I’m having trouble figuring out what the question actually is. On the face of it (taking, for now, only things that Martha and Marco and their computer actually affirm and ignoring one thing that seems at first like obvious hyperbole) it seems that what we’re told is: there’s a (presumably finite) set P of pizza-types and a set O of subsets of P (“orders”), whenever two sets are in O so is their union, and O contains no singleton sets. This obviously isn’t enough to tell us anything interesting.
But then there’s also this stuff about whether there’s an element of P that’s in more than half the elements of O. Marco thinks probably yes (are we supposed to infer something from this? it seems like we’d need to know a lot more than we’re told about exactly how much Marco knows and how good he is at reasoning) and he says “Today’s state of the art mathematics isn’t powerful enough to ascertain that this is actually the case!” (are we supposed to take this literally? not powerful to ascertain this given what information?)
Is the idea that given |P| or |O| or both along with the statements in the first paragraph, but no other information, present-day mathematics is not able to determine whether or not there is necessarily an element of P in more than half the elements of O?
If so, then it seems like this is about the union-closed sets conjecture, which is apparently known to be true when |P| is at most 11 or |O| is at most 46 or when O contains either 1-element or 2-element sets. So I guess you are looking for the number 12? (For all I know, the conjecture might be known to hold whenever, say, |P|<=13 and |O|<=60, in which case having at least 12 types of pizza is not sufficient. But, across all possible states of Marco’s knowledge, assuming he knows nothing more than |P| and maybe |O|, the smallest |P| could be to make his statement true is 12.)
You are right. It’s 12 or more different kinds of pizza. If it was 1 kind of pizza served, he could be certain, that one kind of pizza was at the majority (in that case at all minus one) orders. Since somebody could order a salad. But only one without pizza, to avoid equal orders by pizza kind.
Even if there were 11 different pizza kinds on the menu, Marco could be sure, there is the majority kind of pizza there. Since this Fraenkel conjecture has been proved up to 11 by now.
But for 12 or more, no one really knows yet. Probably it’s true, but who knows. Congratulation, you were rather quick. Despite the fact, the problem formulation looks vague to you.
After some time, a new math puzzle.
https://protokol2020.wordpress.com/2021/04/17/pizzeria-at-the-end-of-the-world/
I’m having trouble figuring out what the question actually is. On the face of it (taking, for now, only things that Martha and Marco and their computer actually affirm and ignoring one thing that seems at first like obvious hyperbole) it seems that what we’re told is: there’s a (presumably finite) set P of pizza-types and a set O of subsets of P (“orders”), whenever two sets are in O so is their union, and O contains no singleton sets. This obviously isn’t enough to tell us anything interesting.
But then there’s also this stuff about whether there’s an element of P that’s in more than half the elements of O. Marco thinks probably yes (are we supposed to infer something from this? it seems like we’d need to know a lot more than we’re told about exactly how much Marco knows and how good he is at reasoning) and he says “Today’s state of the art mathematics isn’t powerful enough to ascertain that this is actually the case!” (are we supposed to take this literally? not powerful to ascertain this given what information?)
Is the idea that given |P| or |O| or both along with the statements in the first paragraph, but no other information, present-day mathematics is not able to determine whether or not there is necessarily an element of P in more than half the elements of O?
If so, then it seems like this is about the union-closed sets conjecture, which is apparently known to be true when |P| is at most 11 or |O| is at most 46 or when O contains either 1-element or 2-element sets. So I guess you are looking for the number 12? (For all I know, the conjecture might be known to hold whenever, say, |P|<=13 and |O|<=60, in which case having at least 12 types of pizza is not sufficient. But, across all possible states of Marco’s knowledge, assuming he knows nothing more than |P| and maybe |O|, the smallest |P| could be to make his statement true is 12.)
You are right. It’s 12 or more different kinds of pizza. If it was 1 kind of pizza served, he could be certain, that one kind of pizza was at the majority (in that case at all minus one) orders. Since somebody could order a salad. But only one without pizza, to avoid equal orders by pizza kind.
Even if there were 11 different pizza kinds on the menu, Marco could be sure, there is the majority kind of pizza there. Since this Fraenkel conjecture has been proved up to 11 by now.
But for 12 or more, no one really knows yet. Probably it’s true, but who knows. Congratulation, you were rather quick. Despite the fact, the problem formulation looks vague to you.
I searched for <<family of subsets closed under union “more than half”>> and the Wikipedia page about the conjecture was the first result :-).
It’s nothing wrong with the Googling method. Besides, one could search for <<pizzeria at the end of the world papa mamma>>. Should work now, too.
Maybe, next year’s solution to this problem will be “at least 13”.