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Puzzles and Paradoxes


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A thread for any logical puzzles or paradoxes you want to try out on others. Paradoxes are fun, if a bit philosophical.  Here's a few to start.

The Sheriff of Tombstone Arizona rides into town on thursday,  opens his office up, he has a bed in there and only stays in town for one night. But when he leaves it's Sunday.  How come?

 

Two guys are in a restaurant,  at the end of the meal the waiter gives them the bill, for £30. The internet is down,  so they have to pay cash. Each has exactly £15 only on them...luckily.  The waiter takes the money and hands it to the manager who is working the till. The manager examines the bill and says the bill is for £27 only, and instructs the waiter to.hand back £3. On his way back to the men he decides to pocket £1, and tell the men the bill cost £28. The two men leave with their £1 each. They both entered with £15, and left with £1 ,spending £14 each which equals £28, plus the £1 the waiter pilfered equals £29. Who has the missing pound ?

 

THE GOD /BOULDER PARADOX. 

God by definition can do anything,  nothing is beyond such a being. Can God create a boulder so heavy that He cannot lift it? If he can, then there's something he cannot do ie lift the boulder,  if he can't then there's something he cannot do . Either way the definition of God is in trouble. 

THE LAWYER/STUDENT PUZZLE 

A lawyer helps a student qualify to become a lawyer,  but generously waives his fee, for now. The student will pay the tuition fee when he has won his first case. Months go by and the student , now qualified,  hasn't taken up a single case. The Lawyer anxious for his fee sues the Student  , reasoning that if the court finds for him,he has won his case, and if he loses then the Student has won a case, either way the Student must pay up. The Student reasons that if the court finds for the tutor, he has lost and needn't pay, and if he wins then the tutor's case is thrown out and he still needn't pay. Who is right,  what should the court do?

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PARADOX OF THE RAVENS.

All ravens are black , another clumsy way of saying this is ' All non black things are non ravens ' These hypothesis say the same things. The equivalence rule in logic states that anything confirming one hypothesis automatically confirms an equivalent hypothesis.  My white pen confirms the second hypothesis,  ie it's not black or a raven, but surely it can't confirm that all ravens are black. The details of this paradox discussed here;

The Sleeping Beauty Puzzle. Not easy to get your head around this. After seeing this video, my guess is that credence and probability are being conflated. Beauty doesn't know what day it is, but there's two  tails awakening to one heads awakening so her credence should be 1 in 3.

 

Edited by HawkMan
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4 hours ago, HawkMan said:

A thread for any logical puzzles or paradoxes you want to try out on others. Paradoxes are fun, if a bit philosophical.  Here's a few to start.

The Sheriff of Tombstone Arizona rides into town on thursday,  opens his office up, he has a bed in there and only stays in town for one night. But when he leaves it's Sunday.  How come?

 

Two guys are in a restaurant,  at the end of the meal the waiter gives them the bill, for £30. The internet is down,  so they have to pay cash. Each has exactly £15 only on them...luckily.  The waiter takes the money and hands it to the manager who is working the till. The manager examines the bill and says the bill is for £27 only, and instructs the waiter to.hand back £3. On his way back to the men he decides to pocket £1, and tell the men the bill cost £28. The two men leave with their £1 each. They both entered with £15, and left with £1 ,spending £14 each which equals £28, plus the £1 the waiter pilfered equals £29. Who has the missing pound ?

 

 

The Sherrif rides a horse called Thursday.

The cost of the meal should have been £13.50  each.  So each should have been given £1.50 change.

Ron Banks

Midlands Hurricanes and Barrow

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Outside a room on the wall are three numbered light switches that operate three light bulbs inside the room. The door is solid with no window in it.
Is it possible to identify which switch operates each bulb ? You may only enter the room ONCE after doing whatever you want with the switches and you can only touch TWO of the three switches.

 

You are accompanying a lion and a goat on a journey , also you have a huge cabbage. You come to a river and see a rowing boat moored ready for use. However you can only take across one animal at a time, or indeed the cabbage. The boat is only big enough to carry you and one of the other three. You must get all three across one at a time, HOWEVER, if you leave the lion alone with the goat it will eat it, also if you leave the goat with the cabbage it will eat it. How can you get all three across safely, remembering that you have to be on board the boat each time you cross the river, either alone, or along with one animal or the cabbage. You can cross the river any number of times.

PRISONER HAT PUZZLE.

 

 

Edited by HawkMan
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15 hours ago, HawkMan said:

You are accompanying a lion and a goat on a journey , also you have a huge cabbage. You come to a river and see a rowing boat moored ready for use. However you can only take across one animal at a time, or indeed the cabbage. The boat is only big enough to carry you and one of the other three. You must get all three across one at a time, HOWEVER, if you leave the lion alone with the goat it will eat it, also if you leave the goat with the cabbage it will eat it. How can you get all three across safely, remembering that you have to be on board the boat each time you cross the river, either alone, or along with one animal or the cabbage. You can cross the river any number of times.

