P6184 [USACO08OCT] Building A Fence G

Hardworking Farmer John wants to build a fence with four sides to keep the cows inside. He now has a long wooden board of length $N$ ($4 \leq N \leq 2,500$). He needs to cut this board into four pieces, each with a positive integer length, so that he can build a fence.

How many different cutting methods are there such that the four cut boards can form a four-sided fence. Note: 1. Do not consider symmetry. You do not need to remove symmetric cases or deal with other similar complicated issues. 2. The area enclosed by the fence must be greater than $0$. 3. The result fits in a 32-bit integer.

One integer $N$.

The number of ways Farmer John can split the board and make a quadrilateral.

Farmer John has $10$ ways to cut the board into four pieces: - (1, 1, 1, 3); - (1, 1, 2, 2); - (1, 1, 3, 1); - (1, 2, 1, 2); - (1, 2, 2, 1); - (1, 3, 1, 1); - (2, 1, 1, 2); - (2, 1, 2, 1); - (2, 2, 1, 1); - (3, 1, 1, 1)。 Among them, there are four cases that cannot form a quadrilateral: - (1, 1, 1, 3), - (1, 1, 3, 1), - (1, 3, 1, 1), - (3, 1, 1, 1)。 Translated by ChatGPT 5