## Money and fruit

A forum for good logic/math puzzles.

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### Money and fruit

Here's two puzzles, in case one has been posted before (sorry if it has!)

Puzzle 1 - There's a gentleman on your right who makes you the following offer - you shall make any statement. If it's true, he will give you \$10. If it's false, he will give you some amount of money OTHER than \$10 (perhaps more, perhaps less, it's his decision). There's another gentleman to your left who has offered that you make a statement, and whether it is true or false, he will give you \$15. You may only accept one of these offers. Now these are prim and proper gentlemen, so if you should be so rude as to make a sentence that cannot be true or false, you will get nothing.

a) Which offer should you take?
b) Why is the answer to part a) that you should take the right gentleman's offer?

Puzzle 2 - All of the following sentences are either true or false. (And use the material "if", if you know about that sort of thing)

If this sentence is true or this sentence is false, then this sentence is true and this sentence is false.

If either this sentence is true, the previous sentence true, or the next sentence is false, then the previous sentence is true.

If I absolutely love apples, then I absolutely love bannanas.

How do I feel about apples and bannanas?
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

4=5
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### Re: Money and fruit

Spoiler:
you do not absalutly love them
and I would take the first gentleman's offer and lie, because there is an infinite amount of mony abouve ten dollars

Token
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### Re: Money and fruit

Question 1:
Spoiler:
I'd take the first offer, and say "This statement is true."

Question 2:
Spoiler:
For the first statement, we have a trivially true antecedent and a trivially false consequent. Therefore it is false.

Since the first statement is false, the second statement is necessarily false. This must mean that the antecedent is true. Since we know both the first and second statements are false, the third statement must also be false.

Therefore for the third statement, we have a true antecedent and a false consequent. So you absolutely love apples, but you do not absolutely love bananas.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

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### Re: Money and fruit

4=5: For the first question,
Spoiler:
You're assuming the amount of money will be chosen in some uniform random way, of which you have no guarantee. Perhaps I could phrase the question another way: a) Which of the two offer-giving gentlemen would you rather BE, and b) Why is the answer to question a) the second gentleman?

I'm not sure how you got your answer for the second question; it's not correct. Check out Token's post.

Token: Your solution to the second question is correct, though you either over-simplified or left out a bunch of steps of reasoning when you said

Spoiler:
Since the first statement is false, the second statement is necessarily false.

As to the first question,
Spoiler:
You've made \$10, which would clearly be inferior to taking the second gentleman's offer and saying something like "this statement is true" (which would give you \$15). That makes a it tough to answer question b).
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

Avin
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### Re: Money and fruit

Actually for the first puzzle,

Spoiler:
I would take the first offer, and say something like the following sentence (depending on what I thought the man might be financially capable of fulfilling):

"You will give me less than 1 million dollars, but not exactly 10 dollars"

The man will be forced to give me 1 million dollars or more in order to satisfy his promise, because if he gave me \$10 my statement would be untrue, which would violate his agreement, and if he gave me some other amount of money less than a million dollars my statement would be true, in which case he was supposed to give me \$10. Therefore in order to fulfill his promise he is forced to give me a million dollars (or more).

jestingrabbit
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### Re: Money and fruit

Avin wrote:Actually for the first puzzle,

Spoiler:
I would take the first offer, and say something like the following sentence (depending on what I thought the man might be financially capable of fulfilling):

"You will give me less than 1 million dollars, but not exactly 10 dollars"

The man will be forced to give me 1 million dollars or more in order to satisfy his promise, because if he gave me \$10 my statement would be untrue, which would violate his agreement, and if he gave me some other amount of money less than a million dollars my statement would be true, in which case he was supposed to give me \$10. Therefore in order to fulfill his promise he is forced to give me a million dollars (or more).

That definitely seems like the right answer.

Welcome to the forums! Regarding your intro post... non-mods often seem to want to take a holier than thou approach. Mods take a much more lenient position. Mostly, its a response to the incredible amounts of spam we were getting before we went to phpBB 3.
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Buttons
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### Re: Money and fruit

MartianInvader wrote:Token: Your solution to the second question is correct, though you either over-simplified or left out a bunch of steps of reasoning when you said
Spoiler:
Since the first statement is false, the second statement is necessarily false.

He didn't over-simplify that much, but I'll fill in the not-so-obvious gaps anyway:
Spoiler:
Since the first statement is false, the consequent of the second statement is false, so either the implication is false (and therefore the second statement is false) or the antecedent is false (and therefore the second statement is false), so the second statement is false.

