2009 cards | combinatorics

Question

Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number k such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40$. The last player who can make a legal move wins.

(a) Does the game necessarily end?

(b) Does there exist a winning strategy for the starting player?

Answer 1

There is a solution on page 780 of Dusan Djukic, Vladimir Jankovic, Ivan Matic, Nikola Petrovic, The IMO Compendium; A Collection of Problems Suggested for The International Mathematical Olympiads, 1959-2009, which I found on Google Books.

Answer 2

It would be worth telling us what you have tried.

Some hints:

• Are there are a finite number of possible positions (and if so what is an upper bound)?

• Can there be a cycle of positions (consider the card with the smallest number turned over)?

• Will the starting player have a winning strategy if initially there are only 41 cards? 42? etc?