(In reply to re: (Happy's post)
Alternatively you could start at exactly the point at whic Happy left off:
Adding the next odd number of counters to the square array is eqivalent to adding another row and another colomn.
Given an N x N square array (of N² counters) adding a new row means adding N counters and acheiving an (N = !) x N array of N² + N = N(N + 1) counters. Adding a new column means addind another (N + 1) counters and acheiving an (N + 1) x (N + 1) array of (N + 1)² = N² + 2N +1 counters. The total number of counters added is 2N + 1, which is an odd number. It is easy to show that the N x N square can be considered to be a (N - 1) x (N - 1) square with an added row and column totalling 2N - 1 counters, so the added numbers are consecutive odd numbers.
Since Happy showed the first few examples to work out imperically, this completes the inductive proof.
Posted by TomM
on 2002-08-08 21:53:20