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beercanman
Joined: 16 May 2009
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 Posted: Sat Nov 28, 2009 5:42 am Post subject: A little logic problem |
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(stolen from China off-topic forum)
GIVEN: The number of dogs in the world is greater than the number of hairs on any one dog.
PROVE: That there are (or or not) two dogs with the same number of hairs. |
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tfunk
Joined: 12 Aug 2006
Location: Dublin, Ireland
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| Posted: Sat Nov 28, 2009 6:00 am Post subject: Re: A little logic problem |
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| beercanman wrote: |
(stolen from China off-topic forum)
GIVEN: The number of dogs in the world is greater than the number of hairs on any one dog.
PROVE: That there are (or or not) two dogs with the same number of hairs. |
You can't prove the second statement with the first claim. Unless 'the same number of hairs' refers to 'the number of hairs on any one dog', in which case the second statement must be true if the first claim is true. |
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Fox
Joined: 04 Mar 2009
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| Posted: Sat Nov 28, 2009 6:51 am Post subject: Re: A little logic problem |
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| beercanman wrote: |
(stolen from China off-topic forum)
GIVEN: The number of dogs in the world is greater than the number of hairs on any one dog.
PROVE: That there are (or or not) two dogs with the same number of hairs. |
I assume you casual, persuasive logic rather than formal logic.
We have X number of possible hair counts, where a hair count is a number of hairs spread over the entire body.
We have Y number of dogs, and each dog has some hair count as described above.
Because Y is greater than X, some dogs must share a hair count; there aren't enough possible hair counts for every dog to have a unique number of hairs, because the number of dogs outweighs the number of possible hair counts.
Therefore, at least two dogs have the same number of hairs. |
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guava
Joined: 02 Sep 2009
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| Posted: Sat Nov 28, 2009 6:57 am Post subject: |
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| The only dogs that must have the same number of hairs are Mexican Hairless. |
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tfunk
Joined: 12 Aug 2006
Location: Dublin, Ireland
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| Posted: Sat Nov 28, 2009 7:02 am Post subject: Re: A little logic problem |
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| Fox wrote: |
Therefore, at least two dogs have the same number of hairs. |
Very cunning, fox! Seriously, smart thinking (I'm talking about the main part of your reasoning, not the part I quoted).
Three dogs could have the same number of hairs, by your reasoning.
Edit: Wait a minute! If three dogs have the same number of hairs then the statement 'two dogs have the same number of hairs' is still true! Right? Fox, you're a genius (or else I'm an idiot). |
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caribmon
Joined: 26 Oct 2009
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| Posted: Sat Nov 28, 2009 7:19 am Post subject: |
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The probability distribution is not unique hair counts in a row, it is absolute hair counts with no order whatsoever. You won't have 1 dog with 1 hair, 1 dog with 2 hairs, 1 dog with 3 hairs, etc. So even though Y is greater than X, it does not guarantee that 2 dogs will have the same hair count. Guava is right, is what I'm saying.
My answer is simple. Yes, there are 2 or more dogs with the same number of hairs. You did not specify alive or dead dogs. Any dead dog cooked over a barbecue has no hair and therefore at any given time, provided there are more than 1 dog(s) being eaten, can be deemed to have the same number of hairs as another dead dog being cooked over the barbecue. I don't eat dogs. That's whack.
