HMMT 十一月 2011 · 团队赛 · 第 5 题

HMMT November 2011 — Team Round — Problem 5

[ 3 ] Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs. C C C C C C C C C C C C C C C C

解析

[ 3 ] Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs. C C C C C C C C C C C C C C C C Answer: 1296 For each quadrant, we have three distinct cases based on the number of Cs in each color: • Case 1: all four the same color: 2 configurations (all red or all blue) • Case 2: 3 of one color, 1 of the other: 2 configurations (three red or three blue) • Case 3: 2 of each color: 2 configurations (red squares adjacent or opposite) 4 Thus, since there are 4 quadrants, there are a total of (2 + 2 + 2) = 1296 possible grids.