Question 4
An plays a game in which an ordered triple is written on a board from left to right. Initially the triple is $(1,1,1)$. On each turn, An performs one of these operations on the current triple $(x,y,z)$:
(i) erase it and write $(y,z,x+z)$;
(ii) erase it and write $(x+z+1,x+y+z+1,x+y+2z+1)$.
(a) Prove that An needs exactly four turns to write a triple $(a,b,6)$.
(b) Find the smallest nonnegative integer $k$ for which some play sequence lets An write $(c,d,129)$ after $k$ turns.
(i) erase it and write $(y,z,x+z)$;
(ii) erase it and write $(x+z+1,x+y+z+1,x+y+2z+1)$.
(a) Prove that An needs exactly four turns to write a triple $(a,b,6)$.
(b) Find the smallest nonnegative integer $k$ for which some play sequence lets An write $(c,d,129)$ after $k$ turns.