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Exploring the Knight’s Tour

Part 1: What is a Knight’s tour?

A Knight’s Tour is a sequence of moves by a knight on a chessboard in which the knight visits every square exactly once.

Closed Tour: Reach all of the boxes on the 8x8 square and end where your Knight started.
Open Tour: Reach all of the squares on the 8x8 square but dont stop where the Knight started.

Part 2: Try it out!

When you open the applet, all of the checkboxes should be turned off. If they aren’t, go ahead and turn them off before you begin. The Knight should be sitting anywhere within the chess board, you can start your Knights tour from where it is, or you can try starting from a different square on the board! Whatever suits your fancy!

Then write down what your strategy is going into this, before you start.
My strategy:

Part 3: Understanding Patterns

Reload your applet so everything resets. Starting at the very top left corner of the chess board, count 5 squares to the right and 2 boxes down and put your Knight there for the starting position. Press the check box for Hint #1 and Hint #2.

What shape do the yellow squares in each quadrant look like?

Can we hit every one of the red boxes in one quadrant before we move on to the next quadrant? Try it.

Keep going until all of the red squares are hit. Once you have landed on all of the red squares (hint #1 squares) on the whole chess board then press the check box for Hint #3.

What shape do the blue squares seem to create in each quadrant? In a similar fashion from above we now want to land on all of the blue squares in each quadrant before moving onto the next quadrant.

Once all of the blue squares are hit go ahead and press the checkbox for Hint #4.

What shape do the green squares look like in each quadrant?

Go around landing on each green square before moving onto the next quadrant. Are you noticing anything interesting? What patterns are you seeing?

Finally press the check box for the last hint, Hint #5 and move your Knight so it reaches every yellow before moving onto the next quadrant. What do you notice about where you ended?

Is your Knights Tour a Closed Tour or an Open Tour?

Volume of a Cone, Cylinder, and Sphere

Use the sliders to adjust the height and radius of the various 3d shapes. Toggle on and off the different shapes and their respective volumes.

After you are done exploring the different sliders and checkboxes, make it so only the cone is showing. Pick a set radius and height of your choice. Write an equation for the volume of a cone.
Hint: think of its base and height.

Now toggle on the Cylinder. Compare the volume of the Cone with the Cylinder. How many Cones could fit into this Cylinder? Write the equation of the Cylinder.

Now lastly, toggle on the Sphere. By comparing the volume of a Cone with the Sphere, how many Cones could fit into the volume of a Sphere? Now write the equation of a Sphere with the volume of a Cone in mind.

Rectangular and Polar Coordinates

Move point A to different positions.
a. What happens to the rectangular coordinates (x,y)?
b. What happens to the polar coordinates (r,θ)?

Click the check box that says "Show Polar Coordinates" so you can no longer see it on the graph.
We know a way to find the missing length of a triangle, using the other side lengths. With that being said how does r relate to (x,y)? Write an equation expressing the polar coordinates in terms of its rectangular coordinates.

Click the "Show Polar Coordinates" so now you can see it again. We can find the measure of θ or side length by using trig.
Using that idea, write the rectangular coordinates in terms of polar coordinates.

Sin Transformation Applet

Link Collection

Applet Collections

Title	Description
GeoGebra	This is a collection of probability applets such as coin flips, pascals triangles, binomial probability
Math Learning Center	This collections of applets is based on fractions, geometry, clocks etc. Seems to be geared more towards younger demographic.
Math Insite	This collections of applets is all about planes on a graph in both 2 dimentions and 3 dimentions.
Interactive Mathematics	This is collections of applets that can be used in math classes in highschool, there is a wide variety of different topics.
Strapplet	This is collections of applets that is used for statistical understanding and exploration.

Applets

Title	Description
Knights Tour Chess Applet	This applet discusses what a knight tour is, how many different ways you can achieve a knights tour and then allows you to try on your own to to complete a knights tour.
Chess Hub	This applet is a chess analysis applet so it allows you to move a pawn and then will show you the ideal next move.
A Knights Tour	This applet specifically looks at what moves a knight can make in chess. This is a helpful applet because it focuses on knights only and takes away all of the other chess pieces allowing you to understand a knights tour much easier.
Geogebra Chess	This is just a standard applet that allows you to learn how and to play chess.

Resources

Title	Description
The Mathematics of Chess	This is an analysis essay that dives into the different mathematicl topics and history that comes from the historic game of chess.
Mathematical Concepts Embedded in Chess	TThis website is a great resourse, it not only teaches you about chess but it also talks about chess and the relationship between its analytical pursuits
The Mathematics in Chess Video	This is a video talking about the mathematics found in math and what it has to do with sacrificing positions in the game. This video is a part of a playlist that goes through even more ideas such as geometry etc. on chess.
Chess and Math: A Closer Look at the Knights Tour	There are a lot of mathematical concepts found in a Knight't Tour that directly correlate to Chess as a whole. This website talks about the connection between a Knights Tour mathematics and the Mathematics in Chess.
Game Theory and Chess	This paper is an analysis that uses the game chess to illuistrate strategy and game theory.