Similarly, who cares about hairs on a dog, counting them and comparing to how many dogs there are. I cut my own hair tonight with kitchen shears and no mirror to see the back of my head, so I hope it looks good, probably uneven though as I was cutting by feel (which is what I've done for years actually). I cut my finger with the scissors though so I had a huge mess of hair everywhere and my finger was bleeding and I used those 50 cent razors to do the neck hair which caused a huge rash all over my neck. Man what a mess. |
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tfunk
Joined: 12 Aug 2006
Location: Dublin, Ireland
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| Posted: Sat Nov 28, 2009 7:34 am Post subject: |
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| caribmon wrote: |
My answer is simple. Yes, there are 2 or more dogs with the same number of hairs. You did not specify alive or dead dogs. Any dead dog cooked over a barbecue has no hair and therefore at any given time, provided there are more than 1 dog(s) being eaten, can be deemed to have the same number of hairs as another dead dog being cooked over the barbecue. |
That's not proof, that's speculation. The dogs could have been Mexican hairless dogs, the dogs could have been dead. Sure, they also could have all had the same number of hairs. |
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ChopChaeJoe
Joined: 05 Mar 2006
Location: Seoul
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| Posted: Sat Nov 28, 2009 9:36 am Post subject: |
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| Seems pretty obvious that 2 have the same number of hairs. look at the opposite case. If you set the number of hairs on the dog with the most hairs as x and the number of dogs as y, since y > x then for each z <= x we can assign at most one dog, but then we run out of numbers! So at some point we must double up. |
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beercanman
Joined: 16 May 2009
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| Posted: Sat Nov 28, 2009 3:51 pm Post subject: |
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| With ten dogs, the number of hairs could range from 0 to 9. |
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VanIslander
Joined: 18 Aug 2003
Location: Geoje, Hadong, Tongyeong,... now in a small coastal island town outside Gyeongsangnamdo!
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| Posted: Sat Nov 28, 2009 11:53 pm Post subject: |
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| guava wrote: |
| The only dogs that must have the same number of hairs are Mexican Hairless. |
Indeed.
There is no logical entailment unless one makes the dubious assumption that hairless dogs don't exist. |
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guava
Joined: 02 Sep 2009
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| Posted: Sun Nov 29, 2009 1:17 am Post subject: |
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| caribmon wrote: |
Guava is right, is what I'm saying.
I cut my own hair tonight with kitchen shears and no mirror to see the back of my head, so I hope it looks good, probably uneven though as I was cutting by feel (which is what I've done for years actually). I cut my finger with the scissors though so I had a huge mess of hair everywhere and my finger was bleeding and I used those 50 cent razors to do the neck hair which caused a huge rash all over my neck. Man what a mess. |
I'll trim your back if you'll trim mine. I got clippers. Actually, I still got an old clippers with no comb attachment. You can have it if you want it. |
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Goku
Joined: 10 Dec 2008
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| Posted: Sun Nov 29, 2009 3:47 pm Post subject: |
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I think you would have to first prove dogs exist.
Because essentially the given is only a rule about the number of dogs. But there's not statement actually saying dogs exist.
plus in the "prove", it says: "That there are (or or not)....
the parenthetical statement makes the possibility that there are no dogs in the world. all the more likely.
Maybe I'm being TOO formal logical. |
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aboxofchocolates
Joined: 21 Mar 2008
Location: on your mind
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| Posted: Sun Nov 29, 2009 4:18 pm Post subject: |
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| Ak! beercanman, give me the answer before my brain melts! |
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beercanman
Joined: 16 May 2009
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| Posted: Sun Nov 29, 2009 4:38 pm Post subject: |
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There seems to be disagreement on the answer. On the China off-topic forum one guy says he uses the "pigeonhole principle" - if you have more pigeons than holes than at least one hole must have at least 2 birds. I figured since the question didn't say all dogs must have hair, then one dog can be bald. So the number of hairs can range from 0 to one less than the number of dogs, meaning two dogs need not have the same number of hairs, yet still fitting the question's needs. If all dogs must have at least one hair, then at least two will have the same number of hairs.
All cleared up? |
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Fox
Joined: 04 Mar 2009
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| Posted: Sun Nov 29, 2009 4:43 pm Post subject: |
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| beercanman wrote: |
There seems to be disagreement on the answer. On the China off-topic forum one guy says he uses the "pigeonhole principle" - if you have more pigeons than holes than at least one hole must have at least 2 birds. I figured since the question didn't say all dogs must have hair, then one dog can be bald. So the number of hairs can range from 0 to one less than the number of dogs, meaning two dogs need not have the same number of hairs, yet still fitting the question's needs. If all dogs must have at least one hair, then at least two will have the same number of hairs.
All cleared up? |
My answer is more or less the same as the "pigeonhole principle." However, I admit that I did assume in my answer that all dogs would have hair. If we grant that dogs can have no hair, I feel your answer is correct, and with just the given information we can't prove anything. |
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