You are in control!
Which offer is better?
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You are to make a statement. If the statement is true, you get exactly $10. If the statement is false, you get either less than or more than $10 but not exactly $10.
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You are to make a statement. Regardless of whether the statement is true or false, you get a random number of dollars uniformly chosen from [$10, $1000000].
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6 solutions
Moderator note:
Simple standard approach.
This reminds me of the question that a compassionate king posed to a wise man that he had to sentence to death. The king said "You can make a statement. If the statement is true, you will be fed to the lions. If the statement is false, you will be thrown off the cliff". How can the wise man avoid being sentenced to death?
Your answer is wrong as option #2 is far superior to option #1. You have to think in terms of how much money you are likely to get.
If you choose option #1 and say a true statement, you get $10. If you choose option #2 and say a true statement, you get more than $10. Therefore if your statement is true, option #2 is guaranteed to be better.
If you choose option #1 and say a false statement, you can get just about any number of dollars as long as it is not exactly $10. Therefore there is a possibility that you could get less than $10 if you say a false statement. If you choose option #2 and say a false statement, you are guaranteed to get more than $10. Therefore if your statement is false, option #2 is likely to be better as there is no possibility of getting less than $10.
So if your statement is true, option #2 is better and if your statement is false option #2 is more likely to be better so therefore in terms of probability you should always go with option #2.
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You're half-right, but you don't really get why.
Option 1 works for getting yourself a precise amount of money. If you want the guarantee of $1000000000, you can have it with option 1, but you might not get it with option 2 since there's no way to rig the payout.
On the other hand, since there is no upper limit specified for the money you can receive but you cannot be given less than or equal to $10, option 2 will pay you more on average. This is because, no matter what finite number you specify for option 1, there will be an infinite number of larger numbers payable through option 2 and a finite number of smaller numbers.
Now, that said, this question was pretty clearly about using logic, and thus the question creator (if they were aware of this flaw) would probably say something like "The amount of money you can potentially receive is finite, bounded by the total amount of money in the world." That's why I chose option 1, despite knowing that by the literal interpretation of the question option 2 works better.
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Yeah, I think you were not using logic, because the rules are only "get less or more than $10" and "get more than $10". It is your feeling declaring there are boundaries about the amount of money other than the two I say above. Definitely #2 is the better answer. Because everything I state will be the right statements, and I will get more than $10.
I can also say "I will receive $100,000" on statement #2, and say it is a true statement, so I can get $100,000. Seems a difficult question, but actually not a good one. :(
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@Vegi Toenlioe – And I could say it is a false statement, and just pay you $11. Lo, and behold, I was right! Option 2 is just weak.
Edit: Since the problem was modified, Option 2 is still weak. If you're lucky, you can get $1 million. With Option 1, you can get $10M instead, by saying, "I will get neither $10 nor $10,000,000."
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@Whitney Clark – The problem is changed. At first it does not have random number of dollars at option #2. At first, option #2 just say that I will get money more than $10. Suck.
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@Vegi Toenlioe – True, but there's no mention of how much more. For example, $11 is more than $10. You have no control in that case.
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@Whitney Clark – So does option #1. I have no control in how much I will get the money. Still suck.
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@Vegi Toenlioe – Maybe not with that attitude, anyway. Just say, "I will get neither $10 nor $1 billion." Proof. That can only be true if you get neither - for example, if you don't get $10. But a true statement gets $10, contradicting the givens, so it cannot be true.
Thus, it must be false, and you get either $10 or $1 billion. But you cannot get exactly $10 for a false statement, so you MUST get $1 billion. QED.
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@Whitney Clark – Who said that I will get $1 billion if I use option 1? The rule just say I will get either less or more than, not exactly $10. If I said "I will get neither $10 nor $1 billion", then if we follow your argument that this statement is false, it does not mean I will get $1 billion. The rule just say I will get either less or more than $10, thus I can get $9, NOT EXACTLY $1 BILLION. SUCK.
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@Vegi Toenlioe – I wish you'd stop saying "suck"; it's not an argument. And who says I will get $1 billion is logic, that's who. You cannot POSSIBLY get $9 if the statement, "I will get neither $10 nor $1 billion," is false, because getting $9 makes that statement true, and true statements demand $10 exactly. Right? The truth is that the offer can be manipulated if you know how.
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@Whitney Clark – No, the statement is not automatically become the prize. The rules did not say anything like 'all the words that we say are related with the prize'. The one that related is the condition, whether it is true or false, NOT THE STATEMENT. So if I said "I will get neither $10 nor $1 billion" and then I decide this statement is false, then the prizes I can get ARE NOT AUTOMATICALLY $10 BILLION IF I CHOOSE #1, because the statement does not have any effect to the prize. I WILL STILL CAN GET LESS OR MORE THAN $10, NOT EXACTLY $1 BILLION JUST BECAUSE I SAID THE STATEMENT BEFORE. YOU WANT THE "F" WORD? I am not arguing here, it's protesting.
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@Vegi Toenlioe – My point from before is that swearing AT ALL is not an argument. And what is the protesting for? What good does that do?
Anyway, what does it mean if the statement, "I will get neither $10 nor $1 billion," is false? It means that I WILL get either $10 nor $1 billion, doesn't it? Those are the ONLY two possibilities that make the statement false. And a false statement does not get exactly $10, leaving $1 billion as the ONLY possibility.
Now, here's a question. Why do you say the statement has no effect on the prize? I have already shown that it does. If a statement is false, its opposite must be true, and at the same time, a true statement gets exactly $10, while a false statement gets something else. You combine the facts, and the conclusion is clear.
I almost forgot: The key to this is that the statement is false because you said it. Given two statements, say S1: "I will get neither $10 nor $10 million," and S2: "I will get neither $10 nor $1 billion," whichever one you say is false, but the other is true, simply because that is the statement you chose. Get it?
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@Whitney Clark – Nope, I cannot see the correlation between the statement and the prize. For me, the statement is used to determine the result. This quiz does not say that the statement has direct correlation with the prize, but the result of the statement does. Maybe you are a better interpreter than me, that's all. There is no reason anymore to argue(?) with you, since you have more capabilities to understand the question, to see the real mean beyond the statement that given.
Why am I protesting? See beyond my statement, because I think you good at it. FO.
Bye.
