The Number of Possible Code Patterns on a 15 Disk Omnigraph

15! = 1.31 x 10¹²

Each disk can be placed on the stack in any of 5 orientations (i.e. rotated with respect to each other by 0, 72, 144, 216, or 288 degrees).  Each of the possible disk stack orders can have r ^(n-1) variations, where n = the number of disks and r = the number of possible rotations, thus the number of variations on each possible stack is:

5¹⁴ = 6.1 x 10⁹

The total number of distinct stack configurations is:

1.31 x 10¹²  x  6.1 x 10⁹ =  7.98 x 10²¹

The number of "message levers" that are extended determines how many different ways the stack can be "played":

• If 1 lever is extended, there are 5 distinct ways to play the stack
• If 2 levers are extended, there are 10 distinct ways to play the stack
• If 3 levers are extended, there are 10 distinct ways to play the stack
• If 4 levers are extended, there are 5 distinct ways to play the stack
• If 5 levers are extended, there is 1 distinct way to play the stack

Thus, if at least one lever is extended, there are 31 ways to play the stack.

So, the total number of distinct patterns possible for a specific set of 15 disks is:

31  x  7.98 x 10²¹  =  2.47 x 10²³  (247 sextillion)

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