Stepping Off Points: #3 PROBLEMS ON A CHESSBOARD

 Problems on a Chess Board

A regular problem is to place 8 Queens on a chessboard so that no queen threatens another queen.  How can this be done (Remember that a Queen can move in all directions)

 

 

Now:  What is the greatest number of Bishops that can be placed on a chessboard so that no Bishop threatens another Bishop? (Bishops move along Diagonals)

What is the minimum number of Knights that can be placed on a chessboard so as to occupy or threaten every square?(Knights move along horizontal and vertical lines)

What is the minimum number of squares that eight Queens can command(that is to occupy or threaten?)

Change the size of the board to say a 4 x 4, 6, x 6, 7 x 7, 10 x 10, and see if there is a pattern for each of the pieces for each sized board.

 

 

 Dominoes and the Chessboard

It is easy to put dominoes (where each domino covers two squares) on a chessboard and cover all the squares.

But what happens if you have a chess board where two corner squares have been cut off.  Can you now place Dominoes to cover all squares?

 

 

 

Adapted and extended from Points of Departure by Association of Teachers Mathematics  1989

 

Stepping Off Points #2 NUMBER SWITCH

Another Open-Ended activity to challenge you and your students

 

Number Switch

Arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 randomly on a three by three grid.  ( i suggest you have the numbers on slips of paper or use Digit Tiles.

By switching two numbers at a time, investigate how many moves it takes you(or you and your partner) to get the numbers into their numerical(normal) order

e.g.

Can you find a strategy that will give you the least number of moves?

Can you write a sentence(rule) to explain how many moves are needed?

Try this with other layouts- say a 4 x 4, using the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16