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Puzzles and Paradoxes


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On 20/09/2020 at 11:48, HawkMan said:

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?

 

I suspect the answer that the horse is called Thursday is the one you were looking for.

However, we have no way of knowing that you have told us the full story, so an option would be to deduce that, having opened up his office, the sheriff closes it later the same day and rides out of town, returning on Saturday.  He then stays for one night on his office bed and leaves on Sunday.

I suppose what my option demonstrates is the wisdom of the witness oath to "tell the truth, the whole truth, and nothing but the truth." It also reminds us that the modern euphemism, 'to be economical with the truth', was not first used to mean 'to lie', which is how the phrase is often used nowadays, but rather 'to mislead by not telling the whole story.'

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Suppose you want to send in the mail a valuable object to a friend. You have a box which is big enough to hold the object. The box has a locking ring which is large enough to have a lock attached and you have several locks with keys. However, your friend does not have the key to any lock that you have. You cannot send the key, your friend cannot send you a key as there is too much risk of these being lost, stolen or copied. How do you send the valuable object, locked, to your friend - so it may be opened by your friend?

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

Suppose you want to send in the mail a valuable object to a friend. You have a box which is big enough to hold the object. The box has a locking ring which is large enough to have a lock attached and you have several locks with keys. However, your friend does not have the key to any lock that you have. You cannot send the key in an unlocked box since it may be stolen or copied. How do you send the valuable object, locked, to your friend - so it may be opened by your friend?

Get your friend to send you a lock. 

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2 minutes ago, bobbruce said:

Get your friend to send you a lock. 

Or send your friend separately the key for the lock you will be using.

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3 minutes ago, The Hallucinating Goose said:

You can't send the key seperate, says so in the riddle. 

That's a moot point, HG.  It says that you cannot send it in an unlocked box, but that wouldn't preclude sending it in an envelope which is, by definition, not a box!  I used to wonder about having a career as a lawyer...

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4 minutes ago, Wiltshire Warrior Dragon said:

That's a moot point, HG.  It says that you cannot send it in an unlocked box, but that wouldn't preclude sending it in an envelope which is, by definition, not a box!  I used to wonder about having a career as a lawyer...

Alright, let me rephrase that, you can't send the key in any way. 

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You place the object in the box and lock it with a lock that your friend can't open.  Send it to your friend,  who can't open it, but puts another lock on it, and sends it back. When getting it back you remove your lock and send it back again. It will then have only your friend's lock on it. 

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16 minutes ago, HawkMan said:

You place the object in the box and lock it with a lock that your friend can't open.  Send it to your friend,  who can't open it, but puts another lock on it, and sends it back. When getting it back you remove your lock and send it back again. It will then have only your friend's lock on it. 

Correct! 

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14 minutes ago, The Hallucinating Goose said:

Along the edge of a running track there are 12 flags. The flags have equal space between them. A runner runs from the 1st to the 12th flag at the same speed. The runner takes 8 seconds to run from the 1st flag to the 8th flag. How long does it take to run from the 1st to the 12th flag? 

Is it 12.5 seconds? 1st to 8th is 7 flags in 8 seconds is I.142 per flag times 11 .However the speed he ran from start to first isn't stated.

Edited by HawkMan
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35 minutes ago, HawkMan said:

Is it 12.5 seconds? 1st to 8th is 7 flags in 8 seconds is I.142 per flag times 11 .However the speed he ran from start to first isn't stated.

I think you are right, HM.  To be fair to HG, he said the runner runs from the first flag, so even if he ran to get to the first flag, I don't think that is relevant.  

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40 minutes ago, HawkMan said:

Is it 12.5 seconds? 1st to 8th is 7 flags in 8 seconds is I.142 per flag times 11 .However the speed he ran from start to first isn't stated.

That is right and just how you worked it out, there are 12 flags but 11 segments between them so 7 sections between the 1st and 8th, so 8 seconds divided by 7 sections to give you the speed between 1 and then times that by the 11 sections.

The speed ran is not needed because its stated that the speed is the same, the runner does not gain or lose speed along the run. 

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5 minutes ago, Wiltshire Warrior Dragon said:

I think you are right, HM.  To be fair to HG, he said the runner runs from the first flag, so even if he ran to get to the first flag, I don't think that is relevant.  

Correct.  The trick of the puzzle is confusing first flag with start 

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31 minutes ago, The Hallucinating Goose said:

A factory uses metal sheets to make their products. 1 sheet is sufficient to make 1 product. The remains of 6 sheets can be melted down and made into another sheet. How many products can the factory make from 36 sheets of metal? 

43. 36 sheets make 36 products. 36 bits left over make another 6 sheets and 6more  products  , and 6 left over bits make another sheet to make 36 +6+1=43.

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On 21/09/2020 at 10:06, HawkMan said:

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 ?

I've been having a bit of a think about this since I haven't been able to sleep tonight.

I don't know if I've actually worked the answer out but the theory I've been going over is, an infinite number of balls are in the room. If the time between balls being thrown out of the room keeps halving then you will never actually reach noon will you? Time periods will just go on halving forever and so there will be infinite time periods. 

