# Simple Formal Logic Riddle

1. Nov 30, 2005

### Ubern0va

Lets assume that the old saying "nice guys always finish last" is true.

A running race is held around the block with five people competing. Two nice guys are in the race. To finish the race, a runner must cross through a gate fit to hold only one person at a time. Who will finish last? Under what conditions?

OOps forgot to mention, for clarity's sake that all persons competing in the race have the intention of winning.

The first part is pretty obvious if you take everything said above literally, but you must solve the second portion to solve the entire riddle.

I made this up myself, and I'm just learning preliminary formal logic, so forgive me if later it turns out that I phrased something wrong. Though I think everything is ok, I'm really not sure.

Last edited: Nov 30, 2005
2. Nov 30, 2005

### HallsofIvy

Assuming "good guys" finish last

And there are two "good guys", then they both finish last.

As to what that means you will have to decide for yourself. You didn't tell us what "finish last" means here!

3. Nov 30, 2005

### Ubern0va

Yes I should probably define finishing, it's a bit ambiguous at the moment, leading you to the incorrect answer.

Lets pretend that there's a gate fit for one person to pass through at a time. We will call passing through this gate "finishing". Who finishes last?

Not so simple now huh? Also, to clarify, the condition is the answer to the riddle, if you have to leave the condition to be decided by me then you have the wrong answer :P

Last edited: Nov 30, 2005
4. Nov 30, 2005

### AKG

The slowest non-nice guy finishes last, assuming there is a slowest non-nice guy.

5. Nov 30, 2005

### motai

It seems that it depends on how "nice" the two nice guys are. The nicer of the two would be the one who would finish last... or you would end up with a strange "oh no, you go ahead and finish ahead of me" argument between both nice guys.

6. Dec 1, 2005

### Ubern0va

But nice guys always finish last. That is, if there is a thing (in the domain) who is a guy, and who is nice, that guy always finishes last.

7. Dec 1, 2005

### Ubern0va

The statement is not that "the nicest guy always finishes last". The degree of niceness of each guy is not discussed in the saying; this ambiguity is what makes this a riddle. Though it isn't the key to the asnwer, which is purely logical.

That argument can't happen because it is stated in the riddle that it is the intention of each runner to win the race.

8. Dec 1, 2005

### AKG

Your premises are contradictory, thus I can prove that anyone finishes last regardless of the conditions (aside from the ones given in the problem). You state that nice guys always finish last. This means that if x is a nice guy, then x finishes last. There are (at least) two nice guys in the race, so both of them finish last. Therefore, neither one finishes before the other, so they finish at the same time. Thus they both pass through the gate at the same time. But only one person can pass through the gate at a given time, so we've reached a contradiction. So I can prove that anyone finishes last. However, the question itself is unanswerable because there is no situation that could possibly model your situation (because it would have to model a contradiction, i.e. a contradiction would have to be true).

9. Dec 2, 2005

### Ubern0va

Hint: There is a way to bypass the contradiction. You were so close in your explanation, I can't believe you didn't come to the right conclusion. I thought you had it untill I read your post fully. :P

HUGE Hint (via random example): Every person has exactly one father. Bill is the father of Jake, and Bob is the father of Jake. What can be said about bill and bob?

The very fact that this seemingly contradictory riddle has an answer should signal that there is some trickery involved, again, it is a riddle not just a question, think outside the of the box (but still in formal logic terms).

Last edited: Dec 2, 2005
10. Dec 2, 2005

### AKG

Under the condition that 2 = 1, the nice guy finishes last? If this is the answer, you should note tha saying that "two nice guys are in the race" means that there are indeed two (distinct) nice guys in the race, and the above is not a sensible way to say that "there is one nice guy known by two names in the race." Two actually means two, not one with two names.

11. Dec 3, 2005

### croxbearer

since each of the 5 people in the race aim to win the race, and nice guys always finish last, hence, one of the 3 not nice guys will win the race and one of the 2 nice guys will finish last. anyway, its unimportant who will finish last! :yuck:

12. Dec 3, 2005

### Ubern0va

AKG, yes that is more or less the answer. Except it's not that two equals one, that can never be. It's that one guy is the equivalent of the other guy, i.e. they are the same person.

However, your point is dually noted, I may try to change the riddle so that it implies that two guys are in the race instead of just coming right out and saying it. That way we can bypass the fact (which I just realised) that my premises state that 'there is a nice guy in the race, and there is another guy who is nice, and in the race, and is not equal to the first nice guy'. Perhaps I should use something similar to that father example I gave. That way there can be two names for the same person, but I never say that there are two guys (in the domain) such that each of them is in the extension of the predicate 'is nice'.

Thanks for the input AKG. I see you also realized that the answer is right there in the premises for those who didn't get the first part if the premises state that 'nice guys always finish last, and that there can only be one person who finishes last, then the answer can only be that the nice guy finishes last'. It comes directly from the premises.

Last edited: Dec 3, 2005
13. Dec 3, 2005

### AKG

If "they" are the same person, then there are not 2 nice guys in the race, but you said there were. Again, "two" means "two", not "one with two names."
For your interest, you mean "duly", not "dually."
Yeah, that would work better.

