Imagine a semi-infinite grid-line in which every box can hold any number of balls.
+--+--+--+--+--+--+--+--+--+--+--
| | | | | | | | | | | ... (infinity)
+--+--+--+--+--+--+--+--+--+--+--
The balls are a bit special as they can undergo fission and split up into two balls, one going into every direct neighbor box. The reverse process of ball-fusion is also possible! It combines two balls, which have a box in-between, to form one ball which is then added to the in-between box.
--+--+--+---+--+--+-- --+--+---+---+---+--+--
| | | O | | | <--fusion/fission--> | | O | | O | |
--+--+--+---+--+--+-- --+--+---+---+---+--+--
It is important to remember that any box can hold multiple balls.
You are now tasked to find out whether the right order of ball-fission and -fusion can lead form this configuration:
+--+--+--+--+--+--+---+--+--
| | | | | | | O | | ... (only empty ones after this)
+--+--+--+--+--+--+---+--+--
to this one:
+--+---+--+--+--+--+--+--+--
| | O | | | | | | | ... (only empty ones after this)
+--+---+--+--+--+--+--+--+--
Bonus:
What about reaching the end-configuration which has exactly one ball in the very first box, starting from one of the earlier two?
Bonus Bonus:
Actually, can we always say when some configuration is reachable from some other configuration? (Given both only have a finite amount of balls)