Challenge: Find a sequence of sets: S1, S2, S3, … such that the intersection of any finite number of them is non-empty, but such that the intersection of all of them is empty, i.e. S1 ? S2 ? S3 ? … = {}.
44 Pennies, A second look
Aaron’s daughter Sophie is acting up, so Aaron gives her a task to keep her busy: he gives her 44 pennies and material for making a vest, then challenges her to make a vest with ten pockets and put all her pennies in the pockets in such a way that she has a different number of pennies in each pocket. Aaron then settles in with a good book, confident in the knowledge that Sophie will be busy for quite some time. He’s just settling into his nap when Sophie comes in and says she is done. Aaron is a bit annoyed, but he explains, patiently, to his daugher:
“You see, Sophie, I once took Math 300 and I can prove to you that what you claim to have done is impossible. Since you must put a different number of pennies in each pocket, the very smallest number of pennies you will be able to fit into your pockets is achieved when we use the smallest numbers per pocket possible. Since you can’t have a negative number of pennies in a pocket [Sophie laughs], the smallest numbers available to us are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The sum of these numbers is (Niftily speaking) 45. Thus you cannot put fewer than 45 coins in your pockets in such a way that there is a different number of coins in each pocket. In particular, you can’t do it with 44 pennies.
“That’s a great argument, Daddy. You are so smart. Except, um, I must have missed something, because I managed to do what you asked. See!”
At this point, she brandishes a vest with 44 pennies arranged in ten pockets having a different number of pennies in each pocket. Aaron looks at her work for a while and thinks to himself “Either I am an idiot, or my daughter is a genius!” Meanwhile Sophie is saying: “So, when you gave me the 44 pennies, you actually thought it would be impossible, huh?” Aaron is starting to feel somewhat uncomfortable at this point, so he says: “Oh, ha, ha, um, here’s 200 pennies, why don’t you get yourself an ice cream sandwhich?” “Thanks Dad! By the way, if you wanted to make it impossible for me, you should have given me, 0,1,2,3,4,5,6,7 or 8 pennies. As soon as you give me 9 or more pennies, I can make a ten-pocket vest and put the pennies in the pockets so that no two pockets have the same number of coins. Seeya!”
How did Sophie do it?!
p.s. With due acknowledgement to Elementary School Teacher Extraordinaire, Cindy McCarthy.
p.p.s This story serves to stress the importance of phrasing questions in a precise enough way so as not to fall prey to hidden assumptions.
Nifty Numbers Redux
Recall we call a positive integer Nifty if it can be written as a sum of one or more positive integers using each of the digits 0,1, …, 9 exactly once. For example 99 is nifty because 99 = 10+24+36+5+7+8+9.
(A) Show that every nifty number is divisible by 9.
(B) Determine the complete list of all Nifty Numbers.
(C) Call a number 0-Nifty if it can be written as a sum of one or more positive integers using each of the digits 1, …, 9 exactly once; in other words, same concept, but now you can’t use 0. Determine the list of all 0-Nifty numbers and discuss how different this list is from the list of Nifty numbers.
Hello world!
Here’s a radical experiment: a blog for Math 300 Section 2, Fall 2010, at UMass Amherst, taught by Farshid Hajir. This space will be mostly for the students to converse with each other about material related to the class, but I might post things here occasionally. It’s open to the entire public, not just our class, so the whole world can benefit from the fabulous discussions that may take place here. I’m sure that all users of this blog will follow rules of polite behavior and common courtesy. Please keep in mind that the official website for the course is http://www.math.umass.edu/~hajir/m300 — this blog is simply a supplemental place where students can post things they might think other students would find interesting/useful.
One motivation for creating this blog is to have a place where I can put Fun Problems with which to challenge you, the students… the idea is that by collaborating on the problems, more students will in fact be involved in these hopefully enriching experiences. In the past, I have used the name “Extra Credit Problems,” a traditional and handy phrase. Candor and honesty being virtues, however, I am moved to call them by a more accurate name “No Credit Whatsoever Problems.” That’s because working on or even solving these problems will in no way have a direct effect your grade for course. These are just problems you may want to think about, if you are so inclined. So why put them on a blog? Because unlike homework problems, which you really should do on your own to develop your skills, and therefore should not discuss on this blog until after the due date (See the Policy on Collaborating on HW), you are completely free, indeed encouraged, to discuss the NCWPs here to your heart’s content whenever you want. It may be that there are some hard problems that you can’t solve on your own but by putting your ideas here, other students can complete your thought. Collaborative Problem Solving, let’s call it.
So I hope you have fun exploring here. Give me your suggestions for stuff you would like to see here.
Farshid