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@Vegi Toenlioe – True enough, the statement IS used to determine the result. If I pick offer #1, then, "I will get neither $10 nor $1 billion," cannot be true, and can be false only if I get $1 billion, so it is false and I get $1 billion; every other possibility leads to contradiction. You have no such control with #2.
I'll go over it again. If I make that statement, it cannot be true, because then I get neither $10 nor $1 billion, so I don't get $10. But it's true, so I do get $10, a contradiction. So it can't be true.
But then it's false, and I do indeed get either $10 or $1 billion, but a false statement does not get $10, but gets more or less, and $1 billion is more or less than $10. Get it?
If something is wrong with my reasoning, please tell me. Bye,
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@Whitney Clark – His reasoning can't be blamed. His reasoning is not really pure mathematics , but more to reality.
I once argued about it , regarding whether the question should have written some assumptions.
His reasoning is more or less stick to the term "Lie" & "Invalid"
If you lie(make a false statement in a high court for example)
Then your statement becomes invalid and null (as good as you never made any statement)
Subsequently you will be subjected to punishment for making false statements DISREGARDING of the content spoken.(as it cannot be taken into account)
The move to call a statement false , invalid and therefore discarded is because you cannot grant impossible wishes. Infinite money is fine , but if asking to revive the dead is impossible.
(in another word , its approaching this question not in mathematical sense)
Therefore , people can make assumptions that whether the statement true or false , it is independent of the results you will be subjected to , example below :
In offer#1 , if you make a true statement you get $10
like a celebrity show , if the host call it
true = you open the $10 dollar box
false = you spin a roulette and get random value except $10
this is what he means by statement does not affect the result.
Once the host call "true" or "false" , the statement is discarded(becomes null and invalid) and proceding is to take place normally without having any elements from the statement.
Whether the host is mathematical literate or not , if he CLAIMS that your false statement is a mumble jumble and cannot be trusted. Thats the end call. You are as good as never made any statement. Mathematically you fall into false category. (but this whole process may or may not be mathematical at all)
* the false statement here can refer to events other than money like reviving the dead
Since you are considered didnt make statement at all , it cannot be true and it must be false.
You will be subjected to random value other than $10.
However ,option 2 guarantees 10 or more. Regardless of any statement. (yes,the host can apathetically give out $11 whether he listens or not , but still better than having less than $10)
This is why all of us argue for option 2. (under the assumption of approaching the question non-mathematical way just like how you are treated when you lie in court)
There is nothing wrong is your reasoning , its just that there is no way you can argue with Vegi. Human can be male or female but male cannot equal to female. Both of you are like opposite gender in this case.
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@Ne-ko Nya – Weird. It sounds like how my dad might describe liberal thinking. (I agree with him, actually, but this is not a political forum.) Oh well.
that is exactly how i tried to solve it
That's not correct. In #2, the least you can get is $10.01, which is the most the payer can get away with paying. Reread the offer and you'll see that it doesn't make the slightest difference if the statement is true or false.
Edit: Now the least is $10.
I think it's better to think it this way. To account possible events, minimum of 4 samples are needed. Focusing on maximum and minimum prize.
| Sample | Offer 1 | Offer 2 |
| 1 | True - $10 | True - $1M |
| 2 | True - $10 | True - $1M |
| 3 | False - $0 | False - $1M |
| 4 | False - $infty | False - $1M |
| Total | $infty | $4M |
Therefore, Offer 1 is better.
Previously I was thinking likewise. (Option-2) But today after reading question second time. I would go for Option-1.
Option 1 is far Superior than 2.
Overall Gain on both cases can be calculated by multiplying their outcomes with probability. PN: True / False probability is 0.5
For Option-2: (Considering maximum Outcome i.e. $1000000) Total Gain= (Probability of True) (True Outcome) + (Probability of False) (False Outcome) Total Gain= (0.5) ($1000000) + (0.5) ($1000000) Total Gain=$1000000
For Option-1:
Total Gain= (Probability of True) (True Outcome) + (Probability of False) (False Outcome) False Outcome= Either less than $1000000 (Even zero) or may be greater than $1000000. But the probability that this number would be greater than $1000000 is far more high. As probability is "possible outcome / total outcomes" and we have infinite total outcomes. (In case-I we are not bounded till $1000000).
Total Gain= (0.5) ($10) + (0.5) ( (0 (<$1000000))+(1 Infinite) ) Total Gain=5$ + Infinite$ Total Gain=Infinite $
I hope, I was able to express my views now.
Maybe there is something I missed....
(A) now where or how with option 1 are some people coming up with $ values of $1M, $100, 000 or even $1000 when nowhere is it stated or implied anything of the sort?
(B) A true statement WILL guarantee you of precisely $10, a false statement only guarantees that you WON'T get precisely $10 but any other value either larger or smaller but nothing about what the limits of that value could be in either direction or even if the " less than" or "more than" outcome is 50/50 or weighted more or less to either one.
Assumptions have no place in determining which is better.
Otherwise......A false statement guarantees you wll not end up with exactly $10 and lets assume the "more or less" outcome is 50/50 and by a value of $10, 000 then doesn't it stand to reason you may equally get $10, 000 or owe $10, 000! See where I am coming from.
However with no assumptions and using only whats actually stated in either option,. The only statement in option 1 that can guarantee that you will receive something and not owe something is a true statement which will only ever payout $10 per true statement but with option 2 who gives a shit true, false makes no difference they both guarantee a payout but not $10 but any other value greater 100% of the time limited to $1M per statement!
So Alex, I'm with you!
Those who chose different..... Please help me understand (without making any assumptions) why I would want to choose offer #1
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Consider the statement "I will neither get $10 nor $1 billion".
If this is given to Option 2, we will get a random amount, but definitely not greater than $1 million.
If this is given to Option 1, is it true or false?
If it's true, then we would get $10. But this makes the statement false, contradiction.
If it's false, then we would get something other than $10. What will this number be? If it's not $1 billion, our statement becomes true, contradiction. The only option is for we to get $1 billion to get everything to work out.
Thus Option 1 gives $1 billion.
The trick is that you devise the statement, so you can troll with this kind of statements. There's nothing probabilistic if you make the "correct" statement.
