.) 2 fathers, 2 sons.

See?

Could it not also be two people? Each a father and a son to people not in the boat. Although I think the assumption is that the boats occupants are related in that way.

 

[DaveC426913]

Could it not also be two people? Each a father and a son to people not in the boat. Although I think the assumption is that the boats occupants are related in that way.

You are correct. For the sake of brevity, I was sloppy in writing out the riddle. The full riddle is more explicit about the people in the boat being related to each other.