USAMO 2005 — Days 1 and 2 Question 4

Legs $L_1$, $L_2$, $L_3$, $L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered $4$-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i$ and still have a stable table? (The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length $0$ is permitted.)

Proposed by Elgin Johnston (credited in the supplied source).