Baldyville

[rookie1ja]

Baldyville - Back to the Cool Math Games

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 in town than hairs on any individual inhabitant's head.

What is the highest possible number of inhabitants?

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Baldyville - solution

There can live maximum of 518 people in the town. By the way, it is clear that one inhabitant must be baldy, otherwise there wouldn’t be a single man in the town.

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?

[Guest]

What if inhabitant 519 had a split end that broke off? Would that be exactly one hair or a partial hair?

[Guest]

There is no correct answer to this question. Just because no one has exactly a certain number of hairs, doesn't mean that somone doesn't have more hair then that. you can say x is not = to 386 but that does not mean that x is not equal to 387, 388, 389 etc.

[Guest]

no, 518 is the correct answer. take as an example, that there are 519 people in the town. to be orderly, we line them up by the number of hairs on their heads from 0-517. thats 518 people, but by our conditions, the last person may not have 518 hairs because that was specified. it was also specified that there are more people than hairs so the the last person must have less than 519 hairs. yet, if that were true there would be a repeat in number of hairs since we've already accounted for people from bald to 517 hairs and 518 is off limits. so for 519 or more, to paraphrase a quote, 'no soup for us.'

[Guest]

Good explanation. I didn't get it either, but now I do

[Guest]

yea well explained thephife..

[Guest]

i dont understand why there cannot be 520 people, even with your explanation of 519

[Guest]

I have exactly the same issue with this puzzle as sdy4444. Why cant someone have 1000 hairs? Nowhere does it say that you cant have MORE then 518. It says you cant have that exact number ... Am I being dense or does this puzzle need some work?

[Guest]

I have exactly the same issue with this puzzle as sdy4444. Why cant someone have 1000 hairs? Nowhere does it say that you cant have MORE then 518. It says you cant have that exact number ... Am I being dense or does this puzzle need some work?

If someone had 1000 hairs, there would need to be at least 1001 people.

the would have no hairs, the 2nd would have one the 3rd two... the 518th would have 517, the 519th can't have 518 hairs (as stipulated in the rules) but their can't be as many people in the town than the greateast number of hairs on a head. hence 518 is the greatest number of people in the town!

QED

[Guest]

The aside in the solution makes the given solution false. If one inhabitant is bald, then any number of non-repeating hair counts would equal the population of the city. If no one is bald, and there is no one with 518 hairs, then there could be an infinite number of people in the town, given that the person with the most hair has one less hair the the number of people in town

[Guest]

Lets take it like this:

Take these 2 conditions -

1) There are more inhabitants than any inhabitant's hair in the town. Thus, no person can have hair which number more than the total amount of people.

2) No two people have the same amount of hair.

And the 3rd condition -

3) No inhabitant has exactly 518 hairs

Now, if we consider that there is only 1 person in baldyville, he has 2 be bald, according to condition 1. If there are 2 people, 1 of them has to be bald and the other should have 1 hair. We can continue this sequence, till say 20, where there will be 20 people. Thus, the max amt of hair that one can have will be 19 (cond. 1) and, since no 2 people can have the same amount of hair (cond 2), there need to be 20 distinct amounts of hair. This can only be satisfied by the values 0,1,2,...,19.

Now, let us consider 518 people. If we extrapolate the previous example, there will be 517 diff amounts of hair, ranging from 0 to 518. But, if a 519th person to be present, there has to be atleast 1 person with 518 hairs. This goes against the 3rd condition, and hence, the max number of people is 518.

[Guest]

518 is the answer...

[Guest]

Phife: "it was also specified that there are more people than hairs so the the last person must have less than 519 hairs"

That's not true. It says there are more people than on ANY inhabitants head. Therefore, the logic is incomplete, and there are infinite possibilites for the answers.

Those who are saying 518 or 519 are operating under a false assumption that the number of people is limited, or that the hairs are in a numbered sequence.

What you are providing are possible answers, not an absolute answer proven by logic.

Can someone dispute this?

[Guest]

True, there are an infinite number of people possible to inhabit the town. There is no reason someone can't have 519 hairs or many MANY more. The fact that someone does NOT have 518, only means that 518 is right out, if one doesn't have it, then two don't (meeting the stated requirements of town). Just because 519 has the value of 518 in it, does not mean that is where it stops. ( Why can't someone in town NOT have 6 hairs, for instance?

