Barge Math Competition - Fall Semester

Week 3’s Problem: Wild Cards

A stack of 2n cards (labeled 1 through 2n, top to bottom) are sitting on a desk in a neat pile.  We reorder the pile in a very special way, as follows.  First, take the top n cards off and put them in a pile (call it pile A) next to the remaining n cards (which we'll call pile B).  Then, form the new, reordered pile from bottom to top:  First take the top card from pile B and put it down on the table (this is the (n+1)st card from the original pile).  Now take the top card from pile A and put it on top of that card.  Thus, after two steps, we have two cards in our new pile:  card n+1 (on the bottom) and card 1 (above card n+1).  Continue in this manner, alternately placing one card from pile B and then one from pile A to create the new, reordered pile.

Example:  If n = 3, then the 6 cards will be reordered as 3 6 2 5 1 4, where 3 is the new top card.

A card may be in the same position in the reordered pile as it occupied in the original pile.  Call such a card lucky.  

a) (3pts) If n = 2006, are there lucky cards?  If so, what card numbers are they?

b) (3pts) Are there any values of n for which card number 2006 in the stack is lucky? If so, what are they? If not, why not?

c) (4pts) What is the maximum number of lucky cards there can be in a stack? For which value(s) of n does this situation occur?

How to submit your answer:

Bring your solution to the Math Department office in 230 Pardee, or send it by email to Ethan Berkove at berkovee@lafayette.edu. Problem #3 is due by the end of the day on Friday, February 24.