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@Ivan Koswara – Thanks Ivan for the clarification!
simple , you make compound statements. (combine 2 or more statements into 1 statement using conjunction) --> this question gives POWER to language.
by combining 2 statements , you can set CONTRADICTING rules to it such that the statement can only be false or true.
That is why people choose offer 1 , like how corporat business write policy that appears to be give freedom but is in fact a lie(because at the end of the day , you have no choice other than the one he sells)
~Scenario of a Ridiculous Government~ its like telling people : You have the freedom of choice to buy car from overseas or local manufacturer , however if you buy overseas car....
1) The price of the car must not exceed 50K 2) The car price will be subjected to tax/tariff at 1000% of its original value.
Basically it means you can only buy overseas cars priced less than 5K that doesnt exist.
The money argument that was put up regarding $10 and $1billion is the same , since the question limits the output of offer#1 as $10 or not exactly $10.
all you have to do is make 2 contradicting statements around $10 such that you will get any value of money you want because otherwise you will violate the offers.
you don't make single statements like " I will get $1billion" you make compound statements like " I will not get $10 and I will not get $1billion"
I agree with you
Yeah, same, I also root for 2.
In option one you have a probability of 50% to get 10$ and a probability of 25% to get more than 10$ so the probability of getting >= 10$ is 75%. In the second option you get 10$ with 100% percent and according to game theory option 1 here is much better :)
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In option 2 you have 100% a probability of getting more than $10, check the second option in the question again:
"2. You are to make a statement. Regardless of whether the statement is true or false, you get more than $10."
Offer 1 is better, yes, but your reasoning is wrong. There were no mention of probabilities, and it could be that less is more likely, all else being equal.
u r wrong jst bcoz of the fact that u don't realise this fact that formation of the statement is in ur hand....and u can easily form any such false statement which earns u much more money
Exactly as my way thinking..!
You are absolutely right.I think this is a failed attempt to create a problem in Game Theory. (Though I chose Option 1,I strongly think Option 2 is the correct answer.)
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If you "strongly think" it, what is your reasoning? A correct answer can be defended with logic and reason; a wrong one cannot.
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@Whitney Clark – I think I wrote the comment claiming someone to be right.That actually means I claimed his logic to be right.I do not find any confusions in here.
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@Titas Biswas – But logic can be proven, and doesn't need someone to "claim" it.
I agree with you; it doesn't matter what your statement is - only its validity matters.
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But that's the whole point you control the validity of the statement by stating the amount you will receive. By doing that, you can predetermine which side of true or false you will come out with option 1 and the resulting behaviour. You can't do that with option 2.
I wonder what you will get with a paradoxical statement like "This statement is false".
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For a paradox you get a swift kick in the pants.
I think you will get 10$ according to #1.
if the 'statement' is false and you are announcing it as 'this statement is false'.
which means you are not lying about it , and you are speaking a truth(TRUE). You get 10$.
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Wrong. I could say I'm a dog, and that wouldn't be true. Bruce Jenner could say he's a woman, before OR after mutilating himself, and it wouldn't be true. "This statement is false" is a paradox, and you cannot receive $10, OR more, OR less.
If seen as a simple sum of probability, you have a 33.33% chance of getting more than $10 in option 1.
On the other hand the probability of getting more than $10 in the second option is 100%.
Thus, logically, it would be a better option to choose the second offer.
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Agreed. But I believe that (not stated in the question) is the fact that your statement includes the amount of money that you're asking for. It is not random. If it is random (as the wording of the question makes it appear) option 2 is obviously better.
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Since you mentioned in your reply that the question does not state the fact that you will receive the amount you stipulate, we cant assume that it will so the answer should be option 2.
I disagree. There was no mention of probability or randomness, and I think you would receive the least amount possible while still agreeing with the givens. In offer 1, you get $10 for a true statement, more or less than $10 for a false statement. If your false statement is consistent with getting less than $10, you'll probably get it with #1.
Imho, not properly stated
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What more does it need? What is confusing about it? #1 is $10 for true, more OR less than $10 for false. #2 is more than $10 no matter what (revised to $10 or $1 million).
Clever, but offer #1 is a self-referential paradox. Such paradoxes are amusing, but I would not classify this as a problem with a logical solution, and #1 is not a correct answer, nor is it an incorrect answer. It is nonsense, but fun!
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Exactly where's the paradox? Payoff is p = $10 for true statement, p != $10 if false.
Pranjal Jain is correct. The trick here is making the statement. option 2 gives you maximum $1,000,000 for each statement, but making statement in the right way, option 1 will give you the exact amount you want. Just say: "I will receive neither $10 nor $xxxxxxx". You will get $xxxxxxx, which is more than $1,000,000.
I don't understand the logic at all that you are using to "solve" this puzzle. The puzzle does not state that you will receive the amount of money based on what your statement says. I could say something like "The Earth is flat" and would get an amount either less than or greater than $10 according to the problem. Same goes for specifying an amount of money in your statement. Even if you use the example "I will receive neither $10 nor $1000." This problem says nothing about how much you get, so you could say this and only get $20 because it is still false. I think that this problem should be altered because it is very confusing as of right now
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if you get 20$ , then "I will receive neither $10 nor $1000." = TRUE
if it is TRUE , you are supposed to get 10$ , a CONTRADICTION. At least in maths , it is not allowed.
That is false; there is nothing confusing about it. You CANNOT get $20 with that statement, because that is neither $10 nor $1000, making it a true statement, and a true statement gets $10, not $20.
A couple of problems here, as already pointed out. First of all, you are claiming there exist a statement for which the first offer is better and you provided one. However, the second offer doesn't say by which distribution money would be provided. If you were thinking random, random doesn't have to mean uniformly random. I'll also point out that it doesn't specify that you'd necessarily get as low as possible like some suggested. We just don't know and as specified, it could be any amount of money by any distribution with any probability for each distribution. Note that it also doesn't have to be money only. For example, offer 2 could mean 10$ and a house. For all we know, maybe offer 2 could yield us the possession of all known universe with some probability. Altogether, this by itself makes it impossible to determine which offer is really better.