JamesCFraser
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### Re: Money and fruit

Spoiler:
Since the first statement is false, the second statement is necessarily false.

As to the first question,
Spoiler:
You've made \$10, which would clearly be inferior to taking the second gentleman's offer and saying something like "this statement is true" (which would give you \$15). That makes a it tough to answer question b).

Spoiler:
"This statement is true" is both true and false. You would get nothing.

Ashbash
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### Re: Money and fruit

Spoiler:
I was under the impression that you could claim the second statement to be false and thus your preference for apples and bananas could not be found, because even if the third statement was false, the statement that "If either this sentence is true, the previous sentence true, or the next sentence is false, then the previous sentence is true" would not affect the truth of the first statement, because it is false in its entirety. Where did I go wrong?

Perhaps things would make a bit more sense if I knew what antecedents and consequents were. Care to enlighten me?

Buttons
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### Re: Money and fruit

Ashbash: in logic, the statement "if P, then Q" says that whenever P is true, Q is also true. The only way that statement can be false is if P is true and Q is false. P is called the "antecedent" (thing-that-falls-before) and Q is the "consequent" (thing-that-follows-completely).

To see if you understand, ask yourself which of the following statements are true:

A. If 2 + 2 = 4, then 3 * 3 = 9.
B. If 2 + 2 = 4, then 3 * 3 = 6.
C. If 2 + 2 = 5, then 3 * 3 = 9.
D. If 2 + 2 = 5, then 3 * 3 = 6.
Spoiler:
Only B is false.

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### Re: Money and fruit

Avin's got the first one.
Spoiler:
I was actually thiking of "You won't give me \$10, nor will you give me a million dollars", but of course yours is just as good if not better.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

Ashbash
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### Re: Money and fruit

Buttons wrote:Ashbash: in logic, the statement "if P, then Q" says that whenever P is true, Q is also true. The only way that statement can be false is if P is true and Q is false. P is called the "antecedent" (thing-that-falls-before) and Q is the "consequent" (thing-that-follows-completely).

To see if you understand, ask yourself which of the following statements are true:

A. If 2 + 2 = 4, then 3 * 3 = 9.
B. If 2 + 2 = 4, then 3 * 3 = 6.
C. If 2 + 2 = 5, then 3 * 3 = 9.
D. If 2 + 2 = 5, then 3 * 3 = 6.
Spoiler:
Only B is false.

Thanks for clarification, but I still don't understand why the statement is not considered false if the antecedent is false.

Buttons
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### Re: Money and fruit

Ashbash wrote:Thanks for clarification, but I still don't understand why the statement is not considered false if the antecedent is false.

Well, because that's what "if" means, at least the material if that's being used here. It's natural to be confused by this, because the use of "if" in logic is somewhat different from what you're used to in regular speech. We tend to think of such statements as causal, so the statement "if I whistle, laser beams shoot out of my eyes" would seem absurd. However, it's true; I can't whistle.* The only way for the statement to be false would be for me to whistle and have laser beams not shoot out of my eyes. Therefore by false antecedent, this sentence is true.

Counterintuitive? Maybe, but when you do a lot of proofs, you'll find that this is really the only useful definition.

*But I really should learn. Those lasers would be a neat party trick.

Blatm
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### Re: Money and fruit

Spoiler:
I don't see why the second statement is false.

"If either this sentence is true, the previous sentence true, or the next sentence is false, then the previous sentence is true." is false only if both "If either this sentence is true, the previous sentence true" and "[if] the next sentence is false, then the previous sentence is true." are both false.

Because "the previous sentence is true" is false, the only way for "If either this sentence is true, the previous sentence true" and "the next sentence is false, then the previous sentence is true." to be false is for "this sentence is true" to be true and "the next sentence is false" to be true, but then "this sentence is true" would be false, which creates a contradiction.

Therefore, statement 2 must be true, so "If either this sentence is true, the previous sentence true" is false, which forces "the next sentence is false, then the previous sentence is true." to be true, which is only the case if "the next sentence is false" is false, which leads me to conclude that you have the same opinion of apples as you do bananas.

Why did I get a different answer?

Token
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### Re: Money and fruit

Blatm wrote:
Spoiler:
I don't see why the second statement is false.

