MathDB

You have the following pieces: one

4\times 1

rectangle, two

3\times 1

rectangles, three

2\times 1

rectangles, and four

1\times 1

squares. Ariel and Bernardo play the following game on a board of

n\times n

, where

n

is a number that Ariel chooses. In each move, Bernardo receives a piece

R

from Ariel. Next, Bernardo analyzes if he can place

R

on the board so that it has no points in common with any of the previously placed pieces (not even a common vertex). If there is such a location for

R

, Bernardo must choose one of them and place

R

. The game stops if it is impossible to place

R

in the way explained, and Bernardo wins. Ariel wins only if all

10

pieces have been placed on the board. a) Suppose Ariel gives Bernardo the pieces in decreasing order of size. What is the smallest n that guarantees Ariel victory? b) For the

n

found in a), if Bernardo receives the pieces in increasing order of size, is Ariel guaranteed victory?
Note: Each piece must cover exactly a number of unit squares on the board equal to its own size. The sides of the pieces can coincide with parts of the edge of the board.

Problem Statement