Another point to undermine your explanation more directly, it is false to assume that if some statement is false that it necessarily follows that the statement is true. There are statements that are neither true or false, such as "It is raining tonight." The statement doesn't say where so we don't know. There are also statements that are paradoxes. Further, I will show that your own statement “I will receive neither $10 nor $1000” doesn't actually have to be true or false. Let's say that the person you are telling your statement doesn't have 1000$. In that case, the statement can't be true since that would imply contradiction like you showed. But it can't be false either, since the person in question doesn't have 1000$ or provisionally high number as you suggest and therefore you won't be getting the money you asked for. Therefore, in that case your statement is neither true or false and offer 1 doesn't say what happens then. Offer 2 however would still give more then 10$, by completely unknown distribution.
The conclusion is that the problem isn't stated well enough, or that we can't determine which offer is better.
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The statement, “I will receive neither $10 nor $1000 [now $10 million],” MUST be true or false, even if we don't know it yet. If I were to predict hail tomorrow in a specific location at a specific time, it would be true or false, even though, like with Schrodinger's cat, we wouldn't know which until then.
The puzzle says the correct answer is 1 but that simply cannot be. Not the way it's worded. Maybe there is some other parameter that if applied makes the answer 1, but the way it's worded it can only be two. Pranji, you say, "But since you cannot receive $10 if the statement is false, you must receive $1000.". That's completely ignoring the fact that if the statement is false with option 1 you can receive LESS than $10. The puzzle basically says with option 1 you can have exactly $10 or ANY other amount. Option 2 you get more than $10 no matter what. Option 2 is better 100%. This puzzle is flawed and whoever wrote it either needs to reexamine their logic, or their ability to speak English.
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Actually, no. If “I will receive neither $10 nor $1000” is false, then you CANNOT receive LESS than $10, because its opposite is "I will receive either $10 or $1000," neither of which is less than $10. (With the revised version, the solution should be modified to read, "$10 or $10 million.")
How about I create statement "I will receive $1000." and use 2?
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So, in that case if statement if statement is true then you may receive 1000 but if it is assumes false then in that case you may get 11$ , it is not safe to make statement 2 safe is 1 only because if you make 1 you are sure to get that amount but in 2 it not sure it depends
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but on taking 2 ...one may also get 100000 $ ...its also greater then 10 $
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@Harshal Choudhary – you may but it is not sure but in case 1 it is sure to get
"you get either less than or more than $10 but not exactly $10" . If it's false there is a chance that you will get lower than $10 but statement 2 is guaranteed to give you more than $10 right??
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compare his statement carefully , if he receive let say $9 , then the statement
“I will receive neither $10 nor $1000.” = TRUE , because he receive $9
but if the statement is true , then he should receive $10 , not $9. this creates a CONTRADICTION.
therefore the statement can only be false.
"I will receive neither $10 nor $1000" = false
the outcome we will get is , "I will receive $10 or $1000" , but since he can only obtain value ANYTHING OTHER THAN $10 , he will get $1000.
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But , what If I use this statement - " I will receive neither $10 nor $9"
let say he gets $1000
"I will receive neither $10 nor $9" =TRUE, because he receives $1000 but if the statement is true, then he should receive $10, not $1000,this creates a CONTRADICTION.
therefore the statement can only be false.
"I will receive neither $10 nor $9" = false
the outcome we will get is, " I will receive $10 or $9", but since he can only obtain value anything other than $10, he will get $9.
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@Hitesh Parmar – With due respect to the question ......'which offer is better?'
offer1 : you name any amount of money , i'll give you
( if you want $9 , you get $9 , if you want Jesus to be revived , HE WILL BE REVIVED)
offer2 : you name any amount of money , i'll randomly give you higher than $10
..........
Which offer is better?
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@Ne-ko Nya – check the question again and point out to me where does it say that offer 1 will give you the amount you name:
Question: Which offer is better?
You are to make a statement. If the statement is true, you get exactly $10. If the statement is false, you get either less than or more than $10 but not exactly $10.
You are to make a statement. Regardless of whether the statement is true or false, you get more than $10.
@Ne-ko Nya – this post sums up everything :3
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@Fitriana Bachtiar – apparently it doesnt , i posted a new comment in 2nd solution , at bottom of the page because the post closed its ability to post new solution. It seems like I made a blunder in accepting offer 1 as best offer.
offer 2 is capable as well.
@Ne-ko Nya – Thank you for your reply. But in offer 1 and offer 2 , it's only mentioned that you get more than $10. So, if Jesus can be revived in offer 1 it can be revived in offer 2 as well.
OFFER 1 & 2 , BOTH JUST MENTIONS MORE THAN $10....
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@Hitesh Parmar – in option 1 you can control how much money you will get by manipulate the value of your statement.
No where in the question does it state that we will receive the amount we stipulate e.g. $1000 or $1million, it only stipulates more than $10, therefore correct answer is option 2.
Actually, there is no chance at all, if you make the right statement. "I will get neither $10 nor $10,000,000" works fine for #1. If that is true, you get $10, and you get neither $10 nor $10 million, hence you get $10 and you don't, a contradiction; hence it cannot be true.
If it is false, then I will get $10 or $10 million, but not $10, because the statement is false. Thus, the ONLY possibility is that it's false and I get $10 million.
Can you please post more Raymond Smullyan logic puzzle. They are quite intriguing.
Nice solution @Pranjal Jain ! :)
Both the puzzle and Pranji say the correct answer is 1 but that simply cannot be. Not the way it's worded. Maybe there is some other parameter that if applied makes the answer 1, but the way it's worded it can only be two. Pranji, you say, "But since you cannot receive $10 if the statement is false, you must receive $1000.". That's completely ignoring the fact that if the statement is false with option 1 you can receive LESS than $10. The puzzle basically says with option 1 you can have exactly $10 or ANY other amount. Option 2 you get more than $10 no matter what. Option 2 is better 100%. This puzzle is flawed and whoever wrote it either needs to reexamine their logic, or their ability to speak English.
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Actually, no. If “I will receive neither $10 nor $1000” is false, then you CANNOT receive LESS than $10, because its opposite is "I will receive either $10 or $1000," neither of which is less than $10. (With the revised version, the solution should be modified to read, "$10 or $10 million.")
the man says i will be thrown off the cliff
Pranjal, I see you have answered many questions, and you are a moderator. Congratulations! However, this problem really is an example of a self-referential paradox where the statement that is made using offer#1 refers to itself. There are many examples where self-referential language statements contradict themselves. Very simple example: "This sentence is false". Now, to make things even more self-referential and confusing, consider what it means if that sentence is the statement I choose to make for offer #1! In other words, this entire problem contradicts itself and the correct answer is "this problem has no logically consistent solution".
