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# Baldyville

### #31

Posted 11 November 2007 - 01:57 AM

The rules

1. No two inhabitants have the same number of hairs on their head.

2. No inhabitant has exactly 518 hairs.

3. There are more inhabitants than any inhabitant's hair in the town.

In regards to the hair count having to be continuous:

No it isn't stated directly, but it is derived and is absolutely true. This is due to rules 3 and 1.

To understand this; start by picking any number to represent the number of hairs from the inhabitant with the most hairs. Lets use 10 to keep it simple.

Since this inhabitant has 10 hairs, there must be at least 11 inhabitants (rule 3).

And according to rule 1, the numbers for each of the inhabitants hair count must be unique.

We can now prove that there can only be 11 inhabitants and the number of hairs must be continuous.

If there are more than 11 inhabitants and the maximum number of hairs is 10 (value picked above) then there must be two with the same number of hairs (by the way: zero is a number, 7th grade math, maybe earlier) which violates rule 1. If you dispute this, I would like to know how you manage to get more than 11 numbers from 0 to 10 without duplicating a number. (Counting implies integers, preschool math or thereabouts). We therefor know that if the person with the most hairs has 10 hairs, there must be 11 people, no more (just shown), no less (rule 3).

Since we know that if the person with the most hairs has 10 hairs, there are 11 people (above). If there are 11 people and the max number of hairs is 10 (as stated), there can't be any gaps in the sequence. Again, I don't know how you can get 11 numbers between 0 and 10 inclusive and have a gap.

Now, if there is a number of hairs that is not allowed to exist, the sequence must stop since there can't be any gaps. Therefore if there can't be anyone with 518 hairs, there can't be anyone with more either. This means that there can be any number of inhabitants less that 518 except the problem is asking for the maximum number which is 518.

If this, along with the previous explanations, doesn't prove it, I must condeed that there are those who refuse to see the forest for the trees. i.e. too fixated on what they want rather than what is.

### #32

Posted 16 November 2007 - 04:52 PM

### #33

Posted 16 November 2007 - 07:49 PM

1. No two inhabitants have the same number of hairs on their head.

2. No inhabitant has exactly 518 hairs.

3. There are more inhabitants than any inhabitant's hair in the town.

What is the highest possible number of inhabitants?

There is no rule stating that there the people in the town have specific numbers on their head it only says they don't have the same number of hairs as any one else. it also says there are more inhabitants that any inhabitants hair. that means there can be 530 people in the town as long as one person is bald no one has 518. Because even if you had 700 people with hair you would still have one man with no hair always keeping the number of people higher than the numbers of hair.

### #34

Posted 16 November 2007 - 08:14 PM

These are the conditions in Baldyville:

1. No two inhabitants have the same number of hairs on their head.

2. No inhabitant has exactly 518 hairs.

3. There are more inhabitants than any inhabitant's hair in the town.

What is the highest possible number of inhabitants?

There is no rule stating that there the people in the town have specific numbers on their head it only says they don't have the same number of hairs as any one else. it also says there are more inhabitants that any inhabitants hair. that means there can be 530 people in the town as long as one person is bald no one has 518. Because even if you had 700 people with hair you would still have one man with no hair always keeping the number of people higher than the numbers of hair.

if you had 700 people then:

1. No two inhabitants have the same number of hairs on their head ... so 0 hair (1st person), 1 hair (2nd person) ... 699 hairs (700th person) ... each one - out of 700 - will have unique number of hairs

2. No inhabitant has exactly 518 hairs ... oops - I have to take this into account for the line above ... so so 0 hair (1st person), 1 hair (2nd person) ... no one with 518 hairs ... 700 hairs (700th person)

3. There are more inhabitants than any inhabitant's hair in the town ... so there would be 700 people and the last one has 700 hairs ... soooo there can't be 700 of them

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### #35

Posted 19 November 2007 - 07:14 PM

baldy-1 person

one hair+ baldy= 1 hair=2 people

one hair+ 2 hair+ baldy=2 hair=3 people

one hair+ 2 hair + 3 hair+ baldy=3 hair=4 people

one hair + 2 hair + 3 hair + 4 hair+baldy=4 hair= 5 people

one hair + 2 hair + 3 hair + 4 hair+ 5 hair+baldy= 5 hair=6 people

no one can have 6 hairs so we are still dealing with 6 people

okay im wrong but if you read this you are no longer sure that HAIR is a real word//haha

### #36

Posted 21 November 2007 - 12:48 AM

These are the conditions in Baldyville:

1. No two inhabitants have the same number of hairs on their head.

2. No inhabitant has exactly 518 hairs.

3. There are more inhabitants than any inhabitant's hair in the town.

What is the highest possible number of inhabitants?]

With all of the discussion on this one, it surprises me no one caught this. Why did no one notice that the "No inhabitant has exactly 518 hairs" does not specify that those hairs need to be on their head? When you take this into account, you can have an infinite number of body and head hair combinations, as long as they don't equal 518. Therefore, this puzzle does not have an answer because it grows to infinity.

Example:

I am bald. I am the only bald guy in town.

I have 1000 hairs on my palms.

This meets the both Criteria 1 and Criteria 2 unless I am mistaken.

Criteria 3 states that (inhabitants > number of hair on a person).

Therefore the number of inhabitants in my example is any number greater than 1000.

Correct me if I am wrong, but I do not think that I am.

### #37

Posted 21 November 2007 - 12:56 AM

These are the conditions in Baldyville:

1. No two inhabitants have the same number of hairs on their head.

2. No inhabitant has exactly 518 hairs.

3. There are more inhabitants than any inhabitant's hair in the town.

What is the highest possible number of inhabitants?]

With all of the discussion on this one, it surprises me no one caught this. Why did no one notice that the "No inhabitant has exactly 518 hairs" does not specify that those hairs need to be on their head? When you take this into account, you can have an infinite number of body and head hair combinations, as long as they don't equal 518. Therefore, this puzzle does not have an answer because it grows to infinity.

Example:

I am bald. I am the only bald guy in town.

I have 1000 hairs on my palms.

This meets the both Criteria 1 and Criteria 2 unless I am mistaken.

Criteria 3 states that (inhabitants > number of hair on a person).

Therefore the number of inhabitants in my example is any number greater than 1000.

Correct me if I am wrong, but I do not think that I am.

actually, there was 1 person in the village who had some hair (of his dog) left on his head ... I forgot to mention that as well ... and now I remember, one old lady had a wig (from not her own hair)

rookie1ja (site admin)

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### #38

Posted 26 November 2007 - 04:29 PM

### #39

Posted 29 December 2007 - 06:19 PM

### #40

Posted 15 January 2008 - 08:52 PM

Spenglers example is fantastic. For those of you confused by the problem-solving approach here (i.e. the consecutive ordering) I'd suggest doing some research on the "well-ordering principle". I did a cursory search and found pages that probably wouldn't appeal to most; however, this is an extremely powerful tool logicians and mathematicians use when attacking problems like this one that deals with sets.

Hmmmmm, so if a man has one hair on his head, and he goes to the barber and gets the hair cut in half, does that mean the man now has half a hair on his head? Using increments of halves would allow for an infinite town population!

**Edited by Quinten27, 15 January 2008 - 08:53 PM.**

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