1

Part of 2004 CentroAmerican

Problems(2)

A border separates a black square from a white square

Source: CentroAmerican 2004

10\times 10

square board, half of the squares are coloured white and half black. One side common to two squares on the board side is called a border if the two squares have different colours. Determine the minimum and maximum possible number of borders that can be on the board.

geometryrectanglecombinatorics proposedcombinatorics

Players A and B remove the numbers 1 to 9

Source: CentroAmerican 2004

On a whiteboard, the numbers

1

9

are written. Players

A

B

take turns, and

A

is first. Each player in turn chooses one of the numbers on the whiteboard and removes it, along with all multiples (if any). The player who removes the last number loses. Determine whether any of the players has a winning strategy, and explain why.

combinatorics proposedcombinatorics