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Pranjal's solution is perfectly correct. What is wrong with it? If the statement is, "I will receive neither $10 nor $1 billion," is true, then you will receive neither of those two amounts, but you will receive $10, a contradiction. If false, then you will not receive exactly $10 by the givens, but the opposite of the statement is true, and you WILL receive EITHER ten bucks OR a billion. Thus, you won't get exactly $10, and you WILL get either that or $1 billion, so you can only get the billion.
To the Challenge Master.
The wise man should state that he'll be thrown off of a cliff. If that's true, then he'll be fed to lions, but then his statement would be a lie. In that case he'd be thrown off of a cliff. But then his statement would be true and he'd be fed to lions, and so on... Until the king's head explode due to the deadlock.
By saying. I would neither be fed to the lions nor not thrown off the cliff. Is it correct?
so, if i choose the second offer, and whenever i make a statement, i get $1000000, what might be a better offer?
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Wrong again. $10.01 (or $11) is also more than $10, so I'd pay that just to be thrifty. I wouldn't be lying, either.
For number two, how about the statement "if this statement is false then i will receive less than X dollars and if this statement is true then i will receive more than X dollars".
If the whole statement is true then both conditional sub-statements must be true. The first substatent is true regardless of the truth of its conclusion because its hypothesis is false. The second substatement's hypothesis is true so it's conclusion must also be true. Thus, if the whole statement is true i will receive more than X dollars.
If the whole statement is false then the second conditional statement is true since its hypothesis is false. For the whole statement to be false, the first conditional must be false. For this to happen, the hypothesis must be true (it is by supposition), and the conclusion false. Therefore I will receive at least X dollars.
Whether the whole statement is true or false, i receive at least X dollars.
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As long as you allow self-referential statements and logical operators you have a turring complete language, so you can always construct a statement/program which results in pretty much any value you like. It's probably impossible to fix the problem.
Very poor. Offer 2 says that it doesn't matter if the statement is true or false, so I'd just listen to your statement and then apathetically pay you $10.01.
@pranjal you should have given the statement “I will receive neither $10 nor $1000.” and then given the two choices to choose then your answer #1 is correct. But in the above in 2nd case also the amount is not fixed we can take any arbitrary amount more than 10 that may b huge than that of first case too.
let x be the total income. E(x)=SUM(probability of the event gain for the event) Here for the 1st case, E(x)=(0.5 10)+(0.5 k) caseI) k>10 E(x)=5+0.5k>10 caseII) k<10 E(x)=5+0.5k<10 For the second case E(x)=(0.5 10)+(0.5*10)=10 Hence, second case is not better when we consider case I and second case is better when we consider caseII Hence 1st case is better with 0.5 probability but second case is better as 0.5 probability of loss is also there.
I will give you $1 for making your statement false.
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Wrong. If the statement, “I will receive neither $10 nor $1000," is FALSE, then the one who makes the statement WILL receive either $10 or $1000, the logical opposite of it. Paying $1 is NOT an option in that case.
Maybe it should be reworded to "I will receive neither $10 nor an amount greater than what I would have received if I had chosen option 2", otherwise there could be a chance that you still receive even more in 2.
Well, back to title "You are in control!", so this riddle have a chance to let you control yourself. From the usual logic view, statement must be consider as variable (apply to every type), and what you do still in control with the riddle. But in rare logic view, while your statement was nor $10 neither $∞, the riddle can't say the statement true because if you given with $10, it's against your statement, and if the riddle say false, the riddle only can give you only $∞ because the riddle self say will not give $10 and if the riddle not giving $∞, it will make your statement true.
What if your statement "I will receive neither $10 nor $1000" is always true because I(any secondary paying entity) give you $1 every time you make a statement which is why you always receive $10+$1 from me making your statement always true.
Hence, option 2 is correct.
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Wrong again. If it is true, you get $10, but that contradicts the statement, which says you do NOT get $10.
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If "I will receive neither $10 nor $1000" is true, you will receive $10 but then the statement becomes false so you should not receive $10 is your argument...... My argument is what if I add $1 from my personal pocket...making it $11..so that the statement stands correct...You receive $10 from the organiser/paying authority...and $1 from ME(any random secondary source).
The point I'm trying to make is if every angle has to be pursued in trying to answer this question then why not the Logic of English Language. Nowhere is it mentioned in the question that only A will pay B and C cannot contribute. Why Option 2 is also right
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@Prabhat Sharma – Oh for crying out loud. There were no qualifications listed. It did not say, "Someone will pay you," but only, "You get," never mind who from. Taking out a dollar from your own pocket, therefore, will only make things WORSE.
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@Whitney Clark – Ya whatevar, maybe I'm just fighting a lost battle! Cheers!
If the sentence “I will receive neither $10 nor $1000.” is defined as FALSE it does not matter what denands it crave outside of the sentence.
2 will surely give you 10$ or more, 1 might get you more than 10$.
Just like "I always lie" always must be defined as FALSE since it can not be TRUE
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If the sentence “I will receive neither $10 nor $1000,” is defined as FALSE, then the truth is just the opposite, and I WILL receive either $10 or $1000. The offer is more than $10 or less, but not exactly $10, so $1000 is the only one that fits.
Okay, "I will receive neither $10 nor [my upper limit]" will guarantee me the upper limit? Why?
If we take this statement to be true, there will be a contradiction, as you've pointed out, and I'll receive $10, which will contradict my statement, making my statement an incorrect one or, in this case, a lie. So, getting $10 is not possible. So, to satisfy the falsity of my statement, I'll definitely receive my stated upper limit. But, wouldn't that make the first part of my statement correct? After all, I didn't receive $10. So, technically, this statement is partly correct and partly incorrect.
Now, you could say that my upper limit contains $10, which satisfies both the parts, by making the whole statement completely false. But, if my upper limit contains $10, doesn't it contradict my statement that I won't receive $10?
