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Cutting Squares

 Published on Saturday, 26th December 2009, 01:00 am; Solved by 680;
Difficulty rating: 80%

Problem 270

A square piece of paper with integer dimensions $N \times N$ is placed with a corner at the origin and two of its sides along the $x$- and $y$-axes. Then, we cut it up respecting the following rules:

  • We only make straight cuts between two points lying on different sides of the square, and having integer coordinates.
  • Two cuts cannot cross, but several cuts can meet at the same border point.
  • Proceed until no more legal cuts can be made.

Counting any reflections or rotations as distinct, we call $C(N)$ the number of ways to cut an $N \times N$ square. For example, $C(1) = 2$ and $C(2) = 30$ (shown below).

0270_CutSquare.gif

What is $C(30) \bmod 10^8$?