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# Problem 270

Cutting Squares

A square piece of paper with integer dimensions N×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×N square. For example, C(1) = 2 and C(2) = 30 (shown below).

What is C(30) mod 108 ?

• 我们只进行这样的切割：从正方形不同的两条边上选择两个坐标为整数的点，并沿着两点的连线切割。
• 两次切割不能相交，但多次切割可以经过边上的同一个点。
• 一直切割下去，直到不存在合法的切割为止。

C(30) mod 108是多少？
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