Answer
it required j −1 moves to construct the one block,
and k +1−j −1 = k −j moves to construct the
other.
Work Step by Step
--Let P(n) be the statement
-that exactly n − 1 moves are required to assemble a puzzle
with n pieces.
- Now P(1) is trivially true.
Assume that
-p(j) is true for all j ≤ k, and consider a puzzle with k + 1 pieces.
The final move must be the joining of two blocks,
of size j and k + 1 − j for some integer j with 1 ≤ j ≤ k.
-- By the inductive hypothesis,
it required j −1 moves to construct the one block,
and k +1−j −1 = k −j moves to construct the
other.
Therefore,
1+(j −1)+(k−j) = k moves are required in all,
---so P(k +1) is true.