 

The Lion puzzle:

Take the goat across first. 

Go back and get the cabbage and take it across. 

Leave the cabbage and take the goat back to the start. 

Leave the goat at the start and take the lion across to the cabbage. 

Go back and get the goat. 

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21 hours ago, HawkMan said:

THE LAWYER/STUDENT PUZZLE 

A lawyer helps a student qualify to become a lawyer,  but generously waives his fee, for now. The student will pay the tuition fee when he has won his first case. Months go by and the student , now qualified,  hasn't taken up a single case. The Lawyer anxious for his fee sues the Student  , reasoning that if the court finds for him,he has won his case, and if he loses then the Student has won a case, either way the Student must pay up. The Student reasons that if the court finds for the tutor, he has lost and needn't pay, and if he wins then the tutor's case is thrown out and he still needn't pay. Who is right,  what should the court do?

The lawyer is right because the court's decision overrides any previous agreement made by the student or lawyer.

LAWYERS ARGUMENT

1. If the courts finds for the lawyer then the lawyer has successfully sued the student and gets paid. CORRECT. 

2. If the court finds for the student then the student has won a case and the lawyer gets paid. CORRECT. 

STUDENTS ARGUMENT

1. The Student reasons that if the court finds for the tutor, he has lost and needn't pay. WRONG. The court is telling him to pay and their decision overrides the contract and by finding for the tutor the court is acknowledging the existence of the contract. 

2. If the student wins then the tutor's case is thrown out and he still needn't pay. WRONG. By fighting the case in the first place, the student is acknowledging the legitimacy of the contract and so agrees that after winning his first case he will have to pay. 

-------------

I will point out though that the mistake made by the student is reasoning the case as he has done. By acknowledging the two points he puts forward he is agreeing with the lawyer to the terms of the contract. If he had disagreed to the terms of the contract he probably would have won the case AND voided the contract.

While a verbal agreement is considered binding in UK law it is very difficult to prove the terms of said contract without any physical evidence and so the court would have thrown out the case, meaning they would not be acknowledging the existence of the contract and as I said, the courts decision overrides the contract and the court is saying the contract does not exist (if the student just disagreed about it's terms but he hasn't). 

Edited by The Hallucinating Goose
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22 minutes ago, The Hallucinating Goose said:

The lawyer is right because the court's decision overrides any previous agreement made by the student or lawyer.

LAWYERS ARGUMENT

1. If the courts finds for the lawyer then the lawyer has successfully sued the student and gets paid. CORRECT. 

2. If the court finds for the student then the student has won a case and the lawyer gets paid. CORRECT. 

STUDENTS ARGUMENT

1. The Student reasons that if the court finds for the tutor, he has lost and needn't pay. WRONG. The court is telling him to pay and their decision overrides the contract and by finding for the tutor the court is acknowledging the existence of the contract. 

2. If the student wins then the tutor's case is thrown out and he still needn't pay. WRONG. By fighting the case in the first place, the student is acknowledging the legitimacy of the contract and so agrees that after winning his first case he will have to pay. 

-------------

I will point out though that the mistake made by the student is reasoning the case as he has done. By acknowledging the two points he puts forward he is agreeing with the lawyer to the terms of the contract. If he had disagreed to the terms of the contract he probably would have won the case AND voided the contract.

While a verbal agreement is considered binding in UK law it is very difficult to prove the terms of said contract without any physical evidence and so the court would have thrown out the case, meaning they would not be acknowledging the existence of the contract and as I said, the courts decision overrides the contract and the court is saying the contract does not exist (if the student just disagreed about it's terms but he hasn't). 

The simplest answer is that at the time the Lawyer instigated the lawsuit the Student hasn't won a case, so his case is thrown out. The Student has now won a case, and a second lawsuit instigated by the Lawyer will surely succeed. 

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Just now, The Hallucinating Goose said:

Not sure about the light switches but before you start pressing switches could you not just enter the room and leave the door open? 

No.

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The Lord Plato is guarding a bridge over a river when Socrates appears. Socrates begs to be allowed passage.
Plato bellows " If the next thing you say is true I will allow passage, but if the next thing you say is false I will throw you in the water, self referential statements like " I like potatoes,  or my name is Socrates don't count, nor do statements like  ,grass is green, it must be irrefutable provable and logically true"
Socrates being clever says;
" You will throw me in the water"
If Plato does indeed throw Socrates in the water, then Socrates spoke truly and shouldn't have been thrown in, but if he doesn't throw him in, Socrates spoke falsely and should have gone in!
Either way Plato is stuffed!!
But hang on, Plato cannot fulfil his promise, it's logically impossible, so he has no obligation to keep his promise, and assuming free will can throw Socrates in the river if he pleases.
It is argued that statements about future intentions cannot have a truth or false value. Socrates saying " You will throw me in the water " is neither true or false.
However Socrates could have said something avoiding future dependant language that would have avoided a dowsing in the river. What?

