2014 USAMO Problems/Problem 4

Revision as of 16:44, 30 April 2014 by TheMaskedMagician (talk | contribs) (Created page with "==Problem== Let <math>k</math> be a positive integer. Two players <math>A</math> and <math>B</math> play a game on an infinite grid of regular hexagons. Initially all the grid ce...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.

Solution