In other words, for the solution to be true, one more stipulation is required "The number of hairs on the inhabitants hairs is consecutive" or the like.

Thoughts?

[Guest]

You guys still aren't thinking of it correctly. Let me modify the problem to see if it is a little easier for you.

Rules:

1) No two inhabitants have the same number of hairs on their head. (someone may be bald - 0 hairs)

2) No inhabitant has exactly 3 hairs on their head

3) The maximum number of hairs on any one person's head may not be greater than or equal to the number of inhabitants. (simply re-wording the original rule)

- So say there is only one inhabitant, then that person must be bald due to rule 3.

- Now say there are 2 inhabitants. Then one must have 0 hairs, and the other must have exactly 1 hair (they can't be the same by rule 1). If one of them had 2 hairs or more, rule 3 would be violated.

- Now say there are 3 inhabitants. Then one must have 0 hairs, another must have 1 hair, and the third must have 2 hairs. If one of them had 3 hairs or more, rule 3 would be violated.

- Now say there are 4 inhabitants. Then one must have 0 hairs, another must have 1 hair, and another must have 2 hairs. That leaves one more inhabitant who cannot have 0, 1, or 2 hairs on their head (rule 1). Rule 2 states they cannot have 3 hairs on their head, and rule 3 states they cannot have more than 3 hairs on their head. Therefore this situation cannot exist according to the rules since, by assumption, negative numbers of hair cannot exist.

- Now say there are 5 inhabitants. Then one must have 0 hairs, another must have 1 hair, another must have 2 hairs, another must have 4 hairs. But the 5th inhabitant cannot have 3 hairs (by rule 2), and they also cannot have 5 or more hairs. Therefore this situation cannot exist either.

Continue on to infinity and you'll realize that the max number of inhabitants for this problem is 3. So whatever number is specified in rule 2 (whether it's 3 or 518 or X), that is the max number of inhabitants that will satisfy all rules.

[Guest]

The aside in the solution makes the given solution false. If one inhabitant is bald, then any number of non-repeating hair counts would equal the population of the city. If no one is bald, and there is no one with 518 hairs, then there could be an infinite number of people in the town, given that the person with the most hair has one less hair the the number of people in town

I think you're missing an essential element of this puzzle. The number of inhabitants must be more than the greatest number of hairs on a single inhabitants head. Since no two inhabitants can have the same number of hairs and the number 518 is forbidden, the series stops at 519.

in VBA for Excel, not the greatest language but ubiquitous, it looks like this:

Sub countHairs()

Dim h

Dim p


	h= 0

	p = 1


	While h<> 518

		h= h+ 1

		p = p + 1

		ActiveCell.Value = p

	Wend

End Sub

Note that 'p', the number of people, starts out one greater than 'h', the number of hairs, and that 'h' cannot equal 518.

[Guest]

You guys still aren't thinking of it correctly. Let me modify the problem to see if it is a little easier for you.

What you did was change the problem. By rule 3

3) The maximum number of hairs on any one person's head may not be greater than or equal to the number of inhabitants.

was

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

Your explanation changed the parameters.

There is nothing in the rules to say there has to be an order and to skip a number voids a rule.

Someone might not have had 314 hairs.... the number of inhabitants based on these parameters is infinite.

Thoughts?

[Guest]

What you did was change the problem. By rule 3

3) The maximum number of hairs on any one person's head may not be greater than or equal to the number of inhabitants.

was

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

Your explanation changed the parameters.

I cannot see the difference. On the one hand, the first version above, we're saying 'not (h >= p)' while the second version says that 'p > h'.

There is nothing in the rules to say there has to be an order and to skip a number voids a rule.

This must be the most common fallacy in this kind of problem. I missed it myself on this one. It's not stated but it is implicit. Try it. Let's say there are three people. p > h so the greatest number of hairs in the population of 3 is 2. That being the case, someone has 2, someone has 1, and someone has 0. Try it with any number; you can't skip any.

[Guest]

AMEN, Mike

[Guest]

Sweet Jesus... if you guys dont understand from the past explanations (very good by the way), I can't help you. I thought they were extremely concise, and if you don't understand, you're going about it wrong. If it were that simple (that the sequence just skipped 518 and kept going), do you think this website would have posted it? That's exactly what everyone thinks before they look deeper to see the real solution.