"If either this sentence is true, the previous sentence true, or the next sentence is false, then the previous sentence is true." is false only if both "If either this sentence is true, the previous sentence true" and "[if] the next sentence is false, then the previous sentence is true." are both false.

Because "the previous sentence is true" is false, the only way for "If either this sentence is true, the previous sentence true" and "the next sentence is false, then the previous sentence is true." to be false is for "this sentence is true" to be true and "the next sentence is false" to be true, but then "this sentence is true" would be false, which creates a contradiction.

Therefore, statement 2 must be true, so "If either this sentence is true, the previous sentence true" is false, which forces "the next sentence is false, then the previous sentence is true." to be true, which is only the case if "the next sentence is false" is false, which leads me to conclude that you have the same opinion of apples as you do bananas.

Why did I get a different answer?

Spoiler:
You get a different answer because you can't split up the second statement like that. Here is a (completely equivalent) rewording of the second statement that might make things clearer.

"Either (this sentence is false, the previous sentence is false, and the next sentence is true), or the previous sentence is true."
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

Blatm
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### Re: Money and fruit

That makes sense. Thanks.

Ashbash
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### Re: Money and fruit

Right. I understand antecedents and consequents now! Thanks a gigabazillion buttons!

MartianInvader wrote:Puzzle 2 - All of the following sentences are either true or false. (And use the material "if", if you know about that sort of thing)

If this sentence is true or this sentence is false, then this sentence is true and this sentence is false.

If either this sentence is true, the previous sentence true, or the next sentence is false, then the previous sentence is true.

If I absolutely love apples, then I absolutely love bannanas.

How do I feel about apples and bannanas?

So tell me if I'm following correct reasoning here.
Spoiler:
Sentence 1 has a true antecedent and a false consequent, rendering it false.
If sentence 2 is to be false, it needs a true antecedent and false consequent. assume this is so. The consequent is false. For the antecedent to be true, the next sentence must be false (because statement 1 is false, and we are assuming statement 2 is false atm)
Thus statement 3 is false, which means he loves apples, but not bananas.

If statement 2 was to be true on the other hand, it needs a false antecedent in conjunction with its already established false consequent. To make it false, none of the three conditions can be satisfied. However this is impossible as one of the conditions is satisfied when it is considered true, as we are doing now.

Therefore the previous solution is the only solution.

Got it. Thanks heaps again!

Buttons
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### Re: Money and fruit

Ashbash wrote:So tell me if I'm following correct reasoning here.
Yep, that's it. Good recap.

fyjham
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### Re: Money and fruit

With question 2, I think there's no answer:
Spoiler:
If sentence 2 is to be false, it needs a true antecedent and false consequent. assume this is so. The consequent is false. For the antecedent to be true, the next sentence must be false (because statement 1 is false, and we are assuming statement 2 is false atm)

Isn't that a bit wrong? I mean, for a rule to be false it doesn't have to always fail, it has to not always succeed.

EG:
1: If A + A = 4 then A * A = 4
2: If the above is true then B = A else B = A + 1

Sure when A happens to = 2 this evaluates correctly, but the rule is still wrong. Just because it's false doesn't mean it can't still be true under certain applications.

With that said, I'll just consider the option that every rule is false (Perfectly possible), and hence you could love or hate apples and bananas all you want. So everything is a possibility and nothing can be dismissed

Owehn
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### Re: Money and fruit

In your example ("If A+A=4 then A*A=4"), what you're really doing when you consider its truth value is insert a "for all A" at the beginning of the statement. (The same goes for any statement with free variables in it.) Since there are no free variables in the three statements of puzzle 2, they can be analyzed without being subject to your complaint.
[This space intentionally left blank.]

fyjham
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### Re: Money and fruit

Personally I'd call the true/false states of the rules free variables

Token
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### Re: Money and fruit

fyjham wrote:Personally I'd call the true/false states of the rules free variables

Then you'd be wrong.

http://en.wikipedia.org/wiki/Free_variable
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phlip
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### Re: Money and fruit

The definition of "if" in logic makes a lot more sense when you start bringing in the universal quantifiers...

Consider the statement "Everything wearing a hat is human". In logic, this'd be: for all x, if x is wearing a hat, then x is human. Clearly, we want this to be true for all human hat-wearers, but also, we want it to be true for any non-hat-wearers, regardless of their humanity. The only thing that should make the statement as a whole false is the existance of a non-human hat-wearer.

