MathDB

Player Zero and Player One play a game on a

n \times n

board (

n \ge 1

). The columns of this

n \times n

board are numbered

1,2,4,\dots,2^{n-1}

. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a

0

and Player One puts a

1

). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a

1

in that row. For instance, when

n=4

, a row with

0101

yields the number

0 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10

.
a) For which natural numbers

n

can Player One always ensure that at least one of the row numbers is divisible by

4

? b) For which natural numbers

n

can Player One always ensure that at least one of the row numbers is divisible by

3

Problem Statement