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.