This idea of the time periods being so tiny makes me wonder as well, does this mean that the ping pong balls exist in two places at once? In other words both in and out of the room. If we apply our understanding of speed to the situation, as the time periods are getting shorter, the balls are being thrown out of the room faster and faster but there is no way that a person throwing balls into the room could keep up with the pace which means that balls would be coming out of the room quicker than they are going in. You would think from that that the flow of balls coming out would have to stop because you aren't supplying them quick enough as I say but if indeed the ball leaves the room before it's been thrown in that means it has to already exist in the room as well as in the thrower's hand. 

Always enjoy a bit of philosophical thought at 3 in the morning! 😊

Edited by The Hallucinating Goose
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So the hare and tortoise puzzle. If I use a bit of reasoning from my above post, the gaps between the checkpoints are getting smaller, let's say halving like the time in the puzzle above. That means then that between each checkpoint, the hare has to run twice the distance of the tortoise. So if the hare was twice as fast as the tortoise, it would still never catch up because they would both be reaching the next checkpoint at the same time. Something like that. 

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Both the hare and balls puzzle play on people's misconceptions about infinity. Infinity isn't just trillions and trillions and trillions,  infinity is an abstract concept like zero. You can't have half of zero, or different size infinities. 

So, the balls; there are three possible answers,  all plausible. 1. All the balls are thrown in and out before noon,  so noon is actually outside the scope of he puzzle,  so how many at noon is indeterminate.  However if you really could throw an INFINITE number of balls in, then either they'll be an infinite number in the room at noon or NONE AT ALL, because an infinite number would have been thrown out. Half the number are thrown out, but half of infinity is a meaningless concept. The set up of the puzzle itself is impossible. 

As for the hare, this deals with infinity too. It wasn't until the 18th century that mathematicians realised that the sum of an infinite series could be finite. Draw a line on a piece of paper,  that is a finite length,  say 12 inches,  and it can be divided infinitely many times.  A half plus a quarter,  plus an eighth plus a sixteenth etc to infinity still equals one.

So the hare completes the task of reaching an infinite number of points between him and the tortoise in a finite time and overtake him.

This is the book I got some of these from recommended .

 

51he6CM0BAL.jpg

Edited by HawkMan
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A decades long lurker on this site I'd like to say thanks for starting this thread, I'm really enjoying excercising the old grey matter on some of these. 👍

However, I would respectfully disagree with your assertion that "You can't have .... different size infinities"

For example: the set of all integers and the set of even integers are both infinite. But for every member of the second set there are two members of the first set. So both sets are infinite in size  - but one is twice as big as the other 🤪

I don't suggest thinking about this stuff too much though as it is known to drive people mad (  https://www.bbvaopenmind.com/en/science/mathematics/georg-cantor-the-man-who-discovered-different-infinities/ )

Blimey - I never expected to be discussing Cantor's set theory on a rugby league website! Just goes to show we really are the superior code 😉
 

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On 23/09/2020 at 12:29, Don Quijote said:

A decades long lurker on this site I'd like to say thanks for starting this thread, I'm really enjoying excercising the old grey matter on some of these. 👍

However, I would respectfully disagree with your assertion that "You can't have .... different size infinities"

For example: the set of all integers and the set of even integers are both infinite. But for every member of the second set there are two members of the first set. So both sets are infinite in size  - but one is twice as big as the other 🤪

I don't suggest thinking about this stuff too much though as it is known to drive people mad (  https://www.bbvaopenmind.com/en/science/mathematics/georg-cantor-the-man-who-discovered-different-infinities/ )

Blimey - I never expected to be discussing Cantor's set theory on a rugby league website! Just goes to show we really are the superior code 😉
 

Without rereading Cantor's arguments,  which I did years ago, I'd say members of an infinite set are uncountable,  hence Infinite,  so one  infinite set having more members is contradictory. It isn't as if one set reaches infinite first, there is no X marks the spot, reach this and it's infinite . Anyway let's not start  on sets or we'll end up asking stuff like is the set of non self membered sets itself a member of it's own set. If it is a member of it's own set it's self membered so it isn't  , and if it isn't a member of it's own set it's non self membered therefore it is.🤪😜😠  No let's not go there. 

Edited by HawkMan
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Ship Of Theseus. 

Ancient Greek mariner Theseus has a ship  made entirely of wooden planks,  except sails etc. As each plank gets old he replaces it with a new plank. The old planks are stored in a warehouse. One by one he replaces the planks, all the time he has a sea worthy vessel. Eventually all the planks are replaced . Call this the Renovated Ship. Question is is this still the Ship Of Theseus, with no original plank. Now suppose he goes to the warehouse and rebuilds the ship using the original planks, call this the Reconstructed Ship. Which is the Ship of Theseus,  the Reconstructed or Renovated? The Reconstructed has the original planks,  the Renovated has a continuous timeline  of existence back to the original.  This is the puzzle of the Metaphysics of Identity. See vid for more.

 

 

Edited by HawkMan
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