Therapy for $10.00!
Here is how it works!

A therapy session is held for 1000 pennies. The pennies are encouraged to touch as much as possible. The job of the counselor is to arrange them on the floor to allow for the maximum contact. Two pennies have contact if their edges touch (thus they are laying flat next to each other).



Here are questions for consideration:
1. What geometric arrangement or arrangements (if more than one cluster is desirable) allows for maximum contact?
2. What is the maximum number of contact points among the 1000 pennies?
                   Also included-How do these answers vary as the number of pennies varies?


Click each link and explore the possibilities!

Find that all of this not "worth" your time? Try an Extension to this Problem...
After trying and trying to arrange a counseling session for all 1000 pennies to meet, one year passed by.  The outcome was not good. These pennies became even more depressed and ate their sorrows away.  Now, the pennies became shaped like marbles. How can you now arrange the pennies? (assumming that the base is framed, and no one falls)?  Look at the stack with a triangular and square base, Which will shortest assuming we want the maximum contact points in each stack?                                                                   Click the below links to begin your exploration in stacking bloated pennies!
The Bloated Penny Extension:        Click here for a visual in Geometer Sketchpad.        
 Click here for an Excel document on bloated pennies
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GEOMETRIC ARRANGEMENT OF CIRCLES...

Here is the task... Go get 19 pennies and play around with them.  Arrange them in different patterns and see which pattern has the most contact points. Try circles, triangles, squares, hexagons, ect.  What do you notice?   Are there differences among these patterns?   What do you notice about their similarities?

I have created a Geometer's Sketchpad document that
explores these patterns.  Click the box below to download this file
and explore these relationships yourself!


Link to GSP website

Click the Picture to go to a reference that I found to be very helpful!!
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Maximum Contact points & How This is Effected
By the Number of Pennies

From the exploration in Geometer's Sketch Pad and a web source, we can clearly
see that the geometric arrangement for optimizing contact between pennies is the hexagon.
Now, the task at hand to create a pattern within this hexagonal arrangement and find the
maximum number of contact point for 1,000 pennies.
The start of our hexagon will be with one penny.  We can  begin to add layers
to see the pattern created. This allows us to write a formula that will
produce the number of pennies in a complete hexagon arrangement.
                                                The following pictures illustrate this pattern.                                                        
       

In the Microsoft Excel file that I have created, I describe the mathematics behind
calculating the number of connection points by using  the corner, side, and center pennies
within the hexagon arrangement
CLICK HERE TO DOWNLOAD!!
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