91–100 of 116 posts
Earlier quoted context omitted.
The use of "the" isn't the weird part. The weird part is that the article attempts to draw a contrast between "pentagons" and "five-sided polygons", which are exactly the same thing.
Pentagons are 5-sided polygons where all the sides are equal. Not all 5-sided polygons are pentagons. What am I missing?
Earlier quoted context omitted.
Pentagons are 5-sided polygons where all the sides are equal. Not all 5-sided polygons are pentagons. What am I missing?
What you're describing is called a regular pentagon.
>... if the room has at least six people, you can say something about them with absolute mathematical certainty... [that it contains] either a group of three who all know each other, or a group of three who have never met. Maybe I'm incredibly dense, but this seems tautologically true, and not worth mentioning. Could somebody kindly explain what I'm missing?
The 'who have never met' isn't the opposite of 'all know each other', it's no two people of these three know each other. The opposite would be don't all know each other. If you imagine a six vertices arranged around a point and make any number of edges connecting them to represent knows each other. For any choice of edges, you can either find 3 vertices that are fully connected or you can find three vertices that hav…
>... if the room has at least six people, you can say something about them with absolute mathematical certainty... [that it contains] either a group of three who all know each other, or a group of three who have never met. Maybe I'm incredibly dense, but this seems tautologically true, and not worth mentioning. Could somebody kindly explain what I'm missing?
Maybe you're not dense, maybe you're just such a genius that pigeonhole principle proofs seem naturally obvious and tautologically true to you? :-) I found this enlightening (particularly "Sketch of a Proof"), though I also admit that it seemed fairly straightforward, with elegance borne of simplicity rather than of brilliance or cleverness: https://en.m.wikipedia.org/wiki/Theorem_on_friends_and_stran...
Thank you for the link, this is a very good explanation.
>... if the room has at least six people, you can say something about them with absolute mathematical certainty... [that it contains] either a group of three who all know each other, or a group of three who have never met. Maybe I'm incredibly dense, but this seems tautologically true, and not worth mentioning. Could somebody kindly explain what I'm missing?
One could know the Two, Two know Three but Three not know One. That's neither all knowing each other nor them not meeting.
It would help if the article included at least one real world application in layman's terms.
Earlier quoted context omitted.
Big reason is that many high school math teachers don't really understand math well enough for the job.
To clarify, since I can't edit: This is coming from my experience as someone who has taught math to teachers getting their master's degree. It sounds mean but I wanted to emphasize that the state of early math education is not just due to poorly designed curriculum, but because there is little incentive for mathematically competent people to teach children. (Imo, of course).
The only way I know of is to just get after school tuition and ignore the school. This is difficult though - why do I have to do maths after school? no one else does.
>... if the room has at least six people, you can say something about them with absolute mathematical certainty... [that it contains] either a group of three who all know each other, or a group of three who have never met. Maybe I'm incredibly dense, but this seems tautologically true, and not worth mentioning. Could somebody kindly explain what I'm missing?
Perhaps you could imagine reading that statement with both occurrences of "three" replaced with blanks. Do you think you could confidently fill them in without more than a moment's thought?
>... if the room has at least six people, you can say something about them with absolute mathematical certainty... [that it contains] either a group of three who all know each other, or a group of three who have never met. Maybe I'm incredibly dense, but this seems tautologically true, and not worth mentioning. Could somebody kindly explain what I'm missing?
The theorem isn’t really saying “it’s A or not A”. It’s saying: “it has to be A or B and not anything else”.
Earlier quoted context omitted.
The 'who have never met' isn't the opposite of 'all know each other', it's no two people of these three know each other. The opposite would be don't all know each other. If you imagine a six vertices arranged around a point and make any number of edges connecting them to represent knows each other. For any choice of edges, you can either find 3 vertices that are fully connected or you can find three vertices that hav…
I would describe it as follows, maybe a little easier to visualize. Draw six dots, draw either a red or a blue line from every dot to every other dot. You must draw either a completely red triangle or a completely blue triangle. Personally I don't think it sounds intuitive when stated like that!