Going by this logic, wouldn't it be much better to say, "I will receive an amount lying between $0 to [my upper limit], except $10"? The "except $10" won't be invoked because, my principal clause/statement has been determined to be false. Or, "I will receive an amount lying between either $0 to $9.99 or 10.01 to [my upper limit]" (assuming that the minimum amount you can receive is $0.01). Or, "I will receive an amount that lies between $0 to [my upper limit], but, if the number amount lies between $0 and [my upper limit], it won't be 10." I think these are better choices because, they have a condition in it. I will get an amount that lies between $0 to [my upper limit] but, it won't be 10. The first two statements still leave a few gaps. But, the last statement makes it clear, that if the amount lies between $0 to [my upper limit] it can't and won't be $10. We can also add an "if" clause to the beginning of the statements and change our condition accordingly.
Let's test its truth. If the last statement is true, then I'll get $10, which isn't possible, because it contradicts my statement. So, to make this whole statement false, I'll definitely receive an amount greater than my upper limit, which doesn't make the second part of my statement, that I can't and won't get $10, true, because, I've set a condition that only if the amount lies between $0 and [my upper limit], it can't be 10. If the amount doesn't lie between the said numbers, then the need to invoke the second part of the second clause/condition of my statement doesn't arise. The 2nd statement is also good, because you don't say anything about the amount not being $10. But, if the statement is true, then you'll get $10, but $10 isn't there in your statement, thus making it false, or a lie. But, if your statement is a lie, and the amount lies anywhere between $0-9.99 and 10.01-[upper limit] then your statement will again become true, which will result in you receiving $10, which makes your statement a lie, thus making it necessary for the amount to lie above [upper limit]. Please, do tell me if you find my logic flawed. By the way, the $0 and the [upper limit] are inclusive, so you can't receive either of those two.
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You could have saved some writing by stopping at your second paragraph. A statement is true or false, unless there's a paradox; it can't be mixed. True, if I get $1 billion, I don't receive $10, but I do get $1 billion, and my statement was that I'd get NEITHER. If that is false, I get $10 or $1 billion. If true, then I don't get $10, I don't get $1 billion, I just get nothing or something else.
If you prefer, you can reword the statement. "The amount of money I receive in dollars will not be either of the numbers 10 or a billion." Better? Or, "What I receive will be something different from both $10 and $1 billion."
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True, and that would give rise to, as you said, a paradox. That's what I tried to point out. The answer is a paradox. If I say I'll get something (other than $10), and I get something, it'll make my answer true, and I should receive $10. But, that's going to render my statement incorrect, and as given, I'll get less or more than $10. The answer in itself gives rise to a paradox. Like, if Pinocchio says, "my nose will grow". It is either a paradoxical statement or an incomplete one. You can say option 1 fits better but, the answer provided by the one who's put forward the question is in itself a paradox. Oh, yes => If you prefer, you can reword the statement. "The amount of money I receive in dollars will not be either of the numbers 10 or a billion." If this statement is true, then the person shall get $10, which contradicts the condition given. But, if it's taken to be false, then he can receive less than $10, or more than any of the two numbers provided, which would make his statement right. So, paradox. This answer renders both the possibilities impossible. Like me going back in time and shooting my grandfather, and maintaining this timeline. Impossible.
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@Sarthak Rout – Not a paradox at all. If the statement is false, then he gets more than $10, or less, but not exactly $10. But the statement said he would get neither exactly $10, nor exactly $1 billion, so if it's false, he does get one of those two. The only one that is isn't exactly $10 is a billion, so he gets that.
True? If not, please explain.
P.S. I know what a paradox is, already.
To Challenge Master: What would happen if the wise man said, "My statement is false."?
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The universe would cave in.
If you ask me, it would have been better if a false statement would gave gained some money, just not exactly $10. That way, a paradox gets nothing.
Great question and great solution. :-)
The statement would be 'you will throw me off the cliff'
It depend on the giver how much you get.
If I am the giver the cheap solution to #1 is to always give you 0.01 $ if your statement is false and 10 $ if you say " You will give me some money", since there no demand of a random amount of $ and to #2 always will be bigger than 9.99 $
Given that #2 is the better choice.
"I will be thrown off the cliff".
logically a statement is the one which is EITHER TRUE OR FALSE. “I will receive neither $10 nor $1000.” is not a statement. http://en.wikipedia.org/wiki/Statement_%28logic%29
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Actually, "I will receive neither $10 nor $1000" is a statement - it's the conjunction of the negation of the statements, "I will receive $10" and "I will receive $1000." Thus, if one or both of those statements are true (i.e. if I get $10, $1000, or somehow $10 and $1000) then the statement "I will receive neither $10 nor $1000" is entirely false. On the other hand, if both those statements are false (i.e. if I get any sum of money other than $10 or $1000, such as $11) then the statement is true.
I do not know that I might obtain a larger amount in #1 than in #2. How do I know that the amount of money I will recieve more than $10 would not be the similar for both option 1(in case of wrong answer) and option 2(in case of right or wrong)? It is possible that I would recieve $20 on making a false statement for option 1, and it might also be true if option 2 is stated either rightly or wrongly. But for option 1, there is a possibility of recieving, say $5 for a wrong answer, which doesn't exist for option 2. Therefore I believe option 2 would be the right choice.
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It depends on the statement you make in case 1. Look at the solution.
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Yes but how do I know that I can use this statement at the first sight of this question? How is it obvious? Isn't it easier to guess that the extra reward to be obtained from both cases is equal? And how do I know that I cannot use this same statement for case 2?
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@Priontu Chowdhury – In case 2, it doesn't matter at all what statement you make. It solely depends on your 'luck'. Of course, its not obvious that there can be such statement. I think that's what makes it a logic problem. It doesn't even seems i i is real, but it is. And you have to accept it.
If you take Option 1, there is a possibility that you may get exactly $10, or less than $10. With Option 2, you always get MORE than $10. Therefore, Option 2 is the better option, because it does not allow for less than $10.
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Wrong. You take Option 1 and make a statement that leaves nothing to chance, like "I will receive neither $10 nor $10 million." This statement guarantees $10 million, whereas Option 2 has a max possibility of $1 million (formerly no guarantee of anything).
I won't be eaten by the Lions.
This is great except where does it say that you will get the amount of money that you have stated you must receive? There isn't any logic to this unless more information is given. To get to the answer you have provided you must make assumptions that are not provided for in the problem.
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It's a consequence of the rules. You get the billion (or however much) because that's the only possibility that doesn't result in contradiction. (We're assuming the one making the offer isn't lying.)