It's just important to remember that this statement doesn't say anything about the hat-wearing status of humans in general, nor the humanity of non-hat-wearers... these can fall either way. It's just saying that if you're wearing a hat, you're human... and, equivalently, if you're not human, you cannot be wearing a hat (the contrapositive).

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`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

fyjham
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### Re: Money and fruit

I just wrote a full 3 pages trying to make my point (Which I assumed would be wrong, but no easier way to find out why than to post and let it be torn to shreds ), then at the end it sorta finished with "Oh **** I'm an idiot"

I get what you're saying now Owehn. All my logic work has been from the perspective of mathematics/programming, so I almost always work with free variables (Thanks token for clarifying what those were, I was considering it anything I didn't know the value of when I read Owehn's answer as opposed to something which could have any value... a subtle but important different it seems ), so the ability to flip the rule when it was proven false really seemed messed up to me.

But I guess when only 1 variable remains and it's a true/false the the contrapositive, to steal some lingo off someone else so I sound smarter , will actually be the same as the... damn no lingo to steal, I'm about to sound dumb again... inverse/opposite/contradictory/somethingelse rule which is normally not necessarily true in a domain containing free variables

Hix
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### Re: Money and fruit

Meta-solution to Puzzle 2:
Spoiler:
Assuming that the Puzzle has a solution (this makes my solution "meta"), we see that "If I absolutely love apples, then I absolutely love bananas." can't be true, since that sentence provides the only information about apple and banana preferences in the entire puzzle, and yet is satisfied by 3 different combinations of preferences (Love both, Love neither, Love only bananas). Thus, that sentence must be false, and hence he absolutely loves apples, but does not absolutely love bananas.

ASmileWithoutACat
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### Re: Money and fruit

Spoiler:
1. a. Take the man on the right's offer.
b. Because of the possibility of getting more than \$15. He can't take money from you (no negative money) and if there is no upper bound (or a very high upper bound) on the amount he can give you, it's likely that the probable outcome (where the amount is Ai from the set of all amounts he can give, An is the largest amount he will give, and the probability of getting it is P(Ai): sum from i = 0 to i = n of P(Ai) * Ai) is higher than \$15, so you are likely to make more than with the man on the left.

Sorry if the math is messy or wrong... not my best subject.

Spoiler:
2. If you absolutely love apples, then you don't absolutely love bananas (or v.v.).
The first sentence can't be true (can't be both true and false).
Therefore, the second sentence can't be true (if it were, the first sentence would be true, and it can't).
So in order for the second sentence to be false, the third sentence must be false (because if it were true, then all the conditions of the second sentence would be false, as well as its conclusion, and therefore the second sentence would be true (I'm not quite certain of myself here)).
So if you absolutely love apples, you cannot feel the same about bananas. Or the other way around.

Myself, I rather like apples, but I detest bananas...
You know it's all beginning
To feel like pretending...

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DrStalker
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### Re: Money and fruit

First question solution enhancement:

Avin wrote:Actually for the first puzzle,

Spoiler:
I would take the first offer, and say something like the following sentence (depending on what I thought the man might be financially capable of fulfilling):

"You will give me less than 1 million dollars, but not exactly 10 dollars"

The man will be forced to give me 1 million dollars or more in order to satisfy his promise, because if he gave me \$10 my statement would be untrue, which would violate his agreement, and if he gave me some other amount of money less than a million dollars my statement would be true, in which case he was supposed to give me \$10. Therefore in order to fulfill his promise he is forced to give me a million dollars (or more).

Spoiler:
Why settle for 1 million? "You give me less than the maximum amount of money you are able to give me, but not exactly 10 dollars"
There are two types of people in the world: 1) those that can extrapolate from incomplete data.

quintopia
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### Re: Money and fruit

[quote=DrStalker]
Spoiler:
Why settle for 1 million? "You give me less than the maximum amount of money you are able to give me, but not exactly 10 dollars"
[/quote]

Spoiler:
Because then he gets to decide what "able" means. Maybe something like "You will give me less than all the money on earth and you won't mastermind a robbery of DeBeers' warehouses and give me all of the diamonds in it at that time, but neither will you give me 10 dollars."

I put the robbing DeBeers part in, because acquiring all the money in the world is impossible (because it would always be immediately replaced with some other form of currency, resulting in the original acquisition no longer being money), whereas robbing DeBeers is just extremely difficult.