[Guest]

Baldyville

These are the conditions in Baldyville:

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

2. No inhabitant has exactly 3 hairs.

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

What is the highest possible number of inhabitants?

Let's substitute 3 for 518 so it's easier for you people to count. Note the bold line - stop being so damn picky and take it at face value (the intent of the puzzle) - it means that the number of people in town (p) is greater than the number of hairs on any one person's head (h) (p>h).

So person 1 is bald (p=1, h=0).

Person 2 has one hair (p=2, h=1).

Person 3 has 2 hairs (p=3, h=2).

No one can have 3 hairs, no one ELSE can have 0, 1, or 2 hairs on their head, because they're already taken.

So you have 3 people, no one has more than 2 hairs (meeting condition 3). If person 4 were to have 4 hairs, you would have p=4 AND h=4, which cannot happen (condition 3, 4 is not greater than 4). The same applies to any person after this (5, 6, 7, ...). Once you skip a number, assuming you can't repeat, condition 3 prevents the existence of more people, because the number of hairs will be equal to the number of people.

Therefore if the number skipped is 518 (instead of 3), the maximum number of people is 518.

If you still don't get it, I feel sorry for you.

[Guest]

Ok it took me a while to get this but maybe I can explain it a little better, with a different style of thinking.

At first I thought there could be an infinite number. Lets say 1000 for example. The reason there cant be 1000 is there has to be more people than hairs on one head. So if there are 1000 habitants then you have to start counting... Habitant 1 (0 hairs) Habitant 2 (1 hair) Habitant 3 (2 hairs). You will do this all the way up to 518 Habitant (517 hairs) Then try to go forward 519 Habitants (518 hairs, can't do that).

And it has to go in order like there, the rule doesnt say it has to be consecutive but thats the only way it works. If you have 518 Habitants there has to be 518 different hair combinations and none of them can be 518 or higher.

This all "clicked" when I tried to do a number over 518. I thought "well it doesnt say it has to be in order so there can be a bald guy and a guy with 4 hairs and a guy with 1000. But thats where the rule 3 comes in. In order to get a million inhabitants you have to have 1000 combinations, and you cant count to 1000 if you cant go over 518. If you try to skip 518 or any other number before it as some people suggested, when you get to the end you will end up with the 1000th person having 1000 hairs. 3. There are more inhabitants than any inhabitant's hair in the town.

Does this help? Its kind of like explaining a color to a blind person. You either get it or you don't but once you see it, you know what it is.

[Guest]

there is nothing stopping you from skipping a hair count: 0, 1, 3, 4, 20

except if you skip a hair count, then you need to have that many people in town (+ 1). so the above example works well, and shows that their must be 21 people in town. okay easy. back fill the missing hairs ... 0,1,2,3,4,...,20 == 21 people.

this works just fine until you reach the X+1 person. let's say we have 5 people: 0,1,2,3,519. when we attempt to backfill to catchup and have 520 residence, we skip 518, and can no long give the 520th person hair as the only number left is 518; without breaching the 519 mark and having to re-backfill again. 0,1,2,3,...516,517,519 and one extra dude.

if we simply pop off one hair on the last dude, then we can easily start backfilling.

1) 518 people, where no one person has 518 or more hairs.

2) we've already proven that 519 (or any number higher is unpossible)

3) and "skipping" numbers while backfilling would be disadvantageous as you may as well maximize on your population .... and thus satisfy the "what's the maximum population" bit.

[Guest]

You guys have missed the most obvious part of the riddle..... the title.

Baldyville

this title implies that there can be an infinite number of people in the town as long as they were all bald. If they are bald they have no hair to be judged by. They can not have the same amount of hair on their head if they have no hair to begin with. The riddle also says that no two people can have the same amount of hair as any other inhabitant ON THEIR HEAD. So, get rid of the hair and everyone in BALDYville is happy.

Just trying to change it up.

[bonanova]

If they are bald they have no hair to be judged by. They can not have the same amount of hair on their head

Interesting.

0 does not equal 0.

But let's keep this a secret, OK?

Word gets out, imagine the recall of all those text books.

Would be devastating.