- The Encinal game: MWF at 12:30pm (frisbee standard time). Often at Cubberly in Palo Alto
- The Eagle Park game: Thursday at 12:30pm. Often at Eagle Park in Mountain View

Recently I created three custom disc designs for both groups. With these designs, I was attempting to appeal to the somewhat geeky nature of many of our players. I have received my order from Discraft for 25 discs with design 1 as well as 25 disks with design 2, so let me know if you would like to purchase either one of these discs. I'm selling them at my cost, which is $12.50 per disc. If I sell out of this production run and it looks like there is sufficient demand, I will probably order more, possibly including the third design as well. You can make payments to me for the discs you pick up at the field in cash or with a Venmo or Paypal payment to me at or by scanning the Venmo or PayPal QR code at the bottom of this page.

The red and blue curves are known as

The small light green curve in the middle is a member of a class of curves known as a

The equation for this particular Hypotrochoid in the complex plane is as follows:

This is the solution provided by the puzzle's author. I saw this solution before I saw the statement of the puzzle, so I don't know if I would have come up with it or not, but my contribution to this solution was to derive these equations for the radii of the circles that ensures the desired tangencies:

From this work, I realized that there are 3 variations of this solution, one if which is shown here. I don't think it will take you long to see what the other two variants are. None of the variants seemed to be as pleasing to the eye as the original solution, which I briefly considered using for the disc design.

The fact that this puzzle has any solutions at all seemed somewhat remarkable to me, but I thought I would try to find other solutions. And indeed I did find one, which is shown below.

This is the solution of the puzzle that I chose for the disc design because it seemed to be the most surprising. I don't know if there are any more solutions to this puzzle, but if you find one I would greatly appreciate you telling me about it. I found that the radii needed to make this solution work are as follows:

Looking at the tangent point at the bottom where four circles meet, the radii of the red, green, blue, and purple circles are 17/11, 2.2, 3, 4 respectively.

Looking at the tangent point at the far right or far left where four circles meet, the radii of the red, green, blue, and purple circles are 1, 2, 3, 4 respectively. The only requirement for that sequence is that the middle two numbers have the same sum as the end two numbers, so for example 1, 3, 4, 6 would have worked equally well although it might not be as pleasing to the eye.

For the 1, 2, 3, 4 sequence chosen here, notice that there are only 6 different circle sizes. All 3 purple circles are the same size and all 3 blue circles are also the same size. Yet the red circles come in two sizes as do the green circles. Surprising indeed!

Another interesting bonus with this design is that it looks somewhat like a face ... of an alien or perhaps a bug?

After my disc designs came out, one of my fellow frisbee players said he came up with a family of solutions to the 12 circle puzzle similar to my disc design except that it had 120 degree symmetry. His freehand sketch looked something like the solution shown here (on the left side of the plot). The 3 smallest circles have radii of 1, 2, 3 (red,green,blue) although actually those sizes are completely arbitrary. A solution can be built with any sequence, as long as all three numbers are different. The fourth circle size (purple) is determined from the other three, and with this particular sequence (1, 2, 3) the fourth circle size turns out to be approximately 4.11684397. After working out this new solution I discovered another solution (shown here on the right side of the plot) that also has 120 degree symmetry. An interesting aspect of this final solution is that only 3 different circle sizes are used. This gives us a way to make the original problem more difficult. In addition to the requirement that each of the 12 circles must be tangent to 5 others, we can add the requirement that no more than 3 different circle sizes are allowed. This final solution meets that requirement unlike all the earlier solutions. The smallest circles (red and blue) have a radius of 1 and the largest circles (green) have a radius of 5, although these two numbers are arbitrary. The size of the purple circles are determined from the other two and with this choice (1,5) the purple circles turn out to have a radius of approximately 2.30930734142. In the spirit of Paul Erdos, I'm offering a $25 prize (two discs) to the first person to provide an expression for the exact value for the size of this purple circle. If it turns out this value is one of the roots of a polynomial equation, I will accept the coefficients of the polynomial as a solution even if that doesn't provide an exact value for the circle size. (A reasoning that the solution is correct must be provided.)

To solve the puzzle you may chose n to be any positive integer, although I am aware of only the one solution shown here (with n=12). As before, if you discover other solutions to this puzzle, I would love to know about it.

The multi-color circle is created with 180 segments with 180 different colors ... enough that the color changes look continuous unless you zoom in to a small portion of the image.

And here is the Matlab code which I wrote to produce all three of these designs:

After Matlab created the three figures, I printed them from the figure file menu directly to a 600dpi png format using the Bullzip pdf printer which, despite its name, also allows output to other graphics formats.

Below are some old pictures from the Encinal game (taken by Brent Gregory):

Scan codes for disc purchases: