2064: [USACO 2025 US Open Contest, Bronze] Problem 1. Hoof Paper Scissors Minus One

In a game of Hoof Paper Scissors, Bessie and Elsie can put out one of N (1 ≤ N ≤ 3000) different hoof symbols labeled 1…N, each corresponding to a different material. There is a complicated chart of how the different materials interact with one another, and based on that chart, either:

• One symbol wins and the other loses.
• The symbols draw against each other.

Hoof Paper Scissors Minus One works similarly, except Bessie and Elsie can each put out two symbols, one with each hoof. After observing all four symbols that they have all put out, they each choose one of their two symbols to play. The outcome is decided based on normal Hoof Paper Scissor conventions.

Given the M (1 ≤ M ≤ 3000) symbol combinations that Elsie plans to make across each game, Bessie wants to know how many different symbol combinations would result in a guaranteed win against Elsie. A symbol combination is defined as an ordered pair (L, R) where L is the symbol the cow plays with her left hoof and R is the symbol the cow plays with her right hoof. Can you compute this for each game?

Out of the following N lines of input, the ith line consists of i characters  a_i,1 a_i,2… a_i,i where each a_{i, j} ∈ {𝙳,𝚆,𝙻}. If a_i,j = 𝙳, then symbol i draws against symbol j. If a_i,j = 𝚆, then symbol i wins against symbol j. If a_i,j = 𝙻, then symbol i loses against symbol j. It is guaranteed that a_i,i = 𝙳.

The next M lines contain two space separated integers s₁ and s₂ where 1 ≤ s₁, s₂ ≤ N. This represents Elsie's symbol combination for that game.

3 3
D
WD
LWD
1 2
2 3
1 1

0
0
5