A hairy situation
Suppose there is this little town with a finite number of people:
(1) No two inhabitants have exactly the same number of hairs.
(2) No inhabitant has exactly 409 hairs.
(3) There are more inhabitants than there are hairs on the head of any inhabitant.
So, what is the largest possible number of inhabitants in that little town?
The answer is 409.
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2 solutions
A good solution sir . Such solutions are always up voted.
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Same here, I upvoted it too. And a good problem @Rama Devi ! :)
BTW, I am just a little curious. Why of all numbers you picked 409? Does it have any particular significance other than the fact that all its digits are prefect squares?
Forgive my questioning, but condition two states "Exactly." Doesn't that mean that greater than can also be included? I realize this opens up for the possibility of an "Infinite" answer, but it does not prohibit a head that has GREATER than 409 hairs. It prohibits Exactly (i.e. "=")
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I think that condition (3) rules this out. Since we are told that there are a finite number of inhabitants, there exists a hairiest person. Suppose this person has, say, 4 2 0 hairs. Then condition (3) implies that there are in excess of 4 2 0 inhabitants. By condition (1) we can have individuals having 0 to 4 2 0 hairs, i.e., 4 2 1 distinct hair "counts", but by condition (2) a hair count of 4 0 9 is unavailable, leaving us with only 4 2 0 possible hair counts, implying that condition (3) cannot be satisfied if the hairiest person has in excess of 4 0 9 hairs. Since they also cannot have 4 0 9 hairs by condition (2), we arrive at the conclusion outlined in my posted solution.
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Fundamentally and Semantically, the question is asking for the largest "Possible." If you have a set (say, for instance, the finite set of the real positive integers 1 through 100) and simply say that 9 is not present, it does not take away from the amount of values above or below the prohibited value, thus allowing 100 to be the largest possible from our finite set.
If you have an infinite set (all real integers) and specify that 110,475 cannot be chosen or simply ceases to exist, you are still left with the unyielding infinite number of values to be chosen from, including every integer above 110,475.
The wording of "exactly" simply takes a single value away from the set ranging from the value 1 and infinitely extending to the positive side of our number line.
If condition (2) stated "No inhabitant has MORE than 409 hairs," the set of possible values to be chosen from would then have a Maximum (this is because the set has been truncated)
Without the proper truncation of the set, the possible answers are now "x" such that it can sufficiently be expressed as y = ∣ x ∣ 2 + 1 and remembering to exclude 409. There are an infinite amount of sufficient values for x so that y can be found.
simply saying that the set is finite is not sufficient information to extrapolate that the set ends at the single value prohibited.
To make my point clear, please follow me.
- You have a set of finite values (of which I will not tell you the complete set)
- I have prohibited the value 110,475
- Is this enough evidence to support that 110,475 is the largest value of the finite set?
We'll solve this only using logic. Before we start, I must mention that we are always looking to make a scenario with as many inhabitants as we can, so we can meet the rule (3). Let us examine the options:
A) If all the inhabitants have more than 409 hairs. In the best case, 1st inhabitant would have 410 hairs, 2nd would have 411 hairs, 3rd would have 412 hairs, etc. Hence, there will be, at least, 409 less inhabitants then number of hairs of any inhabitant and that is of course contradictory to the rule (3). So, we reject this option.
B) If some inhabitants have less and some more than 409 hairs. In the best case, the maximum number of hairs of any inhabitant will be 410 and the others would have less number of hairs. Hence, there will be, at least, the equal number of inhabitants to the number of hairs of any inhabitant (don't forget that there could be people with no hairs!) and that is, again, contradictory to rule (3). So, we reject this option too.
C) If all inhabitants have less than 409 hairs. This is the only option left. And it really works: In the best case (case where every number of hairs is common for a specific inhabitant) there will always be 1 more number of inhabitants than number of hairs of any inhabitant (because of possibility that one inhabitant could have 0 hairs). Since we are looking for the most possible number of inhabitants, then the "hairest" man would have 408 hairs and the number of inhabitants would be exactly 409.
Suppose that the "hairiest" inhabitant has n hairs. Then by condition (3) there must be at least ( n + 1 ) inhabitants. The only way that this can happen while simultaneously satisfying condition (1) is if there is precisely one inhabitant with k hairs for each integer k such that 0 ≤ k ≤ n . Given this, by condition (2) we see that no inhabitant can have ≥ 4 0 9 hairs, and so the hairiest person can have at most 4 0 8 hairs. This implies that the maximum possible number of inhabitants in the town is 4 0 9 .