I like the question, but I think you should clarify that this is a riddle and not a math logic question since what you give as an example is not a statement (in math). One should be very carefull with circular reference.
"Proof" that your statement is not a statement:
Let S
1: I will receive neither $10 nor $1000
and S
2: I will receive neither $10 nor $1000000
Since if S
1 and S
2 are statement so must
S
1 or S
2
but that is a paradox
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Hi, what is the paradox, please?
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If S1 or S2 is true then you will get 10 dollars hence S1 is false, and S2 is false hence S1 or S2 is false.
If S1 or S1 is false then you will get x != 10 dollars, hence either S1 is true or S2 is true hence S1 or S2 is true
So the paradox is the following is "true" (S1 or S2) <=> not (S1 or S2)
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@Niklas Hjuler – I'm not sure I understand that. There is no paradox, because if you utter either statement, it becomes false because of what it is. The other is true, simply because you never said it. All that matters is the statement you utter, as the offer states. Make either statement and it becomes false and the other true.
Take Raymond Smullyan's logic puzzle which goes, "There is a yes/no question that you yourself cannot answer correctly, even though everyone else can. What is it?" An answer that works is, "Will you answer no to this question?" Yes and no are both false answers to this question if you give it, but any observer can take that information and give the correct answer, simply because he is not the one being asked. The question doesn't change, only the circumstances.
oh my god... the amount of slow witted people in this thread is astounding. let's just put it this way... if you're a total bonehead and can't figure out how to turn a situation to your advantage, then #2 is right, because being the bonehead that you are, you will have no idea what statement to make, so you'll say anything, like.. "my name is earl" and walk away with an amount greater than 10$, while someone who is even dumber than you are would say "my name is earl" to #1 and risk getting an amount less than 10$.
but for those of us who are smart.. we would say the statement described in the solution and 100% guarantee an amount much much larger than 10$. No, the formulation of #1 does not explicitly state that you can claim how much money you want... but a smart person can figure out how to make a statement that would make the only possible outcome a very positive one (any amount he wishes), while another bonehead picks option 2, says "my name is earl" and walks away with probably 17$ or something. good choice.. idiot
Your answer is wrong because the question says nothing, nor implies anything about getting what you say in your statement. It doesn't even say the statement has to be about money. The question says if I make the statement "my name is Shawn" I get $10 under option 1 and between 10 and 1000000 under option 2. I'll take option 2 all day.
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True enough, but you can manipulate the system with option 1. Say, "I will get neither $10 nor $10 million," and the statement is false; because if it is true, then you get neither $10 nor $10 million, so you don't get $10, but you DO get $10 because it's a true statement, so you do and don't, a contradiction.
Therefore, the statement is false, and you DO get either $10 or $10 million. And a false statement gets either less than $10 or more, but not exactly $10 - and $10 million is the only option that fits the criterion. Thus, you get $10M.
Nothing in the statements implies your statement is necessarily about money. Option 2 will always get you more than $10, while option 1 could get you less than or equal to $10. Therefore 2 is better.
You yourself don't know the exact answer but you posted this question and gave wrong option as answer to it.
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He gave the right solution. What exactly is wrong with it?
The expected winnings from statement 2 is $500,005 so $1000 wouldn't beat it.
The problem with this type of question is that it's a question of logic that relates to the real world where all sorts of other factors come in to play. For example, there aren't an infinite number of dollars in existence so there must be an upper limit on what you can be paid back. If you answered “I will receive neither $10 nor $1000000000000000000.” well there aren't that many dollars in existence so what would happen? The problem falls apart - there is no guarantee! How much money does the person posing the question have? If he were a normal person they wouldn't have millions of dollars to just give away. These are all factors normal people will consider when being asked this type of problem. You can probably tell from my comment I got it wrong.
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The thing is, this is just a math-and-logic question, not a real world question. There was a similar problem about a chess board once. There was to be one grain of wheat on the first square, two on the second, four on the third, eight on the fourth, sixteen on the fifth, and so on, doubling each time. In theory, the full board would need more than 18 quintillion grains of wheat. In practice, so one version of the story goes, the king could find no way out other than the subject's execution.
Similarly, this problem has nothing to do with the real world. You are to suppose that the one offering has vastly large amounts of money at his disposal, maybe trillions of dollars, maybe more. Maybe even the power to offer some as credit. Maybe these are all pesos instead of dollars. Who knows, who cares.
The answer is # 1 because in # 2 the range you will get regardless of the falsity of the statement you are to make is [ 1 0 , 1 0 0 0 0 0 0 ] . However, in # 1 the range is [ 0 , 1 0 ] ∪ + ∞ hence there is a larger probability that you will get more than $ 1 0 .
Make the statement "I will receive less than $10", this will cause a paradox unless they give you more than $10 (which is better than option 2).
If the statement is true then they will give you $10 making the statement false. If the statement is false then if they give you less then $10 it would make the statement true. Therefore the only solution is to give you more than $10 in order to fulfil their promise.
Wrong. Option 2 was that you would get more than $10 no matter what. The statement you are to make is a red herring.
Choose 1. All you have to do is make any false statement, then by the rules of the number 1, you will receive some random number of dollars from 0 to infinite dollars. By the nature of infinity, you must receive infinite money. Think of it like this: What proportion of all real numbers are larger than $1000? What proportion of all real numbers are larger than $1,000,000,000,000,000,000,000? All of them, really, 99.9999999...% forever recurring. That's 100%. The amount of money you receive is guaranteed to be larger than any number you can name. ie you receive infinite money. It doesn't even matter what your statement is.
Wrong. Yes, you should choose 1, but the amount of money wasn't given to be random, just something other than exactly $10.
The statement is the answer itself, so you should give the wrong one to get more money, so you give the number 1, which is the wrong choice😉
Dan Stols , pardon me to use your solution as a bump.
Question master. I would like you to clarify a situation.
'what is a statement?'
I find that if we combine 2 statement into 1 statement through conjunction such as 'and' or 'nor' . Both offer becomes equal because we can include one statement that defines and control the $10.
Example in offer 2 : 'I will not receive $10 and I will receive $1000'
TRUE : I get $1000
I did not receive $10 = TRUE , I received $1000 = TRUE , I receive $10++ = TRUE
.
