HMMT 十一月 2011 · 团队赛 · 第 5 题
HMMT November 2011 — Team Round — Problem 5
题目详情
英文原题
- [ 3 ] Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to becolored either red or blue. A quadrant operation on the grid consists of choosing one of the fourtwo-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to theadjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changingthe orientation of the C. Given that two colorings are the considered same if and only if one can beobtained from the other by a series of quadrant operations, determine the number of distinct coloringsof the Cs.
C C C C
C C C C
C C C C
C C C C
解析
英文解析
- [ 3 ] Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to becolored either red or blue. A quadrant operation on the grid consists of choosing one of the fourtwo-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to theadjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changingthe orientation of the C. Given that two colorings are the considered same if and only if one can beobtained from the other by a series of quadrant operations, determine the number of distinct coloringsof 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 eachcolor:
• 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)
Thus, since there are 4 quadrants, there are a total of (2 + 2 + 2) = 1296 possible grids.4