Question 3
One has a long row of glasses and $n$ stones in the central glass (glass $0$). The following movements are allowed:
• Movement $A$
If there is at least one stone in glass $i$ and at least one in glass $i+1$, one may make one stone in glass $i+1$ jump to glass $i-1$, eliminating one stone in glass $i$.
• Movement *B*
If there are at least two stones in glass *i* one may make a stone in glass *i* jump to glass *i* + 2 and make another stone jump to glass *i* - 1.
Prove the following: performing movements A and B for a sufficiently long time, one shall always obtain a configuration from which neither movements are allowed anymore. Besides this final configuration does not depend on the choices of movements during the process.
• Movement $A$
If there is at least one stone in glass $i$ and at least one in glass $i+1$, one may make one stone in glass $i+1$ jump to glass $i-1$, eliminating one stone in glass $i$.
• Movement *B*
If there are at least two stones in glass *i* one may make a stone in glass *i* jump to glass *i* + 2 and make another stone jump to glass *i* - 1.
Prove the following: performing movements A and B for a sufficiently long time, one shall always obtain a configuration from which neither movements are allowed anymore. Besides this final configuration does not depend on the choices of movements during the process.