FALSE : I get $10 (CONTRADICTION) , because
I will not receive $10 and I will receive $1000 = FALSE ,
.
therefore , ' I will receive $10 and I will not receive $1000 '
But since i must receive $10++ , the statement cannot be false. because it CONTRADICTS , so It must be true.
( thus , offer 1 and offer 2 becomes equally capable IF AND ONLY IF we are allowed to use conjunction )
How should this situation be clarified?
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I don't see how this affects the offer in option 2 though. Despite the outcome you are not guaranteed a specific sum, only that it will be $10+.
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Yes, and I would just pay $11, which is not $1000.
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@Whitney Clark – This is perfectly reasonable, assuming that the statement you give must be classified as true or false. If you choose option 2 and give a statement that can only be classified as true without contradiction, then you're away, just as with option 1.
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@Joel Toms – Exactly!! This question really taught me what people said when they say business corporate controls government through a contract paper
'that is so-called poorly written'
@Whitney Clark – which is why i posted to ask clarification....
----------'I will not receive $10 and I will receive $1000'----------
the statement composes of 2 statements using conjunction 'and' , in order for both to be true , i need to receive $1000
if i receive other than $1000 let say $11 , the statement first half is true , 2nd half is false.
in this case we consider whole statement to be FALSE.
and therefore the outcome will be
----------'I will receive $10 and I will not receive $1000'----------
CONTRADICTION occurs , i must receive $10 , how can i receive $11?
note 1 : the question did not specify the usage of english conjunction when making a statement.
note 2 : the definition of false is not mentioned. if false = invalid , which means ignore the whole statement and consider it never exists , then your $11 would fit the image.
Otherwise we would just flip around the whole sentence as in antonym and synonym.
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@Ne-ko Nya – "False" is the opposite of "true". And why "must" you receive $10?
The opposite of 'I will not receive $10 and I will receive $1000' is "I will receive $10 or I will not receive $1000." The opposite of "I am at Antarctica" is "I am not at Antarctica" (possibly the moon), not "I am at the Arctic."
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@Whitney Clark – this is what i mean by note 1 : usage of english is unrestricted.
TRUE : I have a (left hand) and a (right hand).
FALSE1 : I do not have a (left hand) AND i do not have a (right hand).
FALSE 2 : I do not have a (left hand) OR i do not have a (right hand).
In order for TRUE statement with conjunction 'AND' to be false , all it takes is EITHER ONE or BOTH statement to be false.
I chose the FALSE1 version of writing , and you mentioned the FALSE2 version.
and yes , i agree with you that if FALSE2 version of writing is picked for offer 2 i will get $11 and the best offer will be offer 1.
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@Ne-ko Nya – What on earth are you talking about? There is only one opposite to a conjunction, and that is the second one. If, "I am going to the party and John is, too," is a false statement, then either I am not going to the party or John isn't. Right? That's just elementary logic. What else would you call it, if I made such a statement, and then showed up without John?
Or if I told you, "Your friend survived the crash AND you inherited $1 million from your uncle," and you get the money but your friend died, then I LIED. Did I not? That was a single compound statement, not two statements spoken at once.
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@Whitney Clark – "I am going to the party and John is, too,"
so , you are saying the false version of this is
'Only one of us will attend the party or both of us will not attend the party'
I will not receive $10 and I will receive $1000
false : (i will receive 10 or i will not receive 1000) or (i will not receive 10 and i will receive 1000)
if that is the case.....yeap i will end up with $11 , offer 2 is a shit.
btw , 2 or more statement spoken at once = compound statement isnt it?(according to google) as long as it is connected with proper conjunction and sounds logical.
'I have a father and i have a mother' (accepted in maths not in english due to a lot redundancy)
'I have a father and a mother' (accepted in maths and english)
'I am inhaling and exhaling' (not accepted in maths and english but grammatically correct)
The opposite of (1) 'I will not receive $10 and I will receive $1000' is (2) "I either will receive $10 or I will not receive $1000," because the opposite of a conjunction is a disjunction . Either (1) or (2) must be true, but both CANNOT be true. It's like the opposite of "James and John will come to my party" is "Either James or John will be absent from my party." If one is true, the other must be false and vice versa.
Very Well Said! This one was painful.
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apparently it doesnt , Whitney Clark clarified it that offer 2 is a shit.
In compound statement such as A+B to be clarify as false , there are 3 ways.
A false B true .... A true B false ...... A and B false
TRUE : I (will not receive $10) and I (will receive $1000)
FALSE1 : I (will receive $10) AND i (will not receive $1000). (CONTRADICTION)
FALSE2 : I (will receive $10) OR i (will not receive $1000) . (ACCEPTABLE)
ACTUAL FALSE STATEMENT( FALSE1 + FALSE2)
[I will receive $10 OR i will not receive $1000 ] or {I will receive $10 AND i will not receive $1000}
*yes , the false statement MUST INCLUDE all possible false events , which means in this case 3 false events. ( if we do not do it this way , we cannot translate it into mathematical notations such as the set and venn diagram )
Mathematical notations ---> True : { A U B } .... False : { A' U B , A U B' , A' U B' }
Therefore i will get $11 if the statement is false and $1000 if true, Therefore option 2 is shit. This will not occur in option 1.
Write a comment or ask a question...2nd option is much better than 1st one if you consider it in terms of money we can make..as 1st one is risky and 2nd o ne is a sure shot.....a class one child can tell the answer
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Wrong. If you say, "I will get neither $10 nor $10,000,000," for option 1, then you will get $10,000,000, which is much better than $1,000,000. It's the only logical possibility; everything else leads to contradictions.
Nothing in the problem statement identifies the distribution of the payout. Giving $0 for every false answer satisfies the conditions of the problem, yet #1 is supposed to be the better solution?
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Wrong. If you say, "I will get neither $10 nor $10,000,000," for option 1, then you cannot get $0, because that would make your statement true, and for a true statement you are promised exactly $10.
You should take offer #1, because you can guarantee yourself an arbitrarily large amount of money, simply by using the following as your statement:
“I will receive neither $10 nor $1000.”
If we assume the statement is true, this leads to a contradiction because you would receive $10, by the rules.
If we assume the statement is false, then to satisfy the falsity of the statement, you must receive either $10 or $1000. But since you cannot receive $10 if the statement is false, you must receive $1000.
Of course you can use any amount in place of “$1000.”