Making Chess Boards

#### Round 1C, 2010

Submit Solution(Code Jam Page)

### Problem

The chess board industry has fallen on hard times and needs your help. It is a little-known fact that chess boards are made from the bark of the extremely rare Croatian Chess Board tree, (Biggus Mobydiccus). The bark of that tree is stripped and unwrapped into a huge rectangular sheet of chess board material. The rectangle is a grid of black and white squares.

Your task is to make as many large square chess boards as possible. A chess board is a piece of the bark that is a square, with sides parallel to the sides of the bark rectangle, with cells colored in the pattern of a chess board (no two cells of the same color can share an edge).

Each time you cut out a chess board, you must choose the largest possible chess board left in the sheet. If there are several such boards, pick the topmost one. If there is still a tie, pick the leftmost one. Continue cutting out chess boards until there is no bark left. You may need to go as far as cutting out 1-by-1 mini chess boards.

Here is an example showing the bark of a Chess Board tree and the first few chess boards that will be cut out of it.

### Input

The first line of the input gives the number of test cases, T. T test cases follow. Each one starts with a line containing the dimensions of the bark grid, M and N. N will always be a multiple of 4. The next M lines will each contain an (N/4)-character hexadecimal integer, representing a row of the bark grid. The binary representation of these integers will give you a strings of N bits, one for each row. Zeros represent black squares; ones represent white squares of the grid. The rows are given in the input from top to bottom. In each row, the most-significant bit of the hexadecimal integer corresponds to the leftmost cell in that row.

### Output

For each test case, output one line containing "Case #x: K", where x is the case number (starting from 1) and K is the number of different chess board sizes that you can cut out by following the procedure described above. The next K lines should contain two integers each -- the size of the chess board (from largest to smallest) and the number of chess boards of that size that you can cut out.

### Limits

1 ≤ T ≤ 100;
N will be divisible by 4;
Each hexadecimal integer will contain exactly N/4 characters.
Only the characters 0-9 and A-F will be used.

1 ≤ M ≤ 32;
1 ≤ N ≤ 32.

#### Large dataset

1 ≤ M ≤ 512;
1 ≤ N ≤ 512;
The input file will be at most 200kB in size.

### Sample

 Input Output ``` 4 15 20 55555 FFAAA 2AAD5 D552A 2AAD5 D542A 4AD4D B52B2 52AAD AD552 AA52D AAAAA 5AA55 A55AA 5AA55 4 4 0 0 0 0 4 4 3 3 C C 4 4 6 9 9 6``` ``` Case #1: 5 6 2 4 3 3 7 2 15 1 57 Case #2: 1 1 16 Case #3: 2 2 1 1 12 Case #4: 1 2 4```

The first example test case represents the image above.

Points Correct Attempted
18pt 640 836
24pt 226 547