14. Dec 4, 2005

### Ubern0va

Yes, thank you for that reiteration. :P

Again, thank you for correcting me (you take joy in this I imagine), I was not complete until this very moment. Give me a break, its one of those words I've not used twice in my entire life.

15. Dec 4, 2005

### AKG

No, I just figured you'd want to know.

16. Dec 4, 2005

### Cybersteve

Why can't one nice guy put the other on his shoulders?:rofl:

17. Dec 5, 2005

### Ubern0va

Lol, I really need to refine this thing. I realized that one or both of the nice guys can die before the end of the race.

18. Dec 12, 2005

### RandallB

The real problem in this is the definition of "last" as compared to "nice".
Since “nice” can be understood as a group of more then one so must “last” be defined as “among the last”; otherwise the word “nicest” must be used to avoid an illogical set up.

This actually comes up as a real life logic problem that has been solved.
Take your example as a dirt car race with cash payoffs for the top three finishers of a multiple lap race.
Thus when the two nice guys wreck in the first lap, no record is kept of position in the lap they both finish last with zero laps complete and no prize money. ( If too many wreck, say more than half, it’s a ‘no-race’)
BUT, where payoffs are different all the way down to last place that it makes a differance, the race must decide who is the “nicest”, and keep track of the order of finish for each lap completed (position by passing someone in a lap doesn’t count until you complete the lap) in case of wrecks in a lap. Thus crossing a common start/finish line one at a time for each lap is also required. And yes this is accomplished in real life at the call of a single judge with whatever aid’s (photo-finish, etc) available.

Many races are so important that even the start itself must be done one at a time. That way if two nice guys fail to even start, the nicest is scheduled to start behind the other & rewarded with last place money.

Logic really is being used in real life,
- although there are still arguments and that single judge does need some protection.

19. Dec 12, 2005

### AKG

If a nice guy dies, he doesn't finish, and thus doesn't finish last, but say that nice guys always finish last, so he will never die before the end of the race.

20. Dec 12, 2005

### TheStatutoryApe

Considering that we have no distinction being made between the two nice guys we can not say which one comes in last since there is no label given to either.

The only logical answer it seems to me is to say that the fifth person to pass through the gate is the one that finishes last. The problem comes in with the stipulation that nice guys always finish last is literally true. Otherwise my answer would make perfect sense I think.

21. Dec 12, 2005

### Ubern0va

I know, I was just teasing you :)

I should remove the ambiguous version so that people don't keep trying to solve it.

TheStatutoryApe – That’s a nice answer! I will somehow have to avoid it in restructuring this riddle. However, the new riddle will give labels to the nice guys (Bob and Jake). That will make the illusion of two separate guys who are nice but without making each guy distinct from the other.

AKG - Actually, that would only be true had I stated that all participants finished the race (i.e. someone who finishes last must come from the domain of those who have finished, so if someone hadn't finished, they are no longer in that domain, and finishing doesn't apply to them at all), but I didn't, so any participant can withdraw from the race (possibly by dying) and then he will neither finish first nor last, leaving the other nice guy to finish last. It's just another catch that I overlooked. To avoid the withdrawal issue, I will add that, which you assumed was already there (the fact that all runners finish the race). After that is added, it will indeed be true that no runner can die while racing.

RandallB - Sorry, I don't think I followed your proposal 100%. From what I could gather, you state that two or more people can finish last. This goes against what is stated in the riddle. However it is true that the ambiguity you refer to exists in the statement "nice guys always finish last", but it makes no difference since only one person can finish at a time. In fact, this ambiguity is what spawns the apparent contradiction.

In the second paragraph, you have a reward system to determine the "nth place winner" with n > 1. My system of 'finishing' is entirely void of rewards. In fact, nowhere do I mention that anyone ever, technically, 'won'. The riddle doesn't care about who ‘wins’ and who 'doesn't win', but about he who 'finishes last'. Apart from that, you seem to just be reiterating my last post, wherein I suggest that one or both 'nice guys' can withdraw from the race in some unforeseen way.

Anyways, no one should be trying to solve the original riddle anymore; it clearly has its problems. Keep the input coming though, so I can fix it.

Thanks for your involvement everyone! :)

Last edited: Dec 12, 2005
22. Dec 13, 2005

### AKG

The statement "nice guys always finish last" implies that "nice guys always finish," so no, I didn't assume that they finished last. You need not say that all participants finish the race. The question, "which finished last" is identical to the question, "of all those who finished, which finished last?" And since nice guys finish last, they necessarily finish, and thus they are members of the set of those who finished.

The first answer I gave to this problem was actually that the slowest non-nice guy would finish last. This is because your problem, as stated, implied that there were (at least) two nice guys in the race, and so I was thinking that since they can't both finish last, but at the same time, they must both finish last, they must never finish, and thus the only people who can finish are the non-nice guys, and so the slowest of them would finish last. But then I realized that "nice guys finish last" implies that "nice guys finish," so that answer had to go.

23. Dec 18, 2005

### Ubern0va

AKG - Ahh, I see, you were refering to a different part of the riddle. The implication is clear to me now. I see my mistake as well; I interpreted the 'nice guy's always finish last' statement with an extra conditional, which, now that I think about it, was pretty dumb of me to do, because the only reason I added the 'always' (on purpose) was to specify the fact that the 'finishing last' property of the 'nice guys' is universal.

S.M.R.T.