## USACO 2016 US Open Contest, Platinum

## Problem 2. Bull in a China Shop

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The shape of the cow figurine is described by an $N \times M$ grid of characters like the one below ($3 \leq N, M \leq 500$), where lowercase letter characters are each part of the figurine (indicating different colors) and '.' characters are not.

............... ............... x..x........... xxxx........... xxxxaaaaaaa... .xx.aaaaaaaaa.. ....aaaaaaa.aa. ....ll...ll.... ....vv...vv.... ...............

Unfortunately, right before FJ can make his purchase, a bull runs through the shop and breaks not only FJ's figurine, but many of the other glass objects on the shelves as well! FJ's figurine breaks into 3 pieces, which quickly become lost among $K$ total pieces lying on the ground ($4 \leq K \leq 100$). Each of the $K$ pieces is described by a grid of characters, just like the original figurine.

Please help FJ determine how many sets of 3 pieces (out of the $K$ on the floor) could be glued back together to mend his broken figurine.

The pieces on the ground might have been flipped vertically or horizontally, or rotated by some multiple of 90 degrees. Therefore, given the original grid as well as $K$ grids describing pieces, you want to find sets of 3 pieces that can be joined together to form the original picture, allowing the pieces to be translated, flipped, or rotated multiples of 90 degrees. When then superimposed, the 3 pieces should exactly form the original picture, with each colored square in the original picture represented in exactly one of the pieces.

#### INPUT FORMAT (file bcs.in):

The first line contains a single integer $K$. Following that will be $K + 1$ piece descriptions. The first description will describe the original glass cow, the following $K$ descriptions will be of the broken pieces.Each description begins with a line containing two integers $R$ and $C$ ($1 \le R, C \le 100$). The following $R$ lines contain $C$ lowercase alphabet characters describing the color of each cell. Each piece will be horizontally/vertically connected and have at least one non-empty cell.

#### OUTPUT FORMAT (file bcs.out):

Output the number of triples $i, j, k$ ($i < j < k$) such that pieces $i$, $j$, and $k$ can be arranged to form the original glass cow.#### SAMPLE INPUT:

5 5 5 aaaaa ..a.. bbabb ..a.. aaaaa 3 5 ..abb ..a.. aaaaa 5 2 a. a. aa a. a. 1 2 bb 1 5 bbabb 2 5 aaaaa ..a..

#### SAMPLE OUTPUT:

3

The three solutions use pieces $(0, 1, 2)$, $(0, 2, 4)$, $(1, 3, 4)$.

Note that this problem has a time limit of 6 seconds per test case (and twice that for Java and Python submissions).

Problem credits: Brian Dean