Thanks Samuli/Rauli for these good old patterns!
I kept researching and came up with other ideas. I wouldn't be surprised if other people did similar research before, please let me know.
First I need to define the "temperature" of a state, which is the sum of products of the number of balls by the stage number.
For example, the temperature of ground state ("liquid state") with 3 balls is 3x1 + 2x1 + 1x1 = 6.
With 4 balls it's 4x1 + 3x1 + 2x1 + 1x1 = 10.
Actually the "temperature" of liquid state with n balls is n(n+1)/2.
I see 5 types of states:
- gas state, where temperature is above the liquid temperature and the first stage is empty (when you throw a 0). ex: +++-- )
- hot state, where temperature is above liquid temperature but the first stage has at least one ball. ex: ++-+ )
- liquid state, i.e. ground state in traditional terminology. ex: +++ )
- cold state, where temperature is less than liquid temperature but more than the number of balls. ex: +-[++] )
- solid state, where all balls are on the first stage; it means that temperature is equal to the number of balls and it cannot be lower. ex: [+++] )
For a given pattern, you can calculate the average temperature of all states.
temperature of 3 is 6 ;
temperature of 423 is 6.33 ;
for 522 it's 7 ;
531 is 7.33;
22 is 5 ;
3 is 4.66
2 is 4.5
 is 3 i.e. the minimum possible temperature.
You could also calculate the temperature ratio = average temperature / liquid temperature, Liquid temperature would then be 1.
It could be used to categorise patterns by their temperature or use temperature as a composition tool.
Thanks for reading...