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8 minutes ago, HawkMan said:

No.

Okay, another idea. 

Press the first switch and leave it on for ages then turn it off again. 

Press the second switch. 

Go into the room. 

The bulb operated by the first switch will be hot because you left it on for ages. 

The bulb operated by the second switch will be on. 

The bulb operated by the third switch will be off and cold. 

 

 

Edit. Just posted that before you said furniture, don't know what furniture has to do with it. Climbing on a chair to reach the bulb or something? 

Edited by The Hallucinating Goose
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2 minutes ago, The Hallucinating Goose said:

Okay, another idea. 

Press the first switch and leave it on for ages then turn it off again. 

Press the second switch. 

Go into the room. 

The bulb operated by the first switch will be hot because you left it on for ages. 

The bulb operated by the second switch will be on. 

The bulb operated by the third switch will be off and cold. 

 

 

Edit. Just posted that before you said furniture, don't know what furniture has to do with it. Climbing on a chair to reach the bulb or something? 

Correct,  in every sense. 

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1 minute ago, HawkMan said:

Correct,  in every sense. 

I got that when I actually paid more attention to what you wrote. You said you can touch two switches and at first I thought that meant only press switches twice but you can press the two switches as many times as you want as long as its the same two. 

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One minute before noon , you have an INFINITE number of ping pong balls outside a room, all numbered . You throw in balls 1 +2, immediately ball 1 is thrown back out. Half a minute before noon 3+4 are thrown in and number 2 thrown out. At a quarter of a minute to noon balls 5+6 are thrown in and number 3 is thrown out. At an eighth of a minute to noon 7+8 in and 4 out  ad infinitum.... How many balls are in the room at Noon ?

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19 minutes ago, The Hallucinating Goose said:

Could Socrates have said something like, "you are guarding this bridge" to Plato? 

No, that is not logically true, in other words something that cannot be not true. Also mathematical statements are ruled out, such as 2+2=4. I know, Plato is such a grouch !

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1 hour ago, HawkMan said:

No, that is not logically true, in other words something that cannot be not true. Also mathematical statements are ruled out, such as 2+2=4. I know, Plato is such a grouch !

So something that can only be true and nothing else but true?

So what I said is wrong because there is a possibility that Plato is not guarding the bridge? 

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15 minutes ago, The Hallucinating Goose said:

Socrates says the word, "true" 

" true" isn't saying anything,  or could be construed as " I am saying true" which is self referential. Socrates can say something that even if its false will not put him in the river.

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42 minutes ago, HawkMan said:

" true" isn't saying anything,  or could be construed as " I am saying true" which is self referential. Socrates can say something that even if its false will not put him in the river.

He could repeat back to Plato what Plato stated, but reversing the roles of course. 

"If what I say is true you will allow me passage, but if what I say is false you will throw me in the water".

That way, if the statement is true then Socrates must be allowed passage but if the statement is false then the whole statement is false and so he won't be thrown in the water. 

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24 minutes ago, The Hallucinating Goose said:

He could repeat back to Plato what Plato stated, but reversing the roles of course. 

"If what I say is true you will allow me passage, but if what I say is false you will throw me in the water".

That way, if the statement is true then Socrates must be allowed passage but if the statement is false then the whole statement is false and so he won't be thrown in the water. 

 

Very impressive! Correct . If the statement is false then it's false he'll be thrown in .

 

The Tortoise and the Hare.

This puzzle is attributed to Zeno of ancient Athens. 

A tortoise and a hare are in a race. The hare gives the tortoise a head start. The race starts,  first the hare has to reach where the tortoise started,  call that point T1, the tortoise has now moved to T2, the hare reaches T2, but the tortoise is now at T3, and so on, each time the gap is smaller but the tortoise continues to T4,5,6 etc. The hare has to cover each of these points, and the tortoise has always moved on, so cannot be caught. So a faster runner cannot overtake a slower one. Something is very wrong with this reasoning,  but what.? Work this out and the numbered balls puzzle is worked out too. 

Edited by HawkMan
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3 minutes ago, HawkMan said:

Very impressive! Correct . If the statement is false then it's false he'll be thrown in .

Thank you! That really messed with my head! 😁

BTW, in case you're wondering, the reason one of my ideas was to say the word true is because truth itself can't be false because they are opposites.

But then when you said he could say something false I thought that was contradictory. What is contradictory? A paradox. What is a paradox in this case? What Plato said! 

Edited by The Hallucinating Goose
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