# Simple riddle and logic

1. Aug 4, 2008

### 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.

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?

2. Aug 4, 2008

### mathman

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

3. Aug 4, 2008

### DaveC426913

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

4. Aug 4, 2008

### CRGreathouse

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.

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).

5. Aug 4, 2008

### 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

6. Aug 4, 2008

### DaveC426913

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?

7. Aug 4, 2008

### CRGreathouse

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.

8. Aug 5, 2008

### peos69

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

9. Aug 5, 2008

### peos69

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

10. Aug 5, 2008

### peos69

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

11. Aug 5, 2008

### DaveC426913

"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.

12. Aug 5, 2008

### Focus

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

13. Aug 5, 2008

### CRGreathouse

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.

14. Aug 5, 2008

### DaveC426913

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.

15. Aug 5, 2008

### DaveC426913

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

See?

16. Aug 5, 2008

### CRGreathouse

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?

17. Aug 5, 2008

### DaveC426913

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:

18. Aug 5, 2008

### LukeD

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.

Last edited: Aug 5, 2008
19. Aug 5, 2008

### D H

Staff Emeritus
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.

20. Aug 5, 2008

### CRGreathouse

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).