Chessboard Triangles
The game of chess was invented in India in the 500s A.D., making it nearly 1500 years old! Humans - and more recently computers - have become experts at analyzing the nearly infinite moves and strategies that result from the geometry of the board and the possible moves of the different pieces.
In this exploration you will:
1) Explore the ways that three different chess pieces can move around the board 2) Solve chess challenges using the Pythagorean Theorem
Part I: Knights, Queens, and Bishops - Oh my!
When you open the Chessboard Triangles BlocksCAD file you will see the chessboard that appears on the front page of this activity. There are three pieces on the board that we’ll get to in a moment.
We first must make sure that we have a way to label each chessboard square. Each chessboard square is 10mm x 10mm, but the centers are located on the 5’s that are marked along each of the two axes. For example, the green square in the picture below will be called the (5, 5) square because that is where its center is.
What are the coordinates of each of the following squares?
Purple: ( ___ , ___ ) Blue: ( ___ , ___ ) Yellow: ( ___ , ___ )
We’ll start with the bishop piece. In chess, bishops can only travel diagonally. They look like this:
List two squares on the chess board that this bishop could travel to in just one move:
( ___, ___ ) and ( ___, ___ )
Is there anywhere on the board where this bishop could not get in three moves? Justify your answer.
A knight piece is shaped like a horse and looks like this:
The allowable move for a knight is often described as an “L” because they can move two squares in any direction and then one in another:
List two possible squares that the blue knight could move to:
( ___, ___ ) and ( ___, ___ )
You want to move the knight at (-15, -25) two squares up and one square to the right. Use a translation block around the knight block to move the knight so that it ends up where you would like it to be.
What are the knight’s new coordinates? ( ___, ___ )
How far did the knight move in the x direction? ____ mm In the y-direction? ____mm
How far is the knight from its original location? Draw a triangle that represents this situation.
The queen is the most powerful piece on the chessboard because it can move any number of spaces in any direction!
The queen can get anywhere on the board in at most two moves. Describe two moves that would get the queen from the yellow square to (5, 25) and use a translation block to move her there.
How far is the queen from where she started? Draw another triangle and show your work.
Part II: Chess Challenges
Use what you learned in Part I about how the three pieces move and how to calculate how far they traveled to answer the following questions. Draw triangles for each!
1) How far does the bishop need to travel to get from the pink square in the top left corner to the square at the bottom right corner?
2) How far does the bishop need to travel to get from the pink square to the bottom- left-most white square at (-25, -35)?
Hint: Remember, the bishop can only move diagonally! You may need to do multiple calculations.
3) What BlocksCAD transformation could bring the knight to the (-15, 25) square? How many legal chess moves would it take for the knight to get there without the help of the transformation?
4) Which of the three pieces has to travel the farthest to get to the (5, 15) square?
Note: Even though the knight has to move up and then over, count the “distance traveled” for one move as simply the distance between the place the knight started and the place the knight finished. If it takes multiple moves, you can add up those distances!
5) Which square in the first quadrant could the queen and the knight meet at if the queen has one move and the knight has two